Credit risk

M. Jeanblanc i Credit risk T. Bielecki, M. Jeanblanc and M. Rutkowski Lecture of M. Jeanblanc October 2005 TUNIS, UNESCO CHAIR Preliminary Version ii Credit Risk, TUNIS 2005 Contents 1 Structural Approach 1.1 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.3 1.4 1.5 Defaultable Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Risk-Neutral Valuation Formula . . . . . . . . . . . . . . . . . . . . . . . . . Defaultable Zero-Coupon Bond . . . . . . . . . . . . . . . . . . . . . . . . . . Merton’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black and Cox Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 5 6 7 8 8 11 15 17 17 18 21 21 21 24 25 25 26 26 26 27 28 30 34 34 35 35 35 36 36 Classic Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Independent barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comments on Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Hazard process Approach: A Toy Model 2.1 The Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 2.1.2 2.1.3 2.1.4 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.2.9 2.3 Defaultable Zero-coupon with Payment at Maturity . . . . . . . . . . . . . . Defaultable Zero-coupon with Payment at Hit . . . . . . . . . . . . . . . . . . Implied probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hazard Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of a Probability Measure . . . . . . . . . . . . . . . . . . . . . . . . . Incompleteness of the Toy model . . . . . . . . . . . . . . . . . . . . . . . . . Risk Neutral Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . Totally Inaccessible Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toy Model and Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.10 Partial information: Duffie and Lando’s model . . . . . . . . . . . . . . . . . Valuation and Trading Defaultable Claims . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Price dynamics of a survival claim (X, 0, τ ). . . . . . . . . . . . . . . . . . . . iii iv 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.4 2.4.1 2.4.2 Credit Risk, TUNIS 2005 Price dynamics of a recovery claim (0, Z, τ ). . . . . . . . . . . . . . . . . . . . Price dynamics of a defaultable claim (X, Z, τ ). . . . . . . . . . . . . . . . . . Valuation of a Credit Default Swap . . . . . . . . . . . . . . . . . . . . . . . . Price Dynamics of a CDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hedging of a Contingent Claim in the CDS Market . . . . . . . . . . . . . . . Two times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More than two times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 37 37 40 42 46 46 48 49 49 49 50 50 51 52 53 53 53 53 54 54 54 55 55 55 55 56 57 58 58 58 59 60 62 66 66 68 68 70 Successive default times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Cox Processes and Extensions 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Construction of Cox Processes with a given stochastic intensity . . . . . . . . . . . . Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice of filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditional Expectation of F∞ -Measurable Random Variables . . . . . . . . . . . . Defaultable Zero-Coupon Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Term Structure Models 3.8.1 3.8.2 3.8.3 3.8.4 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duffee’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jarrow and Turnbull’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . Vacicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The CIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hazard process Approach: Reference filtration 4.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.3 4.4 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpretation of the intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . Restricting the information . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complete model case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition and Properties of (H) Hypothesis . . . . . . . . . . . . . . . . . . . (H) hypothesis and shrinking filtration . . . . . . . . . . . . . . . . . . . . . . Change of a probability measure . . . . . . . . . . . . . . . . . . . . . . . . . Stochastic Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (H) Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 4.4.2 Information at discrete times . . . . . . . . . . . . . . . . . . . . . . . . . . . Delayed information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Jeanblanc v 70 73 73 73 75 76 77 79 82 83 95 98 99 Dynamics of asset prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Risk-neutral valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unconstrained strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constrained strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Defaultable asset with total default . . . . . . . . . . . . . . . . . . . . . . . . Two defaultable assets with total default . . . . . . . . . . . . . . . . . . . . Defaultable asset with total default . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Intensity approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Hedging 5.1 Semimartingale Model with a Common Default . . . . . . . . . . . . . . . . . . . . . 5.1.1 5.1.2 5.2 5.2.1 5.2.2 5.3 5.3.1 5.3.2 5.4 5.4.1 5.4.2 5.4.3 Trading Strategies in a Semimartingale Set-up Martingale Approach to Valuation and Hedging . . . . . . . . . . . . . . . . . . . . . PDE Approach to Valuation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . Defaultable asset with non-zero recovery . . . . . . . . . . . . . . . . . . . . . 103 Two defaultable assets with total default . . . . . . . . . . . . . . . . . . . . 106 109 6 Indifference pricing 6.1 6.2 6.1.1 6.2.1 6.2.2 6.2.3 6.2.4 6.3 6.3.1 6.3.2 6.4 6.4.1 6.4.2 6.4.3 6.5 6.6 Defaultable Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Hodges Indifference Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 . . . . . . . . . . . . . . . . . . . . 111 Solution of Problem X (PF ) Hodges prices relative to the reference filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Exponential Utility: Explicit Computation of the Hodges Price . . . . . . . . 112 Risk-Neutral Spread Versus Hodges Spreads . . . . . . . . . . . . . . . . . . . 114 Recovery paid at time of default . . . . . . . . . . . . . . . . . . . . . . . . . 115 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Hodges Buying and Selling Prices . . . . . . . . . . . . . . . . . . . . . . . . . 121 Quadratic Hedging with F-Adapted Strategies . . . . . . . . . . . . . . . . . 123 Quadratic Hedging with G-Adapted Strategies . . . . . . . . . . . . . . . . . 124 Jump-Dynamics of Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Optimization Problems and BSDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Quadratic Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 MeanVariance Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Quantile Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 133 7 Dependent Defaults and Credit Migrations 7.1 7.1.1 7.1.2 7.2 7.2.1 7.2.2 Basket Credit Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 The ith -to-Default Contingent Claims . . . . . . . . . . . . . . . . . . . . . . 134 Case of Two Entities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Canonical Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Independent Default Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Conditionally Independent Defaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 vi 7.2.3 7.2.4 7.2.5 7.2.6 7.3 7.3.1 7.3.2 7.3.3 7.4 7.4.1 7.4.2 7.5 7.5.1 7.5.2 7.5.3 7.6 7.7 7.8 7.6.1 Credit Risk, TUNIS 2005 Signed Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Valuation of FDC and LDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 General Valuation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Default Swap of Basket Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Direct Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Indirect Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Simplified Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Construction and Properties of the Model . . . . . . . . . . . . . . . . . . . . 144 Bond Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Kusuoka’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Interpretation of Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Bond Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Extension of Kusuoka’s Construction . . . . . . . . . . . . . . . . . . . . . . . 149 Copula-Based Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Jarrow and Yu Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Extension of Jarrow and Yu Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Dependent Intensities of Credit Migrations . . . . . . . . . . . . . . . . . . . . . . . 148 Dynamics of Dependent Credit Ratings . . . . . . . . . . . . . . . . . . . . . . . . . 151 Defaultable Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.8.1 7.8.2 7.8.3 7.8.4 7.8.5 7.8.6 Standing Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Credit Migration Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Defaultable Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Premia for Interest Rate and Credit Event Risks . . . . . . . . . . . . . . . . 157 Defaultable Coupon Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Examples of Credit Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 161 8 Portfolio management 8.1 8.1.1 8.1.2 8.1.3 8.1.4 8.2 8.2.1 8.2.2 8.2.3 9 Appendix 9.1 9.1.1 Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 CreditMetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 KMV approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 CreditRisk+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 CreditPortfolioView . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Davis and Lo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Portfolio management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 167 Hitting times of a level and law of the maximum for Brownian motion . . . . 167 Portfolio management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Hitting times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 M. Jeanblanc 1 Hitting times for a Drifted Brownian motion . . . . . . . . . . . . . . . . . . 170 Hitting Times for Geometric Brownian Motion . . . . . . . . . . . . . . . . . 172 Other processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Non-constant Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9.1.2 9.1.3 9.1.4 9.1.5 9.1.6 9.2 9.3 Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Poisson processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.3.1 9.3.2 Standard Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Inhomogeneous Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . 182 Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Itˆ’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 o Stopping times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.4 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.4.1 9.4.2 9.4.3 9.5 9.6 9.7 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Ornstein-Uhlenbeck processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 9.6.1 9.7.1 9.7.2 9.7.3 9.7.4 Vacisek model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Cox-Ingersoll-Ross Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 CIR Processes and BESQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Transition Probabilities for a CIR Process . . . . . . . . . . . . . . . . . . . . 193 CIR Processes as Spot Rate Models . . . . . . . . . . . . . . . . . . . . . . . 193 Zero-coupon Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 2 Credit Risk, TUNIS 2005 Introduction These notes are mainly based on the papers of Bielecki, Jeanblanc and Rutkowski: T.R. Bielecki, M. Jeanblanc and M. Rutkowski, Hedging of defaultable claims, Lecture Notes in Mathematics, 1847, pages 1–132,Paris-Princeton , Springer-Verlag, 2004, R.A. Carmona, E. Cinlar, I. Ekeland, E. Jouini, J.E. Scheinkman, N. Touzi, eds. T.R. Bielecki, M. Jeanblanc and M. Rutkowski, Stochastic Methods In Credit Risk Modelling, Valuation And Hedging, Lecture Notes in Mathematics, Frittelli, M. edt, CIME-EMS Summer School on Stochastic Methods in Finance, Bressanone, Springer, 2004. T.R. Bielecki, M. Jeanblanc and M. Rutkowski, Hedging of credit derivatives in models with totally unexpected default, Proceeding of the Ritsumeikan Conference,2005. The goal of this lecture is to present a survey of recent developments in the area of mathematical modeling of credit risk and credit derivatives. Credit risk embedded in a financial transaction is the risk that at least one of the parties involved in the transaction will suffer a financial loss due to decline in the creditworthiness of the counter-party to the transaction, or perhaps of some third party. For example: • A holder of a corporate bond bears a risk that the (market) value of the bond will decline due to decline in credit rating of the issuer. • A bank may suffer a loss if a bank’s debtor defaults on payment of the interest due and (or) the principal amount of the loan. • A party involved in a trade of a credit derivative, such as a credit default swap (CDS), may suffer a loss if a reference credit event occurs. • The market value of individual tranches constituting a collateralized debt obligation (CDO) may decline as a result of changes in the correlation between the default times of the underlying defaultable securities (i.e., of the collateral). The most extensively studied form of credit risk is the default risk – that is, the risk that a counterparty in a financial contract will not fulfil a contractual commitment to meet her/his obligations stated in the contract. For this reason, the main tool in the area of credit risk modeling is a judicious specification of the random time of default. A large part of the present text will be devoted to this issue. Our main goal is to present the most important mathematical tools that are used for the arbitrage valuation of defaultable claims, which are also known under the name of credit derivatives. We also examine the important issue of hedging these claims. In Chapter 1, we provide a concise summary of the main developments within the so-called structural approach to modeling and valuation of credit risk. We also study the random barrier case. Chapter 2 is devoted to the study of a toy model within the hazard process framework. Chapter 3 studies the case of Cox processes. Chapter 4 is devoted to the reduced-form approach. This approach 3 4 Credit Risk, TUNIS 2005 is purely probabilistic in nature and, technically speaking, it has a lot in common with the reliability theory. Chapter 5 studies hedging strategies under assumption that a defaultable asset is traded. Chapter 6 provides an introduction to the area of modeling dependent credit migrations and defaults. Chapter 7 studies different ways to give a price in incomplete market setting. An appendix recalls some notion of stochastic calculus and probability theory. Let us only mention that the proofs of most results can be found in Bielecki and Rutkowski [4], Bielecki et al. [5, 6, 87] and Jeanblanc and Rutkowski [59]. We quote some of the seminal papers; the reader can also refer to books of Bruy`re [80], Bluhm et al. [9], Bielecki and Rutkowski [4], Cossin e and Pirotte [22], Duffie and Singleton [36], Lando [68], Sch¨nbucher [83] for more information. At o the end of the bibliography, we also mention some web addresses where articles can be downloaded. Finally, it should be acknowledged that some results (especially within the reduced form approach) were obtained independently by various authors, who worked under different set of assumptions and within distinct setups, and thus we decided to omit detailed credentials in most cases. We hope our colleagues will accept our apologies for this deficiency, and we stress that this by no means signifies that these results that are not explicitly attributed are ours. Begin at the beginning, and go on till you come to the end. Then, stop. L. Carroll, Alice’s Adventures in Wonderland Chapter 1 Structural Approach In this chapter, we present the structural approach to modeling credit risk (it is also known as the value-of-the-firm approach). This methodology directly refers to economic fundamentals, such as the capital structure of a company, in order to model credit events (a default event, in particular). As we shall see in what follows, the two major driving concepts in the structural modeling are: the total value of the firm’s assets and the default triggering barrier. This was historically the first approach used in this area, and it goes back to the fundamental papers by Black and Scholes [8] and Merton [76]. 1.1 Basic Assumptions We fix a finite horizon date T ∗ > 0, and we suppose that the underlying probability space (Ω, F, P), endowed with some (reference) filtration F = (Ft )0≤t≤T ∗ , is sufficiently rich to support the following objects: • The short-term interest rate process r, and thus also a default-free term structure model. • The firm’s value process V, which is interpreted as a model for the total value of the firm’s assets. • The barrier process v, which will be used in the specification of the default time τ . • The promised contingent claim X representing the firm’s liabilities to be redeemed at maturity date T ≤ T ∗ . • The process C, which models the promised dividends, i.e., the liabilities stream that is redeemed continuously or discretely over time to the holder of a defaultable claim. • The recovery claim X representing the recovery payoff received at time T, if default occurs prior to or at the claim’s maturity date T . • The recovery process Z, which specifies the recovery payoff at time of default, if it occurs prior to or at the maturity date T. 1.1.1 Defaultable Claims Technical Assumptions. We postulate that the processes V, Z, C and v are progressively measurable with respect to the filtration F, and that the random variables X and X are FT -measurable. In addition, C is assumed to be a process of finite variation, with C0 = 0. We assume without mentioning that all random objects introduced above satisfy suitable integrability conditions. 5 6 Credit Risk, TUNIS 2005 Probabilities P and Q. The probability P is assumed to represent the real-world (or statistical ) probability, as opposed to the martingale measure (also known as the risk-neutral probability). The latter probability is denoted by Q in what follows. Default Time. In the structural approach, the default time τ will be typically defined in terms of the value process V and the barrier process v. We set τ = inf { t > 0 : t ∈ T and Vt ≤ vt } with the usual convention that the infimum over the empty set equals +∞. In main cases, the set T is an interval [0, T ] (or [0, ∞) in the case of perpetual claims). In first passage structural models, the default time τ is given by the formula: τ = inf { t > 0 : t ∈ [0, T ] and Vt ≤ v (t)}, ¯ where v : [0, T ] → I + is some deterministic function, termed the barrier. ¯ R Predictability of Default Time. Since the underlying filtration F in most structural models is generated by a standard Brownian motion, τ will be an F-predictable stopping time (as any stopping time with respect to a Brownian filtration): there exists a sequence of increasing stopping times announcing the default time. Recovery Rules. If default does not occur before or at time T, the promised claim X is paid in full at time T. Otherwise, depending on the market convention, either (1) the amount X is paid at the maturity date T, or (2) the amount Zτ is paid at time τ. In the case when default occurs at maturity, i.e., on the event {τ = T }, we postulate that only the recovery payment X is paid. In a general setting, we consider simultaneously both kinds of recovery payoff, and thus a generic defaultable claim is formally defined as a quintuple (X, C, X, Z, τ ). 1.1.2 Risk-Neutral Valuation Formula Suppose that our financial market model is arbitrage-free, in the sense that there exists a martingale measure (risk-neutral probability) Q, meaning that price process of any tradeable security, which pays no coupons or dividends, becomes an F-martingale under Q, when discounted by the savings account B, given as t Bt = exp 0 ru du . We introduce the jump process Ht = 1 {τ ≤t} , and we denote by D the process that models all cash 1 flows received by the owner of a defaultable claim. Let us denote X d (T ) = X1 {τ >T } + X1 {τ ≤T } . 1 1 Definition 1.1.1 The dividend process D of a defaultable contingent claim (X, C, X, Z, τ ), which settles at time T, equals Dt = X d (T )1 {t≥T } + 1 (1 − Hu ) dCu + ]0,t] ]0,t] Zu dHu . It is apparent that D is a process of finite variation, and (1 − Hu ) dCu = ]0,t] ]0,t] 1 {τ >u} dCu = Cτ − 1 {τ ≤t} + Ct 1 {τ >t} . 1 1 1 Note that if default occurs at some date t, the promised dividend Ct −Ct− , which is due to be paid at this date, is not received by the holder of a defaultable claim. Furthermore, if we set τ ∧t = min {τ, t} then Zu dHu = Zτ ∧t 1 {τ ≤t} = Zτ 1 {τ ≤t} . 1 1 ]0,t] M. Jeanblanc 7 Remark 1.1.1 In principle, the promised payoff X could be incorporated into the promised dividends process C. However, this would be inconvenient, since in practice the recovery rules concerning the promised dividends C and the promised claim X are different, in general. For instance, in the case of a defaultable coupon bond, it is frequently postulated that in case of default the future coupons are lost, but a strictly positive fraction of the face value is usually received by the bondholder. We are in the position to define the ex-dividend price St of a defaultable claim. At any time t, the random variable St represents the current value of all future cash flows associated with a given defaultable claim. Definition 1.1.2 For any date t ∈ [0, T [, the ex-dividend price of the defaultable claim (X, C, X, Z, τ ) is given as St = Bt EQ ]t,T ] −1 Bu dDu Ft . (1.1) In addition, we always set ST = X d (T ). 1.1.3 Defaultable Zero-Coupon Bond Assume that C ≡ 0, Z ≡ 0 and X = L for some positive constant L > 0. Then the value process S represents the arbitrage price of a defaultable zero-coupon bond (also known as the corporate discount bond) with the face value L and recovery at maturity only. In general, the price D(t, T ) of such a bond equals −1 D(t, T ) = Bt EQ BT (L1 {τ >T } + X1 {τ ≤T } ) Ft . 1 1 It is convenient to rewrite the last formula as follows: −1 D(t, T ) = LBt EQ BT (1 {τ >T } + δ(T )1 {τ ≤T } ) Ft , 1 1 where the random variable δ(T ) = X/L represents the so-called recovery rate upon default. It is natural to assume that 0 ≤ X ≤ L so that δ(T ) satisfies 0 ≤ δ(T ) ≤ 1. Alternatively, we may re-express the bond price as follows: −1 D(t, T ) = L B(t, T ) − Bt EQ BT w(T )1 {τ ≤T } Ft 1 , where −1 B(t, T ) = Bt EQ (BT | Ft ) is the price of a unit default-free zero-coupon bond, and w(T ) = 1 − δ(T ) is the writedown rate upon default. Generally speaking, the time-t value of a corporate bond depends on the joint probability distribution under Q of the three-dimensional random variable (BT , δ(T ), τ ) or, equivalently, (BT , w(T ), τ ). ˜ Example 1.1.1 Merton [76] postulates that the recovery payoff upon default equals X = VT , where the random variable VT is the firm’s value at maturity date T of a corporate bond. Consequently, the random recovery rate upon default equals δ(T ) = VT /L, and the writedown rate upon default equals w(T ) = 1 − VT /L. Expected Writedowns. For simplicity, we assume that the savings account B is non-random – that is, the short-term rate r is deterministic. Then the price of a default-free zero-coupon bond −1 equals B(t, T ) = Bt BT , and the price of a zero-coupon corporate bond satisfies D(t, T ) = Lt (1 − w∗ (t, T )), 8 Credit Risk, TUNIS 2005 where Lt = LB(t, T ) is the present value of future liabilities, and w∗ (t, T ) is the conditional expected writedown rate under Q. It is given by the following equality: w∗ (t, T ) = EQ w(T )1 {τ ≤T } | Ft . 1 The conditional expected writedown rate upon default equals, under Q, ∗ wt = EQ w(T )1 {τ ≤T } | Ft 1 w∗ (t, T ) = , Q{τ ≤ T | Ft } p∗ t ∗ ∗ where p∗ = Q{τ ≤ T | Ft } is the conditional risk-neutral probability of default. Finally, let δt = 1−wt t ∗ ∗ ∗ be the conditional expected recovery rate upon default under Q. In terms of pt , δt and pt , we obtain ∗ ∗ D(t, T ) = Lt (1 − p∗ ) + Lt p∗ δt = Lt (1 − p∗ wt ). t t t If the random variables w(T ) and τ are conditionally independent with respect to the σ-field Ft ∗ under Q, then we have wt = EQ (w(T ) | Ft ). Example 1.1.2 In practice, it is common to assume that the recovery rate is non-random. Let the recovery rate δ(T ) be constant, specifically, δ(T ) = δ for some real number δ. In this case, the ∗ writedown rate w(T ) = w = 1 − δ is non-random as well. Then w∗ (t, T ) = wp∗ and wt = w for t every 0 ≤ t ≤ T. Furthermore, the price of a defaultable bond has the following representation D(t, T ) = Lt (1 − p∗ ) + δLt p∗ = Lt (1 − wp∗ ). t t t We shall return to various recovery schemes later in the text. 1.2 Classic Structural Models Classic structural models are based on the assumption that the risk-neutral dynamics of the value process of the assets of the firm V are given by the SDE: dVt = Vt (r − κ) dt + σV dWt∗ , V0 > 0, where κ is the constant payout (dividend) ratio, and the process W ∗ is a standard Brownian motion under the martingale measure Q. 1.2.1 Merton’s Model We present here the classic model due to Merton [76]. Basic assumptions. A firm has a single liability with promised terminal payoff L, interpreted as the zero-coupon bond with maturity T and face value L > 0. The ability of the firm to redeem its debt is determined by the total value VT of firm’s assets at time T. Default may occur at time T only, and the default event corresponds to the event {VT < L}. Hence, the stopping time τ equals τ = T 1 {VT 0. T −t 1 LB(t, T ) log . T −t D(t, T ) This agrees with the well-known fact that risky bonds have an expected return in excess of the riskfree interest rate. In other words, the yields on corporate bonds are higher than yields on Treasury bonds with matching notional amounts. Notice, however, when t tends to T, the credit spread in Merton’s model tends either to infinity or to 0, depending on whether VT < L or VT > L. Formally, if we define the forward short spread at time T as F SST = lim S(t, T ) t↑T then F SST (ω) = 0, ∞, if ω ∈ {VT > L}, if ω ∈ {VT < L}. M. Jeanblanc 11 1.2.2 Black and Cox Model By construction, Merton’s model does not allow for a premature default, in the sense that the default may only occur at the maturity of the claim. Several authors put forward structural-type models in which this restrictive and unrealistic feature is relaxed. In most of these models, the time of default is given as the first passage time of the value process V to either a deterministic or a random barrier. In principle, the bond’s default may thus occur at any time before or on the maturity date T. The challenge is to appropriately specify the lower threshold v, the recovery process Z, and to explicitly evaluate the conditional expectation that appears on the right-hand side of the risk-neutral valuation formula −1 Bu dDu Ft , St = Bt EQ ]t,T ] which is valid for t ∈ [0, T [. As one might easily guess, this is a non-trivial mathematical problem, in general. In addition, the practical problem of the lack of direct observations of the value process V largely limits the applicability of the first-passage-time models based on the value of the firm process V . Corporate Zero-Coupon Bond Black and Cox [7] extend Merton’s [76] research in several directions, by taking into account such specific features of real-life debt contracts as: safety covenants, debt subordination, and restrictions on the sale of assets. Following Merton [76], they assume that the firm’s stockholders receive continuous dividend payments, which are proportional to the current value of firm’s assets. Specifically, they postulate that dVt = Vt (r − κ) dt + σV dWt∗ , V0 > 0, where the constant κ ≥ 0 represents the payout ratio, and σV > 0 is the constant volatility. The short-term interest rate r is assumed to be constant. Safety covenants. Safety covenants provide the firm’s bondholders with the right to force the firm to bankruptcy or reorganization if the firm is doing poorly according to a set standard. The standard for a poor performance is set by Black and Cox in terms of a time-dependent deterministic barrier v (t) = Ke−γ(T −t) , t ∈ [0, T [, for some constant K > 0. As soon as the value of firm’s assets ¯ crosses this lower threshold, the bondholders take over the firm. Otherwise, default takes place at debt’s maturity or not depending on whether VT < L or not. Default time. Let us set vt = v (t), ¯ L, for t < T, for t = T . The default event occurs at the first time t ∈ [0, T ] at which the firm’s value Vt falls below the level vt , or the default event does not occur at all. The default time equals ( inf ∅ = +∞) τ = inf { t ∈ [0, T ] : Vt < vt }. The recovery process Z and the recovery payoff X are proportional to the value process: Z ≡ β2 V and X = β1 VT for some constants β1 , β2 ∈ [0, 1]. The case examined by Black and Cox [7] corresponds to β1 = β2 = 1. To summarize, we consider the following model: X = L, C ≡ 0, Z ≡ β2 V, X = β1 VT , τ = τ ∧ τ , ¯ where the early default time τ equals ¯ τ = inf { t ∈ [0, T ) : Vt < v (t)} ¯ ¯ and τ stands for Merton’s default time: τ = T 1 {VT Ke−γ(T −t) }, R R u(Ke−γ(T −t) , t) = β2 Ke−γ(T −t) with the boundary condition and the terminal condition u(v, T ) = min (β1 v, L). Probabilistic approach. For any t < T the price D(t, T ) = u(Vt , t) of a defaultable bond has the following probabilistic representation, on the set {τ > t} = {¯ > t} τ D(t, T ) = EQ Le−r(T −t) 1 {¯≥T, VT ≥L} Ft 1 τ + EQ β1 VT e−r(T −t) 1 {¯≥T, VT 0. Prior to bond’s default, that is: on the set ˆ {τ > t}, the price process D(t, T ) = u(Vt , t) of a defaultable bond equals 2ˆ D(t, T ) = LB(t, T ) N h1 (Vt , T − t) − Rt a N h2 (Vt , T − t) + β1 Vt e−κ(T −t) N h3 (Vt , T − t)) − N h4 (Vt , T − t) 2ˆ + β1 Vt e−κ(T −t) Rt a+2 N h5 (Vt , T − t)) − N h6 (Vt , T − t) θ+ζ θ−ζ + β2 Vt Rt N h7 (Vt , T − t) + Rt N h8 (Vt , T − t) , M. Jeanblanc 13 ν 2 + 2σ 2 (r − γ) and ˆ log (Vt /L) + ν(T − t) √ , σ T −t log v 2 (t) − log(LVt ) + ν(T − t) ¯ √ , σ T −t log (L/Vt ) − (ν + σ 2 )(T − t) √ , σ T −t log (K/Vt ) − (ν + σ 2 )(T − t) √ , σ T −t log v 2 (t) − log(LVt ) + (ν + σ 2 )(T − t) ¯ √ , σ T −t log v 2 (t) − log(KVt ) + (ν + σ 2 )(T − t) ¯ √ , σ T −t log (¯(t)/Vt ) + ζσ 2 (T − t) v √ , σ T −t log (¯(t)/Vt ) − ζσ 2 (T − t) v √ . σ T −t where Rt = v (t)/Vt , θ = a + 1, ζ = σ −2 ¯ ˆ h1 (Vt , T − t) = h2 (Vt , T − t) = h3 (Vt , T − t) h4 (Vt , T − t) = = h5 (Vt , T − t) = h6 (Vt , T − t) = h7 (Vt , T − t) = h8 (Vt , T − t) = Special Cases Assume that β1 = β2 = 1 and the barrier function v is such that K = L. Then ¯ necessarily γ ≥ r. It can be checked that for K = L we have D(t, T ) = D1 (t, T ) + D3 (t, T ) where: 2ˆ D1 (t, T ) = LB(t, T ) N h1 (Vt , T − t) − Rt a N h2 (Vt , T − t) θ+ζ θ−ζ D3 (t, T ) = Vt Rt N h7 (Vt , T − t) + Rt N h8 (Vt , T − t) . Case γ = r. If we also assume that γ = r then ζ = −σ −2 ν , and thus ˆ θ+ζ Vt Rt = LB(t, T ), θ−ζ 2ˆ 2ˆ Vt Rt = Vt Rt a+1 = LB(t, T )Rt a . It is also easy to see that in this case h1 (Vt , T − t) = while h2 (Vt , T − t) = log(Vt /L) + ν(T − t) √ = −h7 (Vt , T − t), σ T −t log v 2 (t) − log(LVt ) + ν(T − t) ¯ √ = h8 (Vt , T − t). σ T −t We conclude that if v (t) = Le−r(T −t) = LB(t, T ) then D(t, T ) = LB(t, T ). This result is quite ¯ intuitive. A corporate bond with a safety covenant represented by the barrier function, which equals the discounted value of the bond’s face value, is equivalent to a default-free bond with the same face value and maturity. Case γ > r. For K = L and γ > r, it is natural to expect that D(t, T ) would be smaller than LB(t, T ). It is also possible to show that when γ tends to infinity (all other parameters being fixed), then the Black and Cox price converges to Merton’s price. Further Developments The Black and Cox first-passage-time approach was later developed by, among others: Brennan and Schwartz [14, 15] – an analysis of convertible bonds, Kim et al. [65] – a random barrier and random interest rates, Nielsen et al. [78] – a random barrier and random interest rates, Leland [70], Leland and Toft [71] – a study of an optimal capital structure, bankruptcy costs and tax benefits, Longstaff and Schwartz [73] – a constant barrier and random interest rates, Brigo [16]. 14 Zhou [89] studies the case where the dynamics of the firm is dVt = Vt− ((µ − λν)dt + σdWt + dXt ) Credit Risk, TUNIS 2005 where W is a Brownian motion, X a compound Poisson process Xt = 1 t eYi − 1 where ln Yi = N (a, b2 ) with ν = exp(a+b2 /2)−1. This choice of parameters implies that V eµt is a martingale. In a first part, Zhou studies Merton’s problem in that setting. In a second part, he gives an approximation for the first passage problem when the default time is τ = inf{t : Vt ≤ L}. One can study the problem τ = inf{t : Vt ≤ L(t)} where L(t) is a deterministic function and V a geometric Brownian motion. However, there exists few explicit results. See the appendix for some references. Optimal Capital Structure We consider a firm that has an interest paying bonds outstanding. We assume that it is a consol bond, which pays continuously coupon rate c. Assume that r > 0 and the payout rate κ is equal to zero. This condition can be given a financial interpretation as the restriction on the sale of assets, as opposed to issuing of new equity. Equivalently, we may think about a situation in which the stockholders will make payments to the firm to cover the interest payments. However, they have the right to stop making payments at any time and either turn the firm over to the bondholders or pay them a lump payment of c/r per unit of the bond’s notional amount. Recall that we denote by E(Vt ) (D(Vt ), resp.) the value at time t of the firm equity (debt, resp.), hence the total value of the firm’s assets satisfies Vt = E(Vt ) + D(Vt ). Black and Cox [7] argue that there is a critical level of the value of the firm, denoted as v ∗ , below which no more equity can be sold. The critical value v ∗ will be chosen by stockholders, whose aim is to minimize the value of the bonds (equivalently, to maximize the value of the equity). Let us observe that v ∗ is nothing else than a constant default barrier in the problem under consideration; the optimal default time τ ∗ thus equals τ ∗ = inf { t ≥ 0 : Vt ≤ v ∗ }. To find the value of v ∗ , let us first fix the bankruptcy level v . The ODE for the pricing function ¯ u = u∞ (V ) of a consol bond takes the following form (recall that σ = σV ) ∞ N law 1 2 2 ∞ V σ uV V + rV u∞ + c − ru∞ = 0, V 2 subject to the lower boundary condition u∞ (¯) = min (¯, c/r) and the upper boundary condition v v V →∞ lim u∞ (V ) = 0. V For the last condition, observe that when the firm’s value grows to infinity, the possibility of default becomes meaningless, so that the value of the defaultable consol bond tends to the value c/r of the default-free consol bond. The general solution has the following form: u∞ (V ) = c + K1 V + K2 V −α , r where α = 2r/σ 2 and K1 , K2 are some constants, to be determined from boundary conditions. We find that K1 = 0, and v α+1 − (c/r)¯α , if v < c/r, ¯ v ¯ K2 = 0, if v ≥ c/r. ¯ Hence, if v < c/r then ¯ u∞ (Vt ) = or, equivalently, u∞ (Vt ) = c r 1− v ¯ Vt c c + v α+1 − v α Vt−α ¯ ¯ r r α +v ¯ v ¯ Vt α . M. Jeanblanc 15 It is in the interest of the stockholders to select the bankruptcy level in such a way that the value of the debt, D(Vt ) = u∞ (Vt ), is minimized, and thus the value of firm’s equity c E(Vt ) = Vt − D(Vt ) = Vt − (1 − qt ) − v qt ¯ ¯¯ r is maximized. It is easy to check that the optimal level of the barrier does not depend on the current value of the firm, and it equals c c α = . v∗ = rα+1 r + σ 2 /2 Given the optimal strategy of the stockholders, the price process of the firm’s debt (i.e., of a consol bond) takes the form, on the set {τ ∗ > t}, D∗ (Vt ) = or, equivalently, 1 c − r αVtα c r + σ 2 /2 α+1 c ∗ ∗ D∗ (Vt ) = (1 − qt ) + v ∗ qt , r ∗ qt = where v∗ Vt α = 1 Vtα c r + σ 2 /2 α . Further Developments We end this section by remarking that other important developments in the area of optimal capital structure were presented in the papers by Leland [70], Leland and Toft[71], Hilberink and Rogers [50] Christensen et al. [20] and LeCourtois and Quittard-Pinon.. It is probably worth noting that Hilberink and Rogers [50] model the firm value process as a diffusion with jumps. The reason for this extension was to eliminate an undesirable feature of previously examined models, in which short spreads tend to zero when a bond approaches maturity date. 1.3 Stochastic Interest Rates In this section, we assume that the underlying probability space (Ω, F, P), endowed with the filtration F = (Ft )t≥0 , supports the short-term interest rate process r and the value process V. The dynamics under the martingale measure Q of the firm’s value and of the price of a default-free zero-coupon bond B(t, T ) are dVt = Vt (rt − κ(t)) dt + σ(t) dWt∗ and dB(t, T ) = B(t, T ) rt dt + b(t, T ) dWt∗ respectively, where W ∗ is a d-dimensional standard Brownian motion. Furthermore, κ : [0, T ] → I R, σ : [0, T ] → I d and b(·, T ) : [0, T ] → I d are assumed to be bounded functions. The forward value R R FV (t, T ) = Vt /B(t, T ) of the firm satisfies under the forward martingale measure PT dFV (t, T ) = −κ(t)FV (t, T ) dt + FV (t, T ) σ(t) − b(t, T ) dWtT where the process WtT = Wt∗ − t ∈ [0, T ], we set Then κ κ dFV (t, T ) = FV (t, T ) σ(t) − b(t, T ) dWtT . t 0 b(u, T ) du, t ∈ [0, T ], is a d-dimensional SBM under PT . For any T t κ FV (t, T ) = FV (t, T )e− κ(u) du . 16 Credit Risk, TUNIS 2005 κ Furthermore, it is apparent that FV (T, T ) = FV (T, T ) = VT . We consider the following modification of the Black and Cox approach: ˜ X = L, Zt = β2 Vt , X = β1 VT , τ = inf { t ∈ [0, T ] : Vt < vt }, where β2 , β1 ∈ [0, 1] are constants, and the barrier v is given by the formula vt = KB(t, T )e L T t κ(u) du for t < T, for t = T, with the constant K satisfying 0 < K ≤ L. Let us denote, for any t ≤ T, T T t κ(t, T ) = t κ(u) du, σ 2 (t, T ) = |σ(u) − b(u, T )|2 du κ where | · | is the Euclidean norm in I d . For brevity, we write Ft = FV (t, T ), and we denote R 1 1 η+ (t, T ) = κ(t, T ) + σ 2 (t, T ), η− (t, T ) = κ(t, T ) − σ 2 (t, T ). 2 2 The following result extends Black and Cox valuation formula for a corporate bond to the case of random interest rates. Proposition 1.3.1 For any t < T, the forward price of a defaultable bond FD (t, T ) = D(t, T )/B(t, T ) equals on the set {τ > t} L N h1 (Ft , t, T ) − (Ft /K)e−κ(t,T ) N h2 (Ft , t, T ) + β1 Ft e−κ(t,T ) N h3 (Ft , t, T ) − N h4 (Ft , t, T ) + β1 K N h5 (Ft , t, T ) − N h6 (Ft , t, T ) + β2 KJ+ (Ft , t, T ) + β2 Ft e−κ(t,T ) J− (Ft , t, T ), where h1 (Ft , t, T ) h2 (Ft , T, t) h3 (Ft , t, T ) h4 (Ft , t, T ) h5 (Ft , t, T ) h6 (Ft , t, T ) = = = = = = log (Ft /L) − η+ (t, T ) , σ(t, T ) 2 log K − log(LFt ) + η− (t, T ) , σ(t, T ) log (L/Ft ) + η− (t, T ) , σ(t, T ) log (K/Ft ) + η− (t, T ) , σ(t, T ) 2 log K − log(LFt ) + η+ (t, T ) , σ(t, T ) log(K/Ft ) + η+ (t, T ) , σ(t, T ) log(K/Ft ) + κ(t, T ) ± 1 σ 2 (t, u) 2 σ(t, u) and for any fixed 0 ≤ t < T and Ft > 0 we set T J± (Ft , t, T ) = t eκ(u,T ) dN . In the special case when κ ≡ 0, the formula of Proposition 1.3.1 covers as a special case the valuation result established by Briys and de Varenne [17]. In some other recent studies of first passage time models, in which the triggering barrier is assumed to be either a constant or an unspecified stochastic process, typically no closed-form solution for the value of a corporate debt is available, and thus a numerical approach is required (see, for instance, Kim et al. [65], Longstaff and Schwartz [73], Nielsen et al. [78], or Sa´-Requejo and Santa-Clara [81]). a M. Jeanblanc 17 1.4 Random Barrier In the case of full information and Brownian filtration, the first hitting time of a deterministic barrier is predictable. This is no longer the case when we deal with incomplete information (as in Duffie and Lando [34]), or when an additional source of randomness is present. We present here a formula for credit spreads arising in a special case of a totally inaccessible time of default. For a more detailed study we refer to Babbs and Bielecki [2]. As we shall see, the method we use here is close to the general method presented in Chapter 3. We suppose here that the default barrier is a random variable D defined on the underlying probability space (Ω, P). The default occurs at time τ where τ = inf{t : Vt ≤ D} , where V is the value of the firm. Note that {τ > t} = {inf Vu > D} . u≤t We shall denote by mV the running minimum of V , i.e. mV = inf u≤t Vu . With this notation, t t {τ > t} = {mV > D} . t 1.4.1 Independent barrier In a first step we assume that, under the risk-neutral probability Q, D is independent of the value of the firm. We denote by FD the cumulative distribution function of the r.v. D, i.e. FD (z) = Q(D ≤ z). We assume that FD is differentiable and we denote fD its derivative. Lemma 1.4.1 Let Ft = Q(τ ≤ t|Ft ) and Γt = − ln(1 − Ft ). Then t Γt = − 0 fD (mV ) u dmV . u FD ((mV ) u Proof: If D is independent of F∞ , Ft = Q(τ ≤ t|Ft ) = Q(mV ≤ D|Ft ) = 1 − FD (mV ) . t t The process mV is decreasing. It follows that Γt = − ln FD (mV ), hence dΓt = t t fD (mV ) t dmV t FD (mV ) t and Γt = − 0 fD (mV ) u dmV . u FD (mV ) u Example 1.4.1 Assume that D is uniformly distributed on the interval [0, 1]. Then, Γt = − ln mV . t The computation of quantities as E(eΓT f (VT )) requires the knowledge of the joined law of the pair (VT , mV ). T We postulate now that the value process V is a geometric Brownian motion with a drift, that is, we set Vt = eΨt , where Ψt = µt + σWt . It is clear that τ = inf {t ≥ 0 : Ψ∗ ≤ ψ}, where Ψ∗ is the t running minimum of the process Ψ: Ψ∗ = inf {Ψs : 0 ≤ s ≤ t}. t We choose the Brownian filtration as the reference filtration, i.e., we set F = FW . Let us denote by G(z) the cumulative distribution function under Q of the barrier ψ. We assume that G(z) > 0 for z < 0 and that G admits the density g with respect to the Lebesgue measure (note that g(z) = 0 for z > 0). This means that we assume that the value process V (hence also the process Ψ) is 18 Credit Risk, TUNIS 2005 perfectly observed. In addition, we suppose that the bond investor can observe the occurrence of the default time. Thus, he can observe the process Ht = 1 {τ ≤t} = 1 {Ψ∗ ≤ψ} . We denote by H the 1 1 t natural filtration of the process H. The information available to the investor is represented by the (enlarged) filtration G = F ∨ H. We assume that the default time τ and interest rates are independent under Q. Then, it is possible to establish the following result (see Giesecke [44] or Babbs and Bielecki [2]). Note that the process Ψ∗ is decreasing, so that the integral with respect to this process is a (pathwise) Stieltjes integral. Proposition 1.4.1 Under the assumptions stated above, and additionally assuming L = 1, Z ≡ 0 ˜ and X = 0, we have that for every t < T S(t, T ) = −1 {τ >t} 1 1 log EP∗ e T −t ∗ T fD (Ψu ) t FD (Ψ∗ ) u dΨ∗ u Ft . In the next chapter, we shall introduce the notion of a hazard process of a random time. For the default time τ defined above, the F-hazard process Γ exists and is given by the formula t Γt = − 0 fD (Ψ∗ ) u dΨ∗ . u FD (Ψ∗ ) u This process is continuous, and thus the default time τ is a totally inaccessible stopping time with respect to the filtration G. 1.5 Comments on Structural Models We end this chapter by commenting on merits and drawbacks of the structural approach to credit risk. Advantages • An approach based on the volatility of the total value of a firm. The credit risk is thus measured in a standard way. The random time of default is defined in an intuitive way. The default event is linked to the notion of the firm’s insolvency. • Valuation and hedging of defaultable claims relies on similar techniques as the valuation and hedging of exotic options in the standard default-free Black-Scholes setup. • The concept of the distance to default, which measures the obligor’s leverage relative to the volatility of its assets value, may serve to reflect credit ratings. • Dependent defaults are easy to handle through correlation of processes corresponding to different names. Disadvantages • A stringent assumption that the total value of the firm’s assets can be easily observed. In practice, continuous-time observations of the value process V are not available. This issue was recently addressed by Crouhy et al.[23], Duffie and Lando [34], Jeanblanc and Valchev [61], who showed that a structural model with incomplete accounting data can be dealt with using the intensity-based methodology. The paper of Guo [48] presents a case with delayed information. • An unrealistic postulate that the total value of the firm’s assets is a tradeable security. • This approach is known to generate low credit spreads for corporate bonds close to maturity. It requires a judicious specification of the default barrier in order to get a good fit to the observed spread curves. M. Jeanblanc 19 Other issues • A major problem with applying structural models is the difficulty with estimation of the volatility of assets value. For the classical Merton’s model, there exists a simple formula that relates this volatility to the volatility of the firm’s equity, which in principle can be easily estimated. However, no such simple expression exists in case of first-passage-time models. Certain market-oriented technologies, such as CreditGrades, attempt to produce such a formula. • Structural models discussed above were at most one-factor models, with the only factor being the short-term interest rate. Two- and three-factor structural models have been also developed and closed-form valuation formulae were derived in some special cases. 20 Credit Risk, TUNIS 2005 Chapter 2 Hazard process Approach: A Toy Model Our goal is to furnish results which cover the reduced form methodology. We provide a detailed analysis of the relatively simple case when the flow of informations available to an agent reduces to the observations of the random time which models the default event. The focus is on the evaluation of conditional expectations with respect to the filtration generated by a default time with the use of the hazard function. In the following chapters, we study the case when an additional information flow - formally represented by some filtration F - is present, with the use of the hazard process. 2.1 The Toy Model We begin with the simple case where a riskless asset, with deterministic interest rate (r(s); s ≥ 0) is t the only asset available in the default-free market. We denote as usual by R(t) = exp − 0 r(s)ds the discount factor. The price of a risk-free zero-coupon bond with maturity T is B(0, T ) = R(T ), whereas its time t price B(t, T ) is t B(t, T ) = RT = exp − def T r(s)ds t . Default occurs at time τ (where τ is assumed to be a positive random variable with density f , constructed on a probability space (Ω, G, P)). We denote by F the cumulative function of the r.v. τ t defined as F (t) = P(τ ≤ t) = 0 f (s)ds and we assume that F (t) < 1 for any t < T , where T is the maturity date (Otherwise there exists t0 < T such that F (t0 ) = 1, and default occurs a.s. before t0 ). We emphasize that the risk is not hedgeable. Indeed, a random payoff of the form 1 {T <τ } cannot 1 be perfectly hedged with deterministic zero-coupon bonds which are the only tradeable assets in our model. To hedge the risk, we shall assume later on that some defaultable asset is traded, e.g., a defaultable zero-coupon bond or a CDS (Credit default swap). Remark 2.1.1 It is not difficult to generalize the study presented in what follows to the case where τ does not admit a density by dealing with the right-continuous version of the cumulative function. The case where τ is bounded can also be studied along the same method. 2.1.1 Defaultable Zero-coupon with Payment at Maturity A defaultable zero-coupon bond (DZC in short)- or a corporate bond- with maturity T and rebate δ paid at maturity, consists of 21 22 Credit Risk, TUNIS 2005 • The payment of one monetary unit at time T if default has not occurred before time T , • A payment of δ monetary units, made at maturity, if τ < T , where 0 < δ < 1. Value of the defaultable zero-coupon bond The “value” of the defaultable zero-coupon bond is defined as D(0, T ) = E(R(T ) [1 {T <τ } + δ1 {τ ≤T } ]) 1 1 = E(R(T ) [1 − (1 − δ)1 {τ ≤T } ]) 1 = B(0, T )[1 − (1 − δ)F (T )] . (2.1) In fact, this quantity is a net present value and is equal to the value of the default free ZC, minus the expected loss, computed under the historical probability. Obviously, this is not a hedging price. The time-t value depends whether or not default has happened before this time. If default has t occurred before time t, the payment of δ will be made at time T , and the price of the DZC is δRT : in that case, the payoff is hedgeable with δ default-free zero-coupon bonds. If the default has not yet occurred, the holder does not know when it will occur. The value D(t, T ) t of the DZC is the expectation of the discounted payoff RT [1 {T <τ } + δ1 {τ ≤T } ] knowing that t < τ , 1 1 that is, D(t, T ) = = = = = t 1 1 E RT (1 {T <τ } + δ1 {τ ≤T } ) t < τ t RT 1 − (1 − δ)P(τ ≤ T t < τ ) t RT P(t < τ ≤ T ) P(t < τ ) F (T ) − F (t) B(t, T ) 1 − (1 − δ) 1 − F (t) 1 − F (T ) F (T ) − F (t) B(t, T ) +δ 1 − F (t) 1 − F (t) 1 − (1 − δ) (2.2) . (2.3) Note that the value of the DZC is discontinuous at time τ , unless F (T ) = 1 (or δ = 1). In the case F (T ) = 1, the default appears with probability one before maturity and the DZC is equivalent to a payment of δ at maturity. If δ = 1, the DZC is in fact a default-free zero coupon bond. Formula (2.2) can be read as D(t, T ) = B(t, T ) − EDLGD × DP where the Expected Discounted Loss Given Default (EDLGD) is defined as B(t, T )(1 − δ) and the Default Probability (DP) is DP = P(t < τ ≤ T ) = P(τ ≤ T |t < τ ) . P(t < τ ) In case the payment is a function of the default time, say δ(τ ), the value of this defaultable zerocoupon is D(0, T ) = = E R(T ) 1 {T <τ } + R(T )δ(τ )1 {τ ≤T } 1 1 T B(0, T ) P(T < τ ) + 0 δ(s)f (s)ds . If the default has not occurred before t, the time-t value D(t, T ) satisfies D(t, T ) = B(t, T )E( 1 {T <τ } + δ(τ )1 {τ ≤T } t < τ ) 1 1 = B(t, T ) 1 P(T < τ ) + P(t < τ ) P(t < τ ) T δ(s)f (s)ds . t M. Jeanblanc 23 Hazard function We introduce the hazard function Γ defined by Γ(t) = − ln(1 − F (t)) and its derivative γ(t) = f (t) where f (t) = F (t), i.e., 1 − F (t) t 1 − F (t) = e−Γ(t) = exp − 0 γ(s)ds = P(τ > t) . The quantity γ(t) is the hazard rate. The interpretation of the hazard rate is the probability that the default occurs in a small interval dt given that the default did not occur before time t γ(t) = lim Note that Γ is increasing. Then, formula (2.3) reads t,d t,d t D(t, T ) = RT + δ(RT − RT ) , h→0 1 P (τ ≤ t + h|τ > t) . h where t,d RT = exp − t T (r + γ)(s)ds . t,d In particular, for δ = 0, D(t, T ) = RT . Hence, the spot rate has to be adjusted by means of a spread (equal to γ) in order to evaluate DZCs. If γ and δ are constant, the credit spread is 1 B(t, T ) 1 ln =γ− ln 1 + δ(eγ(T −t) − 1) T − t D(t, T ) T −t and goes to γ(1 − δ) when t goes to T . The quantity λ(t, T ) = f (t,T ) 1−F (t,T ) where F (t, T ) = P(τ ≤ T |τ > t) and f (t, T ) dT = P(τ ∈ dT |τ > t) is called the conditional hazard rate. One has T F (t, T ) = 1 − exp − t λ(s, T )ds . T In our setting, 1 − F (t, T ) = and λ(s, T ) = γ(s). P(τ > T ) = exp − P(τ > t) t γ(s)ds Remark 2.1.2 In case τ is the first jump of an inhomogeneous Poisson process with deterministic intensity (λ(t), t ≥ 0) (See Appendix if needed), t f (t) = P(τ ∈ dt)/dt = λ(t) exp − 0 t λ(s)ds = λ(t)e−Λ(t) where Λ(t) = 0 λ(s)ds and P(τ ≤ t) = F (t) = 1 − e−Λ(t) , hence the hazard function is equal to the compensator of the Poisson process, i.e. Γ(t) = Λ(t). Conversely, if τ is a random time with density f , setting Λ(t) = − ln(1 − F (t)) allows us to interpret τ as the first jump time of an inhomogeneous Poisson process with intensity the derivative of Λ. 24 Credit Risk, TUNIS 2005 2.1.2 Defaultable Zero-coupon with Payment at Hit Here, a defaultable zero-coupon bond with maturity T consists of • The payment of one monetary unit at time T if default has not yet occurred, • A payment of δ(τ ) monetary units, where δ is a deterministic function, made at time τ if τ < T. Value of the defaultable zero-coupon The value of this defaultable zero-coupon bond is D(0, T ) = = = E(R(T ) 1 {T <τ } + R(τ )δ(τ )1 {τ ≤T } ) 1 1 T P(T < τ )R(T ) + 0 T R(s)δ(s)dF (s) R(s)δ(s)dG(s) , (2.4) G(T )R(T ) − 0 where G(t) = 1 − F (t) = P(t < τ ) is the survival probability. (We do not assume here that F is differentiable.) Obviously, if the default has occurred before time t, the value of the DZC is null (this was not the case for payment of the rebate at maturity), and D(t, T ) = 1 t T |τ > t) = D (t,T)) . Of course, this probability may B(t,T differ from the historical probability. The implied hazard rate is the function λ(t, T ) such that λ(t, T ) = − ∂ D∗ (t, T ) ln . ∂T B(t, T ) = γ ∗ (T ) 2.1.4 Spreads A term structure of credit spreads associated with the zero-coupon bonds S(t, T ) is defined as S(t, T ) = − In our setting, on the set {τ > t} S(t, T ) = − whereas S(t, T ) = ∞ on the set {τ ≤ t}. 1 ln Q∗ (τ > T |τ > t) , T −t 1 D∗ (t, T ) ln . T −t B(t, T ) 26 Credit Risk, TUNIS 2005 2.2 Toy Model and Martingales We now present the results of the previous section in a different form, following closely Dellacherie ([28], page 122). We keep the same notation for the cumulative function and the hazard function, assumed to be continuous. We denote by (Ht , t ≥ 0) the right-continuous increasing process Ht = 1 {t≥τ } and by (Ht ) its natural filtration. The filtration H is the smallest filtration which makes τ 1 a stopping time. The σ-algebra Ht is generated by the sets {τ ≤ s} for s ≤ t (or by the r.v. τ ∧ t) (note that the set {τ > t} is an atom). A key point is that any integrable Ht -measurable r.v. H is of the form H = h(τ ∧ t) = h(τ )1 {τ ≤t} + h(t)1 {t<τ } where h is a Borel function. 1 1 We now give some elementary tools to compute the conditional expectation w.r.t. Ht , as presented in Br´maud [12], Dellacherie [28], Elliott [39]. e 2.2.1 Key Lemma E(X1 {s<τ } ) 1 . P(s < τ ) Lemma 2.2.1 If X is any integrable, G-measurable r.v. E(X|Hs )1 {s<τ } = 1 {s<τ } 1 1 (2.7) Proof: The r.v. E(X|Hs ) is Hs -measurable. Therefore, it can be written in the form E(X|Hs ) = h(τ ∧ s) = h(τ )1 {s≥τ } + h(s)1 {s<τ } for some function h. By multiplying both members by 1 {s<τ } , 1 1 1 and taking the expectation, we obtain E[1 {s<τ } E(X|Hs )] 1 = = E[E(1 {s<τ } X|Hs )] = E[1 {s<τ } X] 1 1 E(h(s)1 {s<τ } ) = h(s)P(s < τ ) . 1 Hence, h(s) = E(X1 {s<τ } ) 1 gives the desired result. P(s < τ ) 2.2.2 Some Martingales τ ∧t Proposition 2.2.1 The process (Mt , t ≥ 0) defined as Mt = Ht − 0 dF (s) = Ht − 1 − F (s) t (1 − Hs− ) 0 dF (s) 1 − F (s) is a H-martingale. Proof: Let s < t. Then: E(Ht − Hs |Hs ) = 1 {s<τ } E(1 {s<τ ≤t} |Hs ) = 1 {s<τ } 1 1 1 which follows from (2.7) with X = 1 {τ ≤t} . 1 On the other hand, the quantity C = E s def t F (t) − F (s) , 1 − F (s) (2.8) (1 − Hu− ) dF (u) Hs , 1 − F (u) is equal to t C = s dF (u) E 1 {τ >u} Hs 1 1 − F (u) F (u) − F (s) dF (u) 1− 1 − F (u) 1 − F (s) s F (t) − F (s) 1 − F (s) t = 1 {τ >s} 1 = 1 {τ >s} 1 which, from (2.8) proves the desired result. M. Jeanblanc 27 2.2.3 Hazard Function t 0 The function dF (s) = − ln(1 − F (t)) = Γ(t) 1 − F (s) is the hazard function. From Proposition 2.2.1, we obtain the Doob-Meyer decomposition of the submartingale Ht as Mt + Γ(t ∧ τ ). The predictable process At = Γt∧τ is called the compensator of H. In particular, if F is differentiable, the process τ ∧t t Mt = Ht − 0 γ(s)ds = Ht − 0 γ(s)(1 − Hs )ds is a martingale, where γ(s) = of τ . f (s) is a deterministic non-negative function, called the intensity 1 − F (s) def t Proposition 2.2.2 The process Lt = 1 {τ >t} exp 1 for t < T , γ(s)ds 0 T is a H-martingale. In particular, E(1 {τ >T } |Ht ) = 1 {τ >t} exp − 1 1 γ(s)ds t . Proof: We shall give 3 different arguments, each of which constitutes a proof. a) Since the function γ is deterministic, for t > s t E(Lt |Hs ) = exp 0 γ(u)du E(1 {t<τ } |Hs ) . 1 From the equality (2.7) E(1 {t<τ } |Hs ) = 1 {τ >s} 1 1 Hence, E(Lt |Hs ) = 1 {τ >s} exp 1 t 0 1 − F (t) = 1 {τ >s} exp (−Γ(t) + Γ(s)) . 1 1 − F (s) s γ(u)du = Ls . b) Another method is to apply Itˆ’s formula (see Appendix 9.3.2 if needed) to the process Lt = o (1 − Ht ) exp 0 t t γ(s)ds dLt = = −dHt exp 0 t γ(s)ds + γ(t) exp 0 γ(s)ds (1 − Ht )dt − exp 0 γ(s)ds dMt . c) A third (sophisticated) method is to note that L is the exponential martingale of M (see Appendix), i.e., the solution of the SDE dLt = −Lt− dMt , L0 = 1. Lemma 2.2.2 Let h be a (bounded) Borel function. Then, E(h(τ )1 τ t} e−Γ(t) 1 1 1 T h(u)dF (u) t (2.9) 28 Credit Risk, TUNIS 2005 Proposition 2.2.3 Assume that Γ is a continuous function. Let h : I + → I be a non-negative R R Borel measurable function such that the random variable h(τ ) is integrable. Then the process Mth = (1 + 1 τ ≤t h(τ )) exp − 1 is a H-martingale. Proof: One notes that Mth t τ t∧τ h(u) dΓ(u) 0 (2.10) = exp − 0 t (1 − Hu )h(u) dΓ(u) + 1 τ ≤t h(τ ) exp − 1 t 0 u h(u) dΓ(u) h(s) dΓ(s) dHu 0 = exp − 0 (1 − Hu )h(u) dΓ(u) + 0 h(u) exp − From Itˆ’s calculus, o dMth t = exp − 0 (1 − Hu )h(u) dΓ(u) (−(1 − Ht )h(t) dΓ(t) + h(t)dHt ) t = h(t) exp − 0 (1 − Hu )h(u) dΓ(u) dMt . Example 2.2.1 In the case where N is an inhomogeneous Poisson process with deterministic int tensity λ and τ is the first time when N jumps, let Ht = Nt∧τ . It is well known that Nt − 0 t∧τ λ(s)ds is a martingale (see Appendix). Therefore, the process stopped at time τ is also a martingale, i.e., Ht − 0 λ(s)ds is a martingale. Furthermore, we have seen in Remark 2.1.2 that we can reduce our attention to this case, since any random time can be viewed as the first time where an inhomogeneous Poisson process jumps. Exercise 2.2.1 In this exercise, F is only continuous on right, and F (t−) is the left limit at point t. Prove that the process (Mt , t ≥ 0) defined as τ ∧t M t = Ht − 0 dF (s) = Ht − 1 − F (s−) t (1 − Hs− ) 0 dF (s) 1 − F (s−) is a H-martingale. 2.2.4 Representation Theorem Proposition 2.2.4 Let h be a (bounded) Borel function. Then, the martingale Mth = E(h(τ )|Ht ) admits the representation t∧τ E(h(τ )|Ht ) = E(h(τ )) − 0 (g(s) − h(s)) dMs , where Mt = Ht − Γ(t ∧ τ ) and g(t) = − 1 G(t) ∞ h(u)dG(u) = − t 1 E(h(τ )1 τ >t ) . 1 G(t) Note that g(t) = Mth on {t < τ }. In particular, any square integrable H-martingale (Xt , t ≥ 0) can t be written as Xt = X0 + 0 xs dMs where (xt , t ≥ 0) is a predictable process. M. Jeanblanc 29 Proof: We give two different proofs. • First proof: From Lemma 2.2.1 Mth = h(τ )1 {τ ≤t} + 1 {t<τ } 1 1 E(h(τ )1 {t<τ } ) 1 P(t < τ ) = h(τ )1 {τ ≤t} + 1 {t<τ } eΓ(t) E(h(τ )1 {t<τ } ) . 1 1 1 An integration by parts leads to eΓt E[h(τ )1 {t<τ } ] = eΓt 1 ∞ ∞ h(s)dF (s) = g(t) t t = 0 h(s)dF (s) − 0 ∞ eΓ(s) h(s)dF (s) + 0 t E(h(τ )1 {s<τ } )eΓ(s) dΓ(s) 1 h 1 Therefore, since E(h(τ )) = 0 h(s)dF (s) and Ms = eΓ(s) E(h(τ )1 {s<τ } ) = g(s) on {s < τ }, the following equality holds on the set {t < τ }: eΓt E[h(τ )1 {t<τ } ] = E(h(τ )) − 1 Hence, t 0 eΓ(s) h(s)dF (s) + 0 t g(s)dΓ(s) . t∧τ 1 {t<τ } E(h(τ )|Ht ) = 1 = 1 {t<τ } E(h(τ )) + 1 1 {t<τ } E(h(τ )) − 1 t∧τ (g(s) − h(s)) 0 t∧τ dF (s) 1 − F (s) , (g(s) − h(s))(dHs − dΓ(s)) 0 where the last equality is due to 1 t<τ 1 0 t∧τ (g(s) − h(s))dHs = 0. ] f10 5 f17 78 08 [ 2 6 ; On the complementary set {t ≥ τ }, we have seen that E(h(τ )|Ht ) = h(τ ), whereas (g(s) − h(s))(dHs − dΓ(s)) =0 (g(s) − h(s))(dHs − dΓ(s)) = 30 Exercise 2.2.2 If Γ is not continuous, prove that t∧τ Credit Risk, TUNIS 2005 E(h(τ )|Ht ) = E(h(τ )) − 0 e∆Γ(s) (g(s) − h(s)) dMs . 2.2.5 Change of a Probability Measure Let P∗ be an arbitrary probability measure on (Ω, H∞ ), which is absolutely continuous with respect to P. We denote by η the H∞ -measurable density of P∗ with respect to P η := dP∗ = h(τ ) ≥ 0, dP ∞ P-a.s., (2.11) where h : I → I + is a Borel measurable function satisfying R R EP (h(τ )) = 0 h(u) dF (u) = 1. We can use Girsanov’s theorem. Nevertheless, we prefer here to establish this theorem in our particular setting. Of course, the probability measure P∗ is equivalent to P if and only if the inequality in (2.11) is strict P-a.s. Furthermore, we shall assume that P∗ (τ = 0) = 0 and P∗ (τ > t) > 0 for any t ∈ I + . Actually the first condition is satisfied for any P∗ absolutely continuous with respect to P. R For the second condition to hold, it is sufficient and necessary to assume that for every t P∗ (τ > t) = 1 − F ∗ (t) = ]t,∞[ h(u) dF (u) > 0, where the c.d.f. F ∗ of τ under P∗ F ∗ (t) := P∗ (τ ≤ t) = [0,t] h(u) dF (u). (2.12) Put another way, we assume that g(t) = eΓ(t) E 1 τ >t h(τ ) = eΓ(t) 1 ]t,∞[ def h(u) dF (u) = eΓ(t) P∗ (τ > t) > 0. We assume throughout that this is the case, so that the hazard function Γ∗ of τ with respect to P∗ is well defined. Our goal is to examine relationships between hazard functions Γ∗ and Γ. It is easily seen that in general we have ln ]t,∞[ h(u) dF (u) Γ∗ (t) = , (2.13) Γ(t) ln(1 − F (t)) since by definition Γ∗ (t) = − ln(1 − F ∗ (t)). Assume first that F is an absolutely continuous function, so that the intensity function γ of τ under P is well defined. Recall that γ is given by the formula γ(t) = f (t) . 1 − F (t) On the other hand, the c.d.f. F ∗ of τ under P∗ now equals F ∗ (t) := P∗ (τ ≤ t) = EP (1 τ ≤t h(τ )) = 1 0 t h(u)f (u) du. so that F ∗ follows an absolutely continuous function. Therefore, the intensity function γ ∗ of the random time τ under P∗ exists, and it is given by the formula γ ∗ (t) = h(t)f (t) h(t)f (t) = . t 1 − F ∗ (t) 1 − 0 h(u)f (u) du M. Jeanblanc 31 To derive a more straightforward relationship between the intensities γ and γ ∗ , let us introduce an auxiliary function h∗ : I + → I given by the formula h∗ (t) = h(t)/g(t). R R, Notice that γ ∗ (t) = h(t)f (t) 1− t 0 h(u)f (u) du = ∞ t h(t)f (t) h(t)f (t) f (t) = −Γ(t) = h∗ (t) = h∗ (t)γ(t). 1 − F (t) e g(t) h(u)f (u) du This means also that dΓ∗ (t) = h∗ (t) dΓ(t). It appears that the last equality holds true if F is merely a continuous function. Indeed, if F (and thus F ∗ ) is continuous, we get dΓ∗ (t) = d(1 − e−Γ(t) g(t)) dF ∗ (t) g(t)dΓ(t) − dg(t) = = h∗ (t) dΓ(t). = ∗ (t) −Γ(t) g(t) 1−F g(t) e To summarize, if the hazard function Γ is continuous then Γ∗ is also continuous and dΓ∗ (t) = h∗ (t) dΓ(t). To understand better the origin of the function h∗ , let us introduce the following non-negative P-martingale (which is strictly positive when the probability measures P∗ and P are equivalent) ηt := dP∗ = EP (η|Ht ) = EP (h(τ )|Ht ), dP |Ht (2.14) so that ηt = Mth . The general formula for ηt reads (cf. (2.2.1)) ηt = 1 τ ≤t h(τ ) + 1 τ >t eΓ(t) 1 1 ]t,∞[ h(u) dF (u) = 1 τ ≤t h(τ ) + 1 τ >t g(t). 1 1 Assume now that F is a continuous function. Then ∞ ηt = 1 τ ≤t h(τ ) + 1 τ >t 1 1 t h(u)eΓ(t)−Γ(u) dΓ(u). On the other hand, using the representation theorem and (5.7), we get h Mth = M0 + h Mu− (h∗ (u) − 1) dMu ]0,t] where h∗ (u) = h(u)/g(u). We conclude that ηt = 1 + ]0,t] ηu− (h∗ (u) − 1) dMu . (2.15) It is thus easily seen that t∧τ ηt = 1 + 1 τ ≤t v(τ )) exp − 1 0 v(u) dΓ(u) , (2.16) where we write v(t) = h∗ (t)−1. Therefore, the martingale property of the process η, which is obvious from (2.14), is also a consequence of Proposition 2.2.3. Remark 2.2.1 In view of (2.15), we have · η t = Et 0 (h∗ (u) − 1) dMu , where E stands for the Dol´ans exponential. Representation (2.16) for the random variable ηt can e thus be obtained from the general formula for the Dol´ans exponential. (See Appendix) e 32 Credit Risk, TUNIS 2005 We are in the position to formulate the following result (all statements were already established above). Proposition 2.2.5 Let P∗ be any probability measure on (Ω, H∞ ) absolutely continuous with respect to P, so that (2.11) holds for some function h. Assume that P∗ (τ > t) > 0 for every t ∈ I + . Then R dP∗ = Et dP |Ht where h∗ (t) = h(t)/g(t), and Γ∗ (t) = g ∗ (t)Γ(t) with g ∗ (t) = ln ]t,∞[ · 0 ∞ (h∗ (u) − 1) dMu , (2.17) g(t) = eΓ(t) t h(u) dF (u), h(u) dF (u) ln(1 − F (t)) . (2.18) If, in addition, the random time τ admits the intensity function γ under P, then the intensity function γ ∗ of τ under P∗ satisfies γ ∗ (t) = h∗ (t)γ(t) a.e. on I + . More generally, if the hazard function Γ R of τ under P is continuous, then the hazard function Γ∗ of τ under P∗ is also continuous, and it satisfies dΓ∗ (t) = h∗ (t) dΓ(t). Corollary 2.2.1 If F is continuous then Mt∗ = Ht − Γ∗ (t ∧ τ ) is a H-martingale under P∗ . Proof: In view Proposition ??, the assertion is an immediate consequence of the continuity of Γ∗ . Alternatively, we may check directly that the product Ut = ηt Mt∗ = ηt (Ht − Γ∗ (t ∧ τ )) follows a H-martingale under P. To this end, observe that the integration by parts formula for functions of finite variation yields Ut = ]0,t] ηt− dMt∗ + ηt− dMt∗ + ηt− dMt∗ + ]0,t] Mt∗ dηt ∗ Mt− dηt + u≤t ∗ Mt− dηt + 1 τ ≤t (ητ − ητ − ). 1 ∗ ∆Mu ∆ηu = ]0,t] ]0,t] = ]0,t] ]0,t] Using (2.15), we obtain Ut = ]0,t] ηt− dMt∗ + ]0,t] ∗ Mt− dηt + ητ − 1 τ ≤t (h∗ (τ ) − 1) 1 = ]0,t] ηt− d Γ(t ∧ τ ) − Γ∗ (t ∧ τ ) + 1 τ ≤t (h∗ (τ ) − 1) + Nt , 1 where the process N, which equals Nt = ]0,t] ηt− dMt + ]0,t] ∗ Mt− dηt is manifestly a H-martingale with respect to P. It remains to show that the process Nt∗ := Γ(t ∧ τ ) − Γ∗ (t ∧ τ ) + 1 τ ≤t (h∗ (τ ) − 1) 1 follows a H-martingale with respect to P. By virtue of Proposition ??, the process 1 τ ≤t (h∗ (τ ) − 1) + Γ(t ∧ τ ) − 1 0 t∧τ h∗ (u) dΓ(u) M. Jeanblanc 33 is a H-martingale. Therefore, to conclude the proof it is enough to notice that t∧τ 0 h∗ (u) dΓ(u) − Γ∗ (t ∧ τ ) = 0 t∧τ (h∗ (u) dΓ(u) − dΓ∗ (u)) = 0, where the last equality is a consequence of the relationship dΓ∗ (t) = h∗ (t) dΓ(t) established in Proposition 2.2.5. By virtue of Proposition ?? if Γ∗ is a continuous function then the process M ∗ = Ht − Γ∗ (t ∧ τ ) follows a H-martingale under P∗ . The next result suggests that this martingale property uniquely characterizes the (continuous) hazard function of a random time. We shall examine this issue in more detail in Section ??. Lemma 2.2.3 Suppose that an equivalent probability measure P∗ is given by formula (2.11) for some function h. Let Λ∗ : I + → I + be an arbitrary continuous increasing function, with Λ∗ (0) = 0. If R R the process Mt∗ := Ht − Λ∗ (t ∧ τ ) follows a H-martingale under P∗ , then Λ∗ (t) = − ln (1 − F ∗ (t)) with F ∗ given by formula (2.17). Proof: The Bayes rule implies EP∗ (Mt∗ |Hs ) = and thus EP∗ (Mt∗ |Hs ) = or equivalently EP∗ (Mt∗ |Hs ) = This means that EP∗ (Mt∗ |Hs ) = where we write J = EP Ht h(τ ) − Ht Λ∗ (t ∧ τ )h(τ ) − (1 − Ht )Λ∗ (t ∧ τ )g(t) Hs . Using (??), we obtain J = Hs h(τ ) − Hs Λ∗ (τ )h(τ ) − (1 − Hs )(1 − F (s))−1 EP 1 {s<τ ≤t} (Λ∗ (τ ) − 1)h(τ ) + 1 {τ >t} Λ∗ (t)g(t) 1 1 ∗ and thus the martingale condition EP∗ (Mt∗ |Hs ) = Ms , is equivalent to the following equality EP (Mt∗ η|Hs ) −1 = ηs EP (Mt∗ ηt |Hs ) EP (η|Hs ) EP (Ht − Λ∗ (t ∧ τ ))(Ht h(τ ) + (1 − Ht )g(t)) Hs , Hs h(τ ) + (1 − Hs )g(s) EP Ht h(τ ) − Ht Λ∗ (t ∧ τ )h(τ ) − (1 − Ht )Λ∗ (t ∧ τ )g(t) Hs . Hs h(τ ) + (1 − Hs )g(s) J , Hs h(τ ) + (1 − Hs )g(s) (1 − Hs )(1 − F (s))−1 EP 1 {s<τ ≤t} (Λ∗ (τ ) − 1)h(τ ) + 1 {τ >t} Λ∗ (t)g(t) = Λ∗ (s)(1 − Hs )g(s). 1 1 Therefore, for every s ≤ t we have EP 1 {s<τ ≤t} (Λ∗ (τ ) − 1)h(τ ) + 1 {τ >t} Λ∗ (t)g(t) = Λ∗ (s)(1 − F (s))g(s) 1 1 so that s t (Λ∗ (u) − 1)h(u) dF (u) + Λ∗ (t)g(t)(1 − F (t)) = Λ∗ (s) s t s ∞ h(u) dF (u), and finally, (Λ∗ (u) − 1) dF ∗ (u) + Λ∗ (t)(1 − F ∗ (t)) = Λ∗ (s)(1 − F ∗ (s)). After simple manipulations involving the integration by parts, we get for s ≤ t t s (1 − F ∗ (u)) dΛ∗ (u) = F ∗ (t) − F ∗ (s), and since Λ∗ (0) = F ∗ (0) = 0, we find that Λ∗ = − ln (1 − F ∗ (t)). 34 Credit Risk, TUNIS 2005 2.2.6 Incompleteness of the Toy model In order to study the completeness of the financial market, we first need to define the tradeable assets. If the market consists only of the risk-free zero-coupon bond, there exists infinitely many e.m.m’s. The discounted asset prices are constant, hence the set Q of equivalent martingale measures is the set of probabilities equivalent to the historical one. For any Q ∈ Q, we denote by FQ the cumulative function of τ under Q, i.e., FQ (t) = Q(τ ≤ t) . The range of prices is defined as the set of prices which do not induce arbitrage opportunities. For a DZC with a constant rebate δ paid at maturity, the range of prices is equal to the set {EQ (RT (1 {T <τ } + δ1 {τ s), f = −G , as soon as the cumulative function of τ is differentiable. Duffie and Lando 1 ∂f have obtained that the intensity is λ(t) = σ 2 (t, 0) (t, 0) where f (t, x) is the conditional density 2 ∂x P(Vt ≤ x, T0 > t) of Vt when T0 > t, i.e. the differential w.r.t. x of , where T0 = inf{t ; Vt = 0}. In P(T0 > t) the case where V is an homogenous diffusion, i.e. dVt = µ(Vt )dt + σ(Vt )dWt , the equality between Duffie-Lando and our result is not so obvious. See Elliott et al. [40] for comments. 2.3 Valuation and Trading Defaultable Claims The goal of this section is to give a brief presentation of general results concerning the valuation and trading of defaultable claims. 2.3.1 Price dynamics of a survival claim (X, 0, τ ). In what follows, we shall refer to a defaultable claim of the form (X, 0, τ ) as a survival claim. By virtue of the risk-neutral valuation formula, the price of the payoff 1 {T <τ } X that settles at time T 1 equals, for every t ∈ [0, T ], Yt = ert EQ (1 {T <τ } e−rT X | Ht ). 1 Note that X is FT -measurable, and thus constant since the σ-field FT is trivial. To find the dynamics of the price process, it suffices to apply Proposition 5.1.1 to the function h(u) = 1 {u>T } e−rT X. For 1 the Q-martingale Mth = e−rt Yt , we thus get, for every t ∈ [0, T ], e−rt Yt = Y0 − 0 t e−ru Yu− dMu . Suppose that Γ(t) = t 0 γ(u) du. Then an application of Itˆ’s formula yields o (2.19) dYt = rYt dt − Yt− dMt = r + 1 {t<τ } γ(t) Yt dt − Yt− dHt . 1 We deal here with an example of a defaultable asset that is subject to the total default, meaning that its price vanishes at and after default. 2.3.2 Price dynamics of a recovery claim (0, Z, τ ). Recall that our standard convention stipulates that the recovery Z is paid at the time of default. Hence, the price process Y of (0, Z, τ ) is given by the expression Yt = ert EQ (1 {T ≥τ } e−rτ Z(τ ) | Ht ). 1 We now have h(u) = 1 {u≤T } e−ru Z(u). Consequently, 1 e−rt Yt = Y0 + 0 t e−ru Z(u) − e−ru Yu− dMu . By applying Itˆ’s formula, we conclude that the dynamics under Q of an asset that delivers Z(τ ) at o time τ are dYt = = rYt− dt + (Z(t) − Yt− ) dMt r + 1 {t<τ } γ(t) Yt dt − 1 {t<τ } Z(t)γ(t) dt + (Z(t) − Yt− ) dHt . 1 1 M. Jeanblanc 37 Exercise 2.3.1 Prove that the price dynamics of a recovery paid at maturity is dSt = rSt dt + (e−r(T −t) Z(t) − St− )dMt . 2.3.3 Price dynamics of a defaultable claim (X, Z, τ ). By combining the formula above with (2.19), and using Remark 5.1.1 together with Girsanov’s theorem, we arrive at the following result. Proposition 2.3.1 The price process Y of a defaultable claim (X, Z, τ ) satisfies under Q dYt = rYt− dt + (Z(t) − Yt− ) dMt with the initial condition Y0 = EQ 1 {T <τ } e−rT X + 1 {T ≥τ } e−rτ Z(τ ) = e−(rT +Γ(T )) X + 1 1 Under the statistical probability P, the price process Y satisfies dYt = rYt− + 1 {t<τ } (Z(t) − Yt− )γ(t)ζ(t) dt + (Z(t) − Yt− ) dMt , 1 where the G-martingale M under P equals t T 0 Z(u)γ(u)e−Γ(u) du. Mt = Mt + 0 1 {u<τ } γ(u)ζ(u) du. 1 Remark 2.3.1 Proposition 2.3.1 can be extended to the case when the recovery is random, and is given in the feedback form as Z(t) = g(t, Yt− ) for some function g(t, y), which is Lipschitz continuous with respect to y. Assume, for instance, that the claim is subject to the fractional recovery of market value, so that Z(t) = δYt− for some constant δ. If, in addition, ζ and γ are constant, then we obtain (cf. (5.6)) dYt = Yt− (r + 1 {t<τ } (δ − 1)γζ) dt + (δ − 1) dMt . 1 Note that here the drift coefficient µt = r +1 {t<τ } (δ −1)γζ in dynamics of Y follows a G-predictable 1 process, but it is not F-predictable. However, the drift of the pre-default value Y is simply r. 2.3.4 Valuation of a Credit Default Swap A credit default swap (CDS) is a contract between two counterparties. B agrees to pay a default payment Z to A if a default of the obligor C occurs. If there is no default until the maturity of the default swap, B pays nothing. A pays a fee for the default protection. The fee can be either a fee paid till the maturity or till the default event. A can not cancelled the contract. He can at any time before the default transfer the contract to D: D will pay the fee and receive the default payment if any. As we shall see, it can happen that D will require an amount of cash to accept to receive the contract. Usually, the fee consists of Ci paid at time Ti (this is the fixed leg). However, here we shall consider a continuous payment. The default payment is called the default leg. A stylized credit default swap is formally introduced through the following definition. Definition 2.3.1 A credit default swap with a constant spread κ and recovery at default is a defaultable claim (0, A, Z, τ ), where Zt ≡ δ(t) and At = −κt for every t ∈ [0, T ]. An RCLL function δ : [0, T ] → I represents the protection payment and a constant κ ∈ I is termed the spread (or the R R premium) of a CDS. For simplicity, we assume that the interest rate r = 0, so that the price of a savings account Bt = 1 for every t. Our results can be easily extended to the case of a constant r. 38 Ex-dividend Price of a CDS Credit Risk, TUNIS 2005 Consider a CDS with the spread κ, which was initiated at time 0 (or indeed at any date prior to the current date t). Its market value at time t does not depend on the past otherwise than through the level of the spread κ. For the moment, we assume that κ is an arbitrary constant. Unless stated otherwise, we assume that the recovery (also known here as the protection payment) is received at the time of default and it is equal δ(t) if default occurs at time t. The ex-dividend price of a CDS maturing at T with spread κ is given by the formula St (κ) = EQ δ(τ )1 {t<τ ≤T } − 1 {t<τ } κ (τ ∧ T ) − t 1 1 Ht . (2.20) Note that in the following Lemma 2.3.1, we do not need to specify the inception date s of a CDS. We only assume that the maturity date T , the spread κ, and the protection payment δ are given. Lemma 2.3.1 The ex-dividend price at time t ∈ [s, T ] of a credit default swap with spread κ and recovery at default equals St (κ) = 1 {t<τ } 1 1 G(t) T T − t δ(u) dG(u) − κ t G(u) du . (2.21) Proof: We have, on the set {t < τ }, St (κ) = = Since t − T t δ(u) dG(u) −κ G(t) T − T t u dG(u) + T G(T ) −t G(t) T 1 G(t)) T − t δ(u) dG(u) − κ T G(T ) − tG(t) − t u dG(u) . T G(u) du = T G(T ) − tG(t) − t u dG(u), we conclude that (2.21) holds. The ex-dividend price of a CDS can also be represented as follows St (κ) = 1 {t<τ } St (κ), 1 ∀ t ∈ [0, T ], (2.22) where St (κ) stands for the ex-dividend pre-default price of a CDS. It is useful to note that formula (2.21) yields an explicit expression for St (κ) and that it is a continuous function provided that G is continuous. Market CDS Spreads Assume now that a CDS was initiated at some date s ≤ t and its initial price was equal to zero. Since a CDS with this property plays an important role, we introduce a formal definition. In Definition 2.3.2, it is implicitly assumed that a recovery function δ is given. Definition 2.3.2 A market CDS started at s is a CDS initiated at time s whose initial value is equal to zero. A T -maturity CDS market spread at time s is the level of the spread κ = κ(s, T ) that makes a T -maturity CDS started at s worthless at its inception. A CDS market spread at time s is thus determined by the equation Ss (κ(s, T )) = 0, where S is defined by (2.21). M. Jeanblanc 39 In our set-up, by virtue of Lemma 2.3.1, the T -maturity market spread κ(s, T ) is a solution to the equation T T δ(u) dG(u) + κ(s, T ) s s G(u) du = 0, and thus we have, for every s ∈ [0, T ], κ(s, T ) = − T s δ(u) dG(u) G(u) du T s . (2.23) Remarks 2.3.1 Let us comment briefly on a model calibration. Suppose that at time 0 the market gives the premium of a CDS for any maturity T . In this way, the market chooses the risk-neutral probability measure Q. Specifically, if κ(0, T ) is the T -maturity market CDS spread for a given recovery function δ then we have κ(0, T ) = − T 0 δ(u) dG(u) G(u) du T 0 . Hence, if credit default swaps with the same recovery function δ and varying maturities are traded at time 0, it is possible to find the implied risk-neutral c.d.f. F (and thus the default intensity γ under Q) from the term structure of CDS spreads κ(0, T ) by solving an ordinary differential equation. Standing assumptions. We fix the maturity date T , and we write briefly κ(s) instead of κ(s, T ). In addition, we assume that all credit default swaps have a common recovery function δ. Note that the ex-dividend pre-default value at time t ∈ [0, T ] of a CDS with any fixed spread κ can be related to the market spread κ(t). We have the following result, in which the quantity ν(t, s) = κ(t) − κ(s) represents the calendar CDS market spread (for a given maturity T ). Proposition 2.3.2 The ex-dividend price of a market CDS started at s with recovery δ at default and maturity T equals, for every t ∈ [s, T ], St (κ(s)) = 1 {t<τ } (κ(t) − κ(s)) 1 or more explicitly, St (κ(s)) = 1 {t<τ } 1 T t T t G(u) du = 1 {t<τ } ν(t, s) 1 G(t) T t G(u) du , G(t) (2.24) G(u) du G(t) T s δ(u) dG(u) G(u) du T s − T t δ(u) dG(u) G(u) du T t . (2.25) Proof: To establish equality (2.25), it suffices to observe that St (κ(s)) = St (κ(s)) − St (κ(t)), and to use (2.21) and (2.23). Remark 2.3.2 A representation of the value of a swap in terms of market swap rates is well known to hold for default-free interest rate swaps. It is especially useful if the calendar spread follows a stochastic process; in particular, it leads to the Black swaption formula within the framework of Jamshidian’s [55] model of co-terminal forward swap rates. Case of a Constant Default Intensity Assume that δ(t) = δ is independent of t, and F (t) = 1 − e−γt for a constant default intensity γ > 0 under Q. In this case, the valuation formulae for a CDS can be further simplified. In view of Lemma 2.3.1, the ex-dividend price of a CDS with spread κ equals, for every t ∈ [0, T ], St (κ) = 1 {t<τ } (δγ − κ)γ −1 1 − e−γ(T −t) . 1 40 Credit Risk, TUNIS 2005 The last formula (or the general formula (2.23)) yields κ(s) = δγ for every s < T , so that the market spread κ(s) is independent of s. As a consequence, the ex-dividend price of a market CDS started at s equals zero not only at the inception date s, but indeed at any time t ∈ [s, T ], both prior to and after default). Hence, this process follows a trivial martingale under Q. As we shall see in what follows, this martingale property the ex-dividend price of a market CDS is an exception, rather than a rule. 2.3.5 Price Dynamics of a CDS In what follows, we assume that t G(t) = Q(τ > t) = exp − 0 γ(u) du where the default intensity γ(t) under Q is deterministic. We first focus on the dynamics of the ex-dividend price of a CDS with spread κ started at some date s < T . Lemma 2.3.2 The dynamics of the ex-dividend price St (κ) on [s, T ] are dSt (κ) = −St− (κ) dMt + (1 − Ht )(κ − δ(t)γ(t)) dt, where the H-martingale M under Q is given by the formula t (2.26) M t = Ht − 0 (1 − Hu )γ(u) du, ∀ t ∈ I +. R (2.27) ¯ Hence, the process St (κ), t ∈ [s, T ], given by the expression ¯ St (κ) = St (κ) + s t t δ(u) dHu − κ s (1 − Hu ) du (2.28) is a Q-martingale for t ∈ [s, T ]. Proof: It suffices to recall that St (κ) = 1 {t<τ } St (κ) = (1 − Ht )St (κ) 1 so that dSt (κ) = (1 − Ht ) dSt (κ) − St− (κ) dHt . Using formula (2.21), we find easily that we have dSt (κ) = γ(t)St (κ) dt + (κ(s) − δ(t)γ(t)) dt. In view of (2.27), the proof of (2.26) is complete. To prove the second statement, it suffices to observe that the process N given by t t Nt = St (κ) − s (1 − Hu )(κ − δ(u)γ(u)) du = − s Su− (κ) dMu is an H-martingale under Q. But for every t ∈ [s, T ] ¯ St (κ) = Nt + s t δ(u) Mu , ¯ ¯ so that S(κ) also follows an H-martingale under Q. Note that the process S(κ) given by (2.28) ¯ represents the cum-dividend price of a CDS, so that the martingale property S(κ) is expected. M. Jeanblanc 41 Equality (2.26) emphasizes the fact that a single cash flow of δ(τ ) occurring at time τ can be formally treated as a dividend stream at the rate δ(t)γ(t) paid continuously prior to default. It is clear that we also have dSt (κ) = −St− (κ) dMt + (1 − Ht )(κ − δ(t)γ(t)) dt. (2.29) It can be useful to reformulate the dynamics of a market CDS in terms of market observables, such as CDS spreads. Corollary 2.3.1 The dynamics of the ex-dividend price St (κ(s)) on [s, T ] are also given as dSt (κ(s)) = −St− (κ(s)) dMt + (1 − Ht ) T t G(u) du dt ν(t, s) − ν(t, s) dt . G(t) (2.30) Proof: In the present set-up, for any fixed s, the calendar spread ν(t, s), t ∈ [s, T ] is a continuous function of bounded variation. In view of (2.26), it suffices to check that T t G(u) du dt ν(t, s) − ν(t, s) dt = (κ(s) − δ(t)γ(t)) dt, G(t) (2.31) where dt ν(t, s) = dt (κ(t) − κ(s)) = dκ(t). Equality (2.31) follows by elementary computations. Trading Strategies with a CDS We shall show that in the present set-up, in order to replicate an arbitrary contingent claim Y settling at time T and satisfying the usual integrability condition, it suffices to deal with two traded assets: a CDS with maturity U ≥ T and a constant savings account B = 1. Since one can always work with discounted values, the last assumption is not restrictive. A strategy φt = (φ0 , φ1 ), t ∈ [0, T ], is self-financing if the wealth process U (φ), defined as t t Ut (φ) = φ0 St (κ) + φ1 , t t satisfies dUt (φ) = φ0 dSt (κ) + φ0 dDt , t t (2.33) where S(κ) is the ex-dividend price of a CDS with the dividend stream D. As usual, we say that a strategy φ replicates a contingent claim Y if UT (φ) = Y . On the set {τ ≤ t ≤ T } the ex-dividend price S(κ) equals zero, and thus the total wealth is necessarily invested in B, so that it is constant. This means that φ replicates Y if and only if Uτ ∧T (φ) = Y . Lemma 2.3.3 For any self-financing strategy φ we have, on the set {τ ≤ T }, ∆τ U (φ) := Uτ (φ) − Uτ − (φ) = φ0 (δ(τ ) − Sτ (κ)). τ (2.34) (2.32) Proof: In general, the process φ0 is G-predictable. In our model, φ0 is assumed to be an RCLL function. The jump of the wealth process U (φ) at time τ equals, on the set {τ ≤ T }, ∆τ U (φ) = φ0 ∆τ S + φ1 ∆τ D τ τ where ∆τ S(κ) = Sτ (κ) − Sτ − (κ) = −Sτ (κ) (recall that the ex-dividend price S(κ) drops to zero at default time) and manifestly ∆τ D = δ(τ ). 42 Credit Risk, TUNIS 2005 2.3.6 Hedging of a Contingent Claim in the CDS Market An HT -measurable random variable can be represented as follows Y = 1 {T ≥τ } h(τ ) + 1 {T <τ } c(T ), 1 1 (2.35) where h : [0, T ] → I is a Borel function, and c(T ) is a constant. For concreteness, we shall deal R with claims Y such that h is an RCLL function, but this restriction is not essential. We first recall a suitable version of the predictable representation theorem. Subsequently, we derive closed-form solution for the replicating strategy for a claim Y given by (2.35) and settling at time T . As tradeable assets, we shall use a CDS started at time 0 and maturing at T , and a savings account. Replication of a Defaultable Claim Assume now that a random variable Y given (2.35) represents a contingent claim settling at T . Formally, we deal with a defaultable claim of the form (X, 0, Z, τ ), where X = c(T ) and Zt = z(t). To deal with such a claim, we shall apply Proposition 2.2.4 to the function h, where h(t) = z(t) for t < T and h(t) = c(T ) for t ≥ T (recall that Q(τ = T ) = 0). In this case, we obtain g(t) = 1 G(t) T − t z(u) dG(u) + c(T )G(T ) , (2.36) and thus for the process MtY = EQ (Y | Ht ), t ∈ [0, T ], we have MtY = EQ (Y ) + (z(u) − g(u)) dMu ]0,t] (2.37) with g given by (2.36). Recall that S(κ) is the pre-default ex-dividend price process of a CDS with spread κ and maturity T . We know that S(κ) is a continuous function of t if G is continuous. Proposition 2.3.3 Assume that the inequality St (κ) = δ(t) holds for every t ∈ [0, T ]. Let φ0 be an RCLL function given by the formula φ0 = t z(t) − g(t) δ(t) − St (κ) , (2.38) and let φ1 = Ut (φ) − φ0 St (κ), where the process U (φ) is given by (2.33) with the initial condition t t U0 (φ) = EQ (Y ), where Y is given by (2.35). Then the self-financing trading strategy φ = (φ0 , φ1 ) is admissible and it s a replicating strategy for a defaultable claim (X, 0, Z, τ ), where X = c(T ) and Zt = z(t). Proof: The idea of the proof is based on the observation that it is enough to concentrate on the formula for trading strategy prior to default. In view of Lemma 2.3.2, the dynamics of the price S(κ) are dSt (κ) = −St− (κ) dMt + (1 − Ht )(κ − δ(t)γ(t)) dt. and thus we have, on the set {τ > t} , dSt (κ) = dSt (κ) = γ(t)St (κ) + κ − δ(t)γ(t) dt. From Corollary ??, we know that the wealth U (φ) of any admissible self-financing strategy is a H-martingale under Q. Since under the present assumptions dBt = 0, for the wealth process Ut (φ) we obtain, on the set {τ > t}, dUt (φ) = φ0 (dSt (κ) − κ dt) = −φ0 γ(t) δ(t) − St (κ) dt. t t (2.39) M. Jeanblanc 43 For the martingale MtY = EQ (Y | Ht ) associated with Y , in view of (2.37) we obtain, on the set {τ > t}, dMtY = −γ(t)(z(t) − g(t)) dt. (2.40) We wish to find φ0 such that Ut (φ) = MtY for every t ∈ [0, T ]. To this end, we first focus on the equality 1 {t<τ } Ut (φ) = 1 {t<τ } MtY for pre-default values. A comparison of (2.39) with (2.40) yields 1 1 φ0 = t z(t) − g(t) δ(t) − St (κ) , ∀ t ∈ [0, T ]. (2.41) Y We thus see that if U0 (φ) = M0 then also 1 {t<τ } Ut (φ) = 1 {t<τ } MtY for every t ∈ [0, T ]. As usual, 1 1 the second component of a self-financing strategy φ is given by (2.32), that is, φ1 = Ut (φ) − φ0 St (κ), t t where U (φ) is given by (2.33) with the initial condition U0 (φ) = EQ (Y ). In particular, we have that φ1 = EQ (Y ) − φ0 S0 (κ). 0 0 To complete the proof, that is, to show that Ut (φ) = Mt for every t ∈ [0, T ], it suffices to compare the jumps of both processes at time τ (both martingales are stopped at τ ). It is clear from (2.37) that the jump of M equals ∆τ M = z(τ ) − g(τ ). Using (2.34), we get for the jump of the wealth process ∆τ U (φ) = φ0 (δ(τ ) − Sτ (κ)) = z(τ ) − g(τ ), τ and thus we conclude that Ut (φ) = MtY for every t ∈ [0, T ]. In particular, φ is admissible and UT (φ) = Uτ ∧T (φ) = z(τ ∧ T ) = Y , so that φ replicates a claim Y . Note that if κ = κ(0) then S0 (κ(0)) = 0, so that φ1 = U0 (φ) = EQ (Y ). 0 Let us now analyze the condition St (κ) = δ(t) for every t ∈ [0, T ]. It ensures, in particular, that the wealth process U (φ) has a non-zero jump at default time for any the self-financing trading strategy such that φ0 = 0 for every t ∈ [0, T ]. It appears that this condition is not restrictive, since t it is satisfied under mild assumptions. Indeed, if κ > 0 and δ is a non-increasing function then the inequality St (κ) < δ(t) is valid for every t ∈ [0, T ] (this follows easily from (2.20)). For instance, if γ(t) > 0 and the protection payment δ > 0 is constant then it is clear from (2.23) that the market spread κ(0) is strictly positive. Consequently, formula (2.20) implies that St (κ(0)) < δ for every t ∈ [0, T ], as was required. To summarize, when a tradeable asset is a market CDS with a constant δ > 0 and the default intensity is strictly positive then the inequality holds. Let us finally observe that if the default intensity vanishes on some set then we do not need to impose the inequality St (κ) = δ(t) on this set in order to equate (2.39) with (2.40), since the requested equality will hold anyway. The method of proof is based on the following observation. Lemma 2.3.4 Let M 1 and M 2 be arbitrary two H-martingales under Q. If for every t ∈ [0, T ] we have 1 {t<τ } Mt1 = 1 {t<τ } Mt2 then Mt1 = Mt2 for every t ∈ [0, T ]. 1 1 Proof: We have Mti = EQ (hi (τ ) | Ht ) for some functions hi : I + → I such that hi (τ ) is R R Q-integrable. Using the well known formula for the conditional expectation EQ (hi (τ ) | Ht ) = 1 {t≥τ } hi (τ ) − 1 {t<τ } 1 1 1 G(t) ∞ t hi (u) dG(u) = 1 {t≥τ } hi (τ ) + 1 {t<τ } gi (t), 1 1 and the assumption that 1 {t<τ } Mt1 = 1 {t<τ } Mt2 , we obtain the equality g1 (t) = g2 (t) for every 1 1 t ∈ [0, T ] (recall that Q(τ > t) > 0 for every t ∈ [0, T ]). Therefore, we have ∞ ∞ h1 (u) dG(u) = t t h2 (u) dG(u), ∀ t ∈ [0, T ]. 44 Credit Risk, TUNIS 2005 This immediately implies that h1 (t) = h2 (t) on [0, T ], almost everywhere with respect to the distribution of τ , and thus we have h1 (τ ) = h2 (τ ), Q-a.s. Consequently, Mt1 = Mt2 for every t ∈ [0, T ]. In our case, Lemma 2.3.4 can be applied to the following H-martingales under Q: M 1 = U (φ) is the wealth process of an admissible self-financing strategy φ and M 2 = M is the conjectured price of a claim Y , as given by the risk-neutral valuation formula. Let us note that the method presented above can be extended to replicate a contingent claim t defaultable claim (X, A, Z, τ ), where X = c(T ), At = 0 a(u) du and Zt = h(t) for some RCLL functions a and h. In this case, it is natural to expect that the cum-dividend price process πt associated with a defaultable claim (X, A, Z, τ ), is given by the formula, for every t ∈ [0, T ], τ πt = Mt + 1 {t≤τ } 1 0 a(u) du + 1 {t<τ } 1 1 G(t) T a(u)G(u) du, t (2.42) where Mt = EQ (Y | Ht ), where Y is given by (2.35). Hence, the pre-default dynamics of this process are dπt = dMt + γ(t)a(t) dt = −γ(t) h(t) − g(t) − a(t) dt, where we set a(t) = (G(t))−1 t a(u)G(u) du. Note that a(t) represents the pre-default value of the future promised dividends associated with A. Therefore, arguing as in the proof of Proposition 2.3.3, we find the following formula for a replicating strategy φ h(t) − g(t) − a(t) φ0 = , ∀ t ∈ [0, T ]. (2.43) t δ(t) − St (κ) It is easy to see that the jump condition at time τ , mentioned in the second part of the proof of Proposition 2.3.3, is satisfied in this case as well. Remark 2.3.3 Of course, if we take as (X, A, Z, τ ) a CDS with spread κ and recovery function δ, then we will get h(t) = δ(t) and g(t) + a(t) = St (κ), so that clearly φ0 = 1 for every t ∈ [0, T ]. t The following immediate corollary to Proposition 2.3.3 is worth stating (let us stress once again that the assumption that a claim is represented by an RCLL function, as opposed to a Borel measurable function, is not essential). Corollary 2.3.2 Assume that St (κ) = δ(t) for every t ∈ [0, T ]. Then the market is complete, in t the sense, that any defaultable claim (X, A, Z, τ ), where X = c(T ), At = 0 a(u) du and Zt = h(t) for some constant c(T ) and RCLL functions a and h, is attainable through continuous trading in a bond and a CDS. The arbitrage price πt of a defaultable claim satisfies, for every t ∈ [0, T ], πt = Ut (φ) = Mt = π0 + ]0,t] T (h(u) − πt− ) dMu , where T π0 = EQ (Y ) + 0 a(t)G(t) dt, and πt = g(t) + a(t) + At is its pre-default price, so that we have, for every t ∈ [0, T ] πt = 1 {t<τ } (g(t) + a(t) + At ) + 1 {t≥τ } (h(τ ) + Aτ ) = 1 {t<τ } πt− + 1 {t≥τ } πτ . 1 1 1 1 M. Jeanblanc 45 Case of a Constant Default Intensity As a partial check of the calculations above, we shall consider once again the case of constant default intensity and constant protection payment. In this case, κ(0) = δγ and St (κ(0)) = 0 for every t ∈ [0, T ], so that dUt (φ) = −φ0 δγ dt = −φ0 κ(0) dt. (2.44) t t Furthermore, for any RCLL function h, formula (2.41) yields φ0 = δ −1 h(t) + eγt t T t h(u) d e−γu − c(T )e−γT . (2.45) Assume, for instance, that h(t) = δ for t ∈ [0, T [ and c(T ) = 0. Then (2.45) gives φ0 = e−γ(T −t) . t Since S0 (κ(0)) = 0, we have φ1 = π0 (Y ) = U0 (φ) = δ(1 − e−γT ). In view of (2.44), the gains/losses 0 from positions in market CDSs over the time interval [0, t] equal, on the set {τ > t}, t Ut (φ) − U0 (φ) = −δγ 0 φ0 du = −δγ u t 0 e−γ(T −u) du = −δe−γT eγt − 1 < 0. Suppose that default occurs at some date t ∈ [0, T ]. Then the protection payments is collected, and the wealth at time t becomes Ut (φ) = Ut− (φ) + φ0 δ = δ(1 − e−γT ) − δe−γT eγt − 1 + δe−γ(T −t) = δ. t The last equality shows that the strategy is indeed replicating on the set {τ ≤ T }. On the set {τ > T }, the wealth at time T equals UT (φ) = δ(1 − e−γT ) − δe−γT eγT − 1 = 0. Since St (κ(0)) = 0 for every t ∈ [0, T ], we have that φ1 = Ut (φ) for every t ∈ [0, T ]. t Short Sale of a CDS As usual, we assume that the maturity T of a CDS is fixed and we consider the situation where the default has not yet occurred. 1. Long position. We say that an agent has a long position at time t in a CDS if he owns at time t a CDS contract that had been created (initiated) at time s0 by some two parties and was sold to the agent (by means of assignment for example) at time s. If s0 = s then the agent is an original counter-party to the contract, that is the agent owns the contract from initiation. If an agent owns a CDS contract, the agent is entitled to receive the protection payment for which the agent pays the premium. The long position in a contract may be liquidated at any time s < t < T by means of assignment or offsetting. 2. Short position. We stress that the short position, namely, selling a CDS contract to a dealer, can only be created for a newly initiated contract. It is not possible to sell to a dealer at time t a CDS contract initiated at time s0 < t. 3. Offsetting a long position. If an agent has purchased at time s0 ≤ s < T a CDS contract initiated at s0 , he can offset his long position by creating a short position at time t. A new contract is initiated at time t, with the initial price St (κ(s0 )), possibly with a new dealer. This short position offsets the long position outstanding, so that the agent effectively has a zero position in the contract at time t and thereafter. 4. Market constraints. The above taxonomy of positions may have some bearing on portfolios involving short positions in CDS contracts. It should be stressed that not all trades involving a CDS are feasible in practice. Let us consider the CDS contract initiated at time t0 and maturing at time 46 Credit Risk, TUNIS 2005 T . Recall that the ex-dividend price of this contract for any t ∈ [t0 , τ ∧ T [ is St (κ(t0 )). This is the theoretical price at which the contract should trade so to avoid arbitrage. This price also provides substance for the P&L analysis as it really marks-to-market positions in the CDS contract. Let us denote the time-t position in the CDS contract of an agent as φ0 , where t ∈ [t0 , τ ∧ T ]. t The strategy is subject to the following constraints: φ0 ≥ 0 t and φ0 ≥ φ00 t t if φ00 ≥ 0 t if φ00 ≤ 0. t It is clear that both restrictions are related to short sale of a CDS. The next simple result shows that under some assumptions a replicating strategy for a claim Y does not require a short sale of a CDS. Corollary 2.3.3 Assume that St (κ) < δ(t) for every t ∈ [0, T ]. Let h be a non-increasing function and let c(T ) ≤ h(T ). Then φ0 ≥ 0 for every t ∈ [0, T ]. t Proof: It is enough to observe that if h be a non-increasing function and c(T ) ≤ h(T ) then it follows easily from the first equality in (??) that for the function g given by (2.36) we have that h(t) ≥ g(t) for every t ∈ [0, T ]. In view of (2.38), this shows that φ0 ≥ 0 for every t ∈ [0, T ]. t 2.4 Successive default times The previous results can easily be generalized to the case of successive default times. We assume in this section that r = 0. 2.4.1 Two times Let us first study the case with two random times τ1 , τ2 . We denote by T1 = inf(τ1 , τ2 ) and i T2 = sup(τ1 , τ2 ), and we assume, for simplicity, that P(τ1 = τ2 ) = 0. We denote by (Ht , t ≥ 0) the 1 2 default process associated with Ti , (i = 1, 2), and by Ht = Ht + Ht the process associated with two defaults. As before, H is the filtration generated by the process H. The σ-algebra Ht is equal to σ(T1 ∧ t) ∨ σ(T2 ∧ t). A Ht -measurable random variable is equal to - a constant on the set t < T1 , - a σ(T1 )-measurable random variable on the set T1 ≤ t < T2 , - a σ(T2 )-measurable random variable on the set T2 ≤ t. Payment at maturity Suppose that a payment of 1 monetary unit is done at maturity if no default has occured, δ1 if one and only one default has occured and δ2 in the remaining case, where 0 ≤ δ2 < δ1 < 1. Therefore, to obtain the value of this claim we have to compute E(1 T s|Ft ) = exp − Λs . Proof: The proof follows from the equality {τ > s} = {Λs < Θ}. From the independence assumption and the Ft -measurability of Λs for s ≤ t, we obtain P(τ > s|Ft ) = P Λs < Θ Ft = exp − Λs . In particular, we have P(τ ≤ t|Ft ) = P(τ ≤ t|F∞ ), (3.1) and, for t ≥ s, P(τ > s|Ft ) = P(τ > s|Fs ). Let us notice that the process Ft = P(τ ≤ t|Ft ) is here an increasing process. 49 50 Remark 3.2.1 If the process λ is not non-negative, we get, for s < t P(τ > s|Ft ) = exp(− sup Λu ) . u≤s Credit Risk, TUNIS 2005 3.3 Choice of filtration We write as before Ht = 1 {τ ≤t} and Ht = σ(Hs : s ≤ t). We introduce the filtration Gt = Ft ∨ Ht , 1 that is, the enlarged filtration generated by the underlying filtration F and the process H. (We denote by F the original Filtration and by G the enlarGed one.) We shall frequently write G = F ∨ H. It is easy to describe the events which belong to the σ-field Gt on the set {τ > t}. Indeed, if Gt ∈ Gt , then Gt ∩ {τ > t} = Bt ∩ {τ > t} for some event Bt ∈ Ft . Therefore any Gt -measurable random variable Yt satisfies 1 {τ >t} Yt = 1 {τ >t} yt , where yt is an 1 1 Ft -measurable random variable. 3.4 Key lemma Proposition 3.4.1 Let Y be an integrable r.v. Then, 1 {τ >t} E(Y |Gt ) = 1 {τ >t} 1 1 E(Y 1 {τ >t} |Ft ) 1 = 1 {τ >t} eΛt E(Y 1 {τ >t} |Ft ). 1 1 E(1 {τ >t} |Ft ) 1 Proof: From the remarks on the Gt -measurability, if Yt = E(Y |Gt ), then there exists an Ft measurable r.v. yt such that 1 {τ >t} E(Y |Gt ) = 1 {τ >t} yt 1 1 and taking conditional expectation w.r.t. Ft of both members, we deduce yt = Corollary 3.4.1 If X is an integrable FT -measurable random variable E(X1 {T <τ } |Gt ) = 1 {τ >t} eΛt E(Xe−ΛT |Ft ) . 1 1 (3.2) E(Y 1 {τ >t} |Ft ) 1 . E(1 {τ >t} |Ft ) 1 Proof: Let X be an FT -measurable r.v. From prop. 3.4.1, E(X1 {τ >T } |Gt ) is equal to 0 on the 1 Gt -measurable set τ < t, whereas E(X1 {τ >T } |Ft ) = E(X1 {τ >T } |FT |Ft ) = E(XeΛT |Ft ). 1 1 Comments 3.4.1 This corollary admits an interesting interpretation. If X1 {T <τ } is some default1 able payoff, its value is the value of the default free payoff X when the interest rate is higher that the spot rate and the difference, i.e., λ can be interpreted as a spread. However, we emphasize that we are not dealing with a risk neutral probability. If the market is assumed to be complete, that means in particular that a defaultable zero-coupon is traded. Then, the intensity has to be evaluated under the risk-neutral probability given by the market. Definition 3.4.1 The process λ is called the intensity of τ . We now compute the expectation of a τ -time value of a predictable process. M. Jeanblanc 51 Lemma 3.4.1 (i) If h is an F-predictable (bounded) process then ∞ E(hτ |Ft ) = E 0 hu λu exp − Λu du Ft and E(hτ |Gt ) = E t ∞ hu λu exp Λt − Λu du Ft 1 {τ >t} + hτ 1 {τ ≤t} . 1 1 ∞ (3.3) In particular E(hτ ) = E 0 hu λu exp − Λu ) du (ii) The process (Ht − t∧τ 0 λs ds, t ≥ 0) is a G-martingale. Proof: Let ht = 1 ]v,w] (t)Bv where Bv ∈ Fv be an elementary predictable process. Then, from 1 Cor. 3.4.1 E(hτ |Ft ) = E 1 ]v,w] (τ )Bv Ft = E E(1 ]v,w] (τ )Bv |F∞ ) Ft 1 1 = It follows that w E Bv P(v < τ ≤ w|F∞ ) Ft = E Bv e−Λv − e−Λw |Ft E(hτ |Ft ) = E Bv v λu e−Λu du Ft = E 0 ∞ hu λu e−Λu du Ft and the result is derived from the monotone class theorem. The martingale property (ii) follows from integration by parts formula. Indeed, let t < s. Then, on the one hand from Cor 3.4.1 E(Hs − Ht |Gt ) = = On the other hand, from part (i) s∧τ ∞ P(t < τ ≤ s|Ft ) P(t < τ |Ft ) 1 {t<τ } E(1 − exp(Λs − Λt )|Ft ) 1 P(t < τ ≤ s|Gt ) = 1 {t<τ } 1 E t∧τ λu du Gt = E Λs∧τ − Λt∧τ |Gt = 1 {t<τ } E 1 hu λu e−(Λu −Λt ) du Ft t where hu = Λ(s ∧ u) − Λ(t ∧ u). Consequently, ∞ t hu λu e−(Λu −Λt ) du s = t (Λu − Λs )λu e−(Λu −Λt ) du + (Λt − Λs ) −(Λs −Λt ) s s ∞ λu e−(Λu −Λt ) du = −(Λs − Λt )e + t λ(u)e −(Λu −Λt ) du + (Λs − Λt )e−(Λs −Λt ) = 1 − e−(Λs −Λt ) . This ends the proof. 3.5 Conditional Expectation of F∞ -Measurable Random Variables E(X|Gt ) = E(X|Ft ) . (3.4) Lemma 3.5.1 Let X be an F∞ -measurable r.v.. Then 52 Credit Risk, TUNIS 2005 Proof: Let X be an F∞ -measurable r.v. To prove that E(X|Gt ) = E(X|Ft ), it suffices to check that E(Bt h(τ ∧ t)X) = E(Bt h(τ ∧ t)E(X|Ft )) for any Bt ∈ Ft and any h = 1 [0,a] . For t ≤ a, the equality is obvious. For t > a, we have from (3.1) 1 E(Bt 1 {τ ≤a} E(X|Ft )) = 1 = as expected. Remark 3.5.1 Let us remark that (3.4) implies that every F-square integrable martingale is a G-martingale. However, equality (3.4) does not apply to any G-measurable random variable; in particular P(τ ≤ t|Gt ) = 1 {τ ≤t} is not equal to Ft = P(τ ≤ t|Ft ). 1 E(Bt E(X|Ft )E(1 {τ ≤a} |F∞ )) = E(E(Bt X|Ft )E(1 {τ ≤a} |Ft )) 1 1 E(XBt E(1 {τ ≤a} |Ft )) = E(Bt X1 {τ ≤a} ) 1 1 3.6 Defaultable Zero-Coupon Bond T From Cor. 3.4.1, for t < T E(1 {T <τ } |Gt ) = 1 {τ >t} E exp − 1 1 λs ds t Ft . Let B(t, T ) be the price at time t of a default-free bond paying 1 at maturity t satisfies T B(t, T ) = EQ exp − t rs ds Ft . The market price Bd (t, T ) of a defaultable zero-coupon bond with maturity T is T Bd (t, T ) = = EQ 1 {T <τ } exp − 1 1 {τ >t} EQ exp − 1 rs ds t T t Gt Ft . [rs + λQ ] ds s If a recovery δ(τ ) is paid at time T , then   − Bd (t, T ) = B(t, T ) − E e  t T s  λQ du  u   rs ds t T ds(1 − δs )λQ e s − t Therefore, given the price of a DZC, we can deduce the risk neutral intensity. The t-time value of a corporate bond, which pays δ at time T in case of default and 1 otherwise, is given by T EQ e− t rs ds (δ1 {τ ≤T } + 1 {τ >T } ) Ft . 1 1 The last quantity is equal to T δB(t, T ) + 1 {τ >t} (1 − δ)E exp − 1 t [rs + λQ ] ds s Ft . In the case where the compensation is time dependent,  T  − ρ(t, T ) = B(t, T ) − E e  rs ds t t T − s t  λu du    ds(1 − δs )λs e M. Jeanblanc 53 3.7 Extension τ = inf{t : Λt ≥ Σ} In Wong [88], the time of default is given as where Σ a non-negative r.v. independent of F∞ . This model reduces to the previous one: if Φ is the cumulative function of Σ, the r.v. Φ(Σ) has a uniform distribution and τ = inf{t : Φ(Λt ) ≥ Φ(Σ)} = inf{t : Ψ−1 [Φ(Λt )] ≥ Θ} where Ψ is the cumulative function of the exponential law. Then, Ft = P(τ ≤ t|Ft ) = P(Λt ≥ Σ|Ft ) = 1 − exp −Ψ−1 (Φ(Λt )) . 3.8 Term Structure Models Some authors choose to model the intensity. A substantial literature proposes to model both the default free term structure and the term structure representing the relative prices of different maturities of default-risky debt, using an extension of the method developed by Heath-Jarrow-Morton. Major papers in this area include Jarrow and Turnbull [57], Sch¨nbucher [84, 85], Hubner [53, 52], o and Bielecki and Rutkowski [4]. Other authors choice to model directly credit spreads Duffie and Singleton [36], Douady and Jeanblanc [32]. 3.8.1 Duffee’s model Duffee [33] assumes that the value of a default free bond is T EQ exp − t rs ds Ft and that the defaultable bond is priced as T EQ where exp − t (rs + γs )ds Ft rt = s1,t + s2,t , ht = β + β1 s1,t + β2 s2,t + s3,t and √ dsi,t = κi (θi − si,t )dt + σi si,t dWi (t) 3.8.2 Jarrow and Turnbull’s model Jarrow and Turnbull consider a situation where the interest rate follows a Vasicek’s dynamics and where the intensity is a linear function of the interest rate and a factor Z, modeled as a Brownian motion. drt = κ(r∞ − rt )dt + σdWt and γt = a0 (t) + a1 (t)rt + a2 (t)Zt . The problem is that γ is not a non-negative value. nevertheless, the corporate bond follows 1 D(t, T ) = δB(t, T ) + (1 − δ) exp(−µ + v) 2 where µ and v are the mean and variance of RT + ΓT . 54 Credit Risk, TUNIS 2005 3.8.3 Vacicek Model In Schonbucher, the dynamics of interest rate and of the intensity are drt dλt = = (k(t) − art )dt + σ(t)dWt (k(t) − aλ(t))dt + σ(t)dBt where W and B are two Brownian motion with correlation ρ. Proposition 3.8.1 The price of a default free zero-coupon with maturity T is B(t, T ) = exp(A(t, T ; a, k, σ) − κ(t, T ; a)rt ) where A(t, T ; a, k, σ) = 1 2 T t T σ 2 (u)κ(t, u; a)2 du − t κ(t, u; a)k(u)du 1 and κ(t, u : a) = a (1 − e−a(T −t) ). The price of a defaultable zero-coupon with maturity T with zero recovery is D(t, T ) = B(t, T )B(t, T ) = exp(A(t, T ; h, σ) − κ(t, T ; a)λt )) with h(t) = k(t) − ρσ(t)σ(t)κ(t, T ; a) Proof: See Appendix for (i). For (ii) write T D(t, T ) = B(t, T )EQT (exp − λs ds) t where QT is the T -forward probability measure. The dynamics of λ under QT are dλt = (h(t) − aλt )dt + σt dBt 3.8.4 The CIR model 3.9 Copula Chapter 4 Hazard process Approach: Reference filtration 4.1 4.1.1 General case The model In reduced form approach, we shall deal with two kinds of information : the information from the asset’s prices, denoted as (Ft , t ≥ 0) and the information from the default time, i.e. the knowledge of the time were the default occured in the past, it the default has appeared. More precisely, this information is modeled by the filtration Ht generated by the default process H. At the intuitive level, F is generated by prices of some assets, or by other economic factors (e.g., interest rates). This filtration can also be a subfiltration of the prices. The case where F is the trivial filtration is exactly what we have studied in the toy example. Though in typical examples F is chosen to be the Brownian filtration, most theoretical results do not rely on such a specification of the filtration F. We denote by Gt = Ft ∨ Ht . Special attention is paid here to the hypothesis (H), which postulates the invariance of the martingale property with respect to the enlargement of F by the observations of a default time. We establish a representation theorem, in order to understand the meaning of complete market in a defaultable world and we deduce the hedging strategies for credit derivatives. The main part of this section can be found in the surveys of Jeanblanc and Rutkowski [59, 60]. For a complete study of credit risk, see the book of Bielecki and Rutkowski [4]. The reader can find other interesting information on the web sites quoted at the end of the bibliography of this document. 4.1.2 Key lemma It is straightforward to establish that any Gt -random variable is equal, on the set {τ > t}, to an Ft -measurable random variable. We denote by Ft = P(τ ≤ t|Ft ) the conditional law of τ given the information Ft . Lemma 4.1.1 Let X be an FT -measurable integrable r.v. Then, E(X1 T <τ |Gt ) = 1 {τ >t} 1 1 where Γt = − ln(1 − Ft ) 55 E(X1 {τ >T } |Ft ) 1 = 1 {τ >t} eΓt E(Xe−ΓT |Ft ). 1 E(1 {τ >t} |Ft ) 1 (4.1) 56 Proof: The proof is exactly the same as Cor. 3.4.1. Indeed, 1 {τ >t} E(X|Gt ) = 1 {τ >t} xt 1 1 Credit Risk, TUNIS 2005 where xt is Ft -measurable, and taking conditional expectation w.r.t. Ft of both members, we deduce xt = E(X1 {τ >t} |Ft ) 1 = 1 {τ >t} eΓt E(Xe−ΓT |Ft ) . 1 E(1 {τ >t} |Ft ) 1 The main point is that here, the process Γ is not necessarily increasing. Lemma 4.1.2 Let h be an F-predictable process. Then, T E(hτ 1 τ t} E( 1 1 1 hu dFu |Ft ) t (4.2) We are not interested with G predictable processes, mainly because any G predictable process is equal, on {t ≤ τ } to an F-predictable process. Lemma 4.1.3 The submartingale Ft admits a decomposition as Z + A where Z is a martingale and A a predictable increasing process. In terms of A, T E(hτ 1 τ t} E( 1 1 1 hu dAu |Ft ) t As we shall see, this elementary result will allow us to compute the value of credit derivatives, as soon as some elementary defaultable asset is priced by the market. Comments 4.1.1 It can be useful to understand the meaning of the lemma in the case where, as in the structural model, the default time is an (Ft ) stopping time. Remark 4.1.1 We emphasize that, in the Cox process approach, the enlarged filtration G = F ∨ H is here the filtration which should be taken into account; the filtration generated by Ft and σ(Θ) is too large. In the latter filtration, in the case where F is a Brownian filtration, τ would be a predictable stopping time. 4.1.3 Martingales Proposition 4.1.1 The process Ft is a submartingale. Proof: We have to prove that ∀t > s, E(Ft |Fs ) ≥ Fs . From definition, and form the increasing property of the process H, E(Ft |Fs ) = ≥ E (E(Ht |Ft ) |Fs ) = E (Ht , |Fs ) E (Hs , |Fs ) = Fs This property imply, from the Doob-Meyer decomposition that Ft = Zt + At where Z is a F-martingale and A a F-predictable increasing process. Proposition 4.1.2 (i) The process Lt = (1 − Ht )eΓ(t) is a G -martingale. (ii) If X is a F-martingale, XL is a G -martingale. M. Jeanblanc 57 (iii) The process Mt = Ht − Γ(t ∧ τ ) is a G -martingale as soon as F (or Γ) is continuous. Proof: (i) From the key lemma, for t > s E(Lt |Gs ) = E(1 {τ >t} eΓt |Gs ) = 1 {τ >s} eΓs E(1 {τ >t} eΓt |Fs ) = 1 {τ >s} eΓs = Ls 1 1 1 1 since E(1 {τ >t} eΓt |Fs ) = E(E(1 {τ >t} |Ft )eΓt |Fs ) = 1. 1 1 (ii) From the key lemma, E(Lt Xt |Gs ) = E(1 {τ >t} Lt Xt |Gs ) 1 = 1 {τ >s} eΓs E(1 {τ >t} eΓt Xt |Fs ) 1 1 = 1 {τ >s} eΓs E(E(1 {τ >t} |Ft )eΓt Xt |Fs ) 1 1 = Ls Xs . (iii) From Itˆ’s Formula (H is a finite variation process, and Γ is continuous): o dLt = (1 − Ht )eΓt dΓt − eΓt dHt and the process Mt = Ht − Γ(t ∧ τ ) can be written Mt ≡ ]0,t] dHu − ]0,t] (1 − Hu )dΓu = − ]0,t] e−Γ(u) dLu and is a G-local martingale since L is G-martingale. 4.1.4 Interpretation of the intensity In this general setting, the process Γ is not with bounded variation. Hence, (iii) does not give the Doob-Meyer decomposition of H. Proposition 4.1.3 We assume for simplicity that F is continuous. The process t∧τ Mt = Ht − 0 dAu du 1 − Fu is a G-martingale. Assuming that A is absolutely continuous wrt the Lebesgue measure, we have proved the existence of a F-adapted process γ, called the intensity such that the process t∧τ t Ht − 0 γu du = Ht − 0 (1 − Hu )γu du is a G-martingale. Lemma 4.1.4 The process γ satisfies γt = lim 1 P(t < τ < t + h|Ft ) h→0 h P(t < τ |Ft ) Proof: The martingale property of M implies that t+h E(1 t<τ t} Y |Gt ) = 1 {τ >t} 1 1 and if Y is a FT -measurable variable, E(1 {τ >T } Y |Gt ) = 1 {τ >t} 1 1 1 Gt E GT Y |Ft . 1 Gt E 1 {τ >t} Y |Ft 1 From EP (1 {τ >T } Y |Gt ) = EP (1 {τ >T } Y |Gt |Gt ), we deduce 1 1 E(1 {τ >T } Y |Gt ) = 1 = E(GT Y |Ft ) Gt Gt 1 E(GT Y |Ft ) 1 {τ >t} E 1 {τ >t} 1 1 Ft Gt Gt E 1 {τ >t} 1 . Hence we have the equality, which holds before default: 1 {τ >t} E(GT Y |Ft ) = 1 {τ >t} E 1 1 1 {τ >t} 1 E(GT Y |Ft ) Ft Gt Remark in fact that last equality holds every time. Indeed, a simple computation shows E 1 {τ >t} E(GT Y |Ft )(Gt )−1 Ft 1 = = = since Y is FT -measurable. As a conclusion, ∀Y FT -measurable, E(GT Y |Ft ) = E 1 {τ >t} 1 E(GT Y |Ft ) Ft Gt E E(1 {τ >t} |Ft )(Gt )−1 E(GT Y |Ft ) Ft 1 E E(GT Y |Ft )| Ft = E GT Y |Ft E E GT |FT Y |Ft = E GT Y |Ft . 4.2 (H) Hypothesis We discuss now the hypothesis on the modeling of default time that we require in order to avoid arbitrages in the defaultable market. 4.2.1 Complete model case Proposition 4.2.1 Let S be a semi-martingale on (Ω, G, P) such that there exists a unique probaS bility Q, equivalent to P on FT , where Ft = Ft = σ(Ss , s ≤ t) such that (St = St Rt , 0 ≤ t ≤ T ) is S ˜ an F -martingale under the probability Q. We assume that there exists a probability Q, equivalent ˜ Then, ((H)) holds to P on GT such that (St , 0 ≤ t ≤ T ) is a G-martingale under the probability Q. ˜ under Q and the restriction of Q to FT is equal Q. Proof: We give a ”financial proof”. Under the hypothesis, any square integrable F − Q martingale can be thought as the discounted value of a contingent claim ξ ∈ FT . Since the same claim exists ˜ in the larger market, which is assumed to be arbitrage free, the claim process is also a G − Q I + R +| F t t t(h)84 6 6 f28 84 1 5 E [(+)0] 7 it n 3 7 6 f238 710 [( )] 6 71 f128 560 [( )]2 12 t t7()) 4 M. Jeanblanc Ff H TD 59 v measurab random martingale. From the uniqueness of price for hedgeable claims, for any contingent claim X ∈ FT ˜ and any G-e.m.m. Q, EQ (XRT |Ft ) = EQ (XRT |Gt ) . ˜ −1 In particular, EQ (Z) = EQ (Z) for any Z ∈ FT ( take t = 0 and X = ZRT ), hence the restriction of ˜ ˜ any e.m.m. Q to the σ-algebra FT equals Q. Moreover, since any square integrable F-Q-martingale ˜ can be written as EQ (X|Ft ) = EQ (X|Gt ), we get that any square integrable F-Q-martingale is a ˜ ˜ G-Q-martingale. Comments 4.2.1 In the literature, it is generally assumed that the defaultable market is complete and arbitrage free. If this assumption means that the set of contingent claims is the set of GT measurable random variable, then, in particular, any FT -measurable random variable is a tradeable contingent claim, and ST is a tradeable asset. 4.2.2 Definition and Properties of (H) Hypothesis We shall now examine the hypothesis (H) which reads: (H) Every F square-integrable martingale is a G square-integrable martingale. This hypothesis implies that the F-Brownian motion remains a Brownian motion in the enlarged filtration. It was studied by Br´maud and Yor [13] and Mazziotto and Szpirglas [75], and for financial e purpose by Kusuoka [66]. This can be written in any of the equivalent forms (see, e.g. Dellacherie and Meyer [29]) : Lemma 4.2.1 Assume that G = F ∨ H, where F is an arbitrary filtration and H is generated by the process Ht = 1 {τ ≤t} . Then the following conditions are equivalent to the hypothesis (H). 1 (i) For any t, h ∈ I + , we have R P(τ ≤ t | Ft ) = P(τ ≤ t | Ft+h ). (i ) For any t ∈ I + , we have R P(τ ≤ t | Ft ) = P(τ ≤ t | F∞ ). (4.4) (4.3) 60 Credit Risk, TUNIS 2005 a local martingale with respect to G, and thus a G-martingale, since L is bounded (recall that any bounded local martingale is a martingale). We conclude that Lt = EP (ξ | Gt ) and thus (iv) holds. Suppose now that (iv) holds. First, we note that the standard truncation argument shows that the boundedness of ξ in (iv) can be replaced by the assumption that ξ is P-integrable. Hence, any F-martingale L is an G-martingale, since L is clearly G-adapted and we have, for every t ≤ s, Lt = EP (Ls | Ft ) = EP (Ls | Gt ). Now, suppose that L is an F-local martingale so that there exists an increasing sequence of Fstopping times τn such that limn→∞ τn = ∞, for any n the stopped process Lτn follows a uniformly integrable F-martingale. Hence, Lτn is also a uniformly integrable G-martingale, and this means that L follows a G-local martingale. Remarks 4.2.1 (i) Equality (4.4) appears in several papers on default risk, typically without any reference to the (H) hypothesis. For example, in the Madan-Unal paper [74], the main theorem follows from the fact that (4.4) holds (See the proof of B9 in the appendix of their paper). This is also the case for Wong’s model [88]. (ii) If τ is F∞ -measurable, and if (4.4) holds, then τ is an F-stopping time. If τ is a F-stopping time, equality (4.3) holds. If F is the Brownian filtration, τ is predictable and Λ = H. (iii) Though condition (H) does not necessarily hold true, in general, it is satisfied when τ is constructed through a standard approach (See Cox processes). This hypothesis is quite natural under the historical probability, and is stable under some change of measure. However, Kusuoka provides an example where (H) holds under the historical probability and does not hold after a change of probability. This counter example is linked with dependency between default of different firms. (iv) Hypothesis (H) holds in particular if τ is independent from F∞ . See Greenfield thesis. [46]. We reduce our attention to the case where ∀t, P (τ ≤ t|Ft ) = P (τ ≤ t|F∞ ) . In that case F is an increasing process. Comments 4.2.2 See Elliott et al. [40] for more comments. The increasing property of F is equivalent to the fact that any F-martingale, stopped at time τ is a G martingale. Quite recently, M. Yor proved that this is equivalent to E(mτ ) = m0 for any bounded F martingale. Proposition 4.2.2 If X is a F-martingale, XL and [L, X] are G -martingales. Proof: Since [L, X] = LX − L− dX − X− dL, the process [L, X] is the sum of three G-martingales. 4.2.3 (H) hypothesis and shrinking filtration If (H) hypothesis holds between F and G, it is in general not true that it holds between F and G. It can hold in very simple cases like for example if (F, F) and (F, G) satisfy (H) hypothesis holds between F and G, and between F and F. Suppose that (H) hypothesis holds between F and G. The process F is increasing. Suppose that t it has an intensity, Ft = 0 fs ds. The process Ft = P(t ≥ τ |Ft ) = E(Ft |Ft ) is a sub martingale which is not increasing anymore, since (H) hypothesis is not supposed to hold between F and F. Doob Meyer theorem implies the existence of Z a F-martingale and A a non decreasing F-predictable process such that: Ft = Zt + At Next lemma allows to link f and A: M. Jeanblanc 61 t 0 Lemma 4.2.2 The compensator of F writes At = E(fs |Fs )ds. t 0 Proof: Let us prove that the process Mt = E(Ft |Ft ) − integrable and F-adapted. Moreover T E(fs |Fs )ds is a F-martingale. It is E(MT |Ft ) = = = = E E(FT |FT ) − 0 E(fs |Fs )ds Ft T E E(FT |FT ) Ft − E 0 t E(fs |Fs )ds Ft T E(FT |Ft ) − E 0 t E(fs |Fs )ds Ft T −E t E(fs |Fs )ds Ft E 0 fs ds Ft t +E t fs ds Ft T −E 0 E(fs |Fs )ds Ft T −E t T E(fs |Fs )ds Ft E(fs |Fs )ds Ft t = = = . Mt + E t T fs ds Ft −E T Mt + t T E fs | Ft ds − t T E E(fs |Fs ) Ft ds E(fs |Ft )ds = Mt . t Mt + t E fs | Ft ds − . 0 Hence F. − 0 E(fs |Fs )ds is a F-martingale and t Doob Meyer theorem implies At = 0 E(fs |Fs )ds. E(fs |Fs )ds is predictable. The uniqueness in Recall that since F is increasing continuous, Λt = Γt = − ln(1 − Ft ) and its intensity is λs = fs /Gs . Indeed, t t t dFs fs Λt = dΛs = = ds 0 0 1 − Fs 0 1 − Fs There is no particular raison that F be increasing F-predictable, and we shall work with the Ft martingale hazard process. As proved in last section Λt = 0 dAs /Gs and last proposition leads to t t dAs fs Λt = = ds Gs Gs 0 0 Where fs denotes E(fs |Fs ). The intensity λ of Λ is equal to λs = E(fs |Fs )/Gs , and not ”as we could think” to E(fs /Gs |Fs ). Note that even if (H) hypothesis holds between F and F, this proof can not be simplified since Ft is increasing but not F-predictable (there is no raison for Ft to have an intensity). This result can be directly proved thanks to Bremaud’s following result: Denoting λs = fs /Gs , t∧τ t∧τ Ht − 0 t∧τ λs ds is a G -martingale =⇒ Ht − 0 t 0 E(λs |Gs )ds is a G -martingale As 0 E(λs |Gs )ds = leads to 1 {s≤τ } E(λs |Gs )ds and 1 {s≤τ } E(λs |Gs ) = E(1 {s≤τ } λs |Gs ), first proposition 1 1 1 = 1 {s≤τ } 1 Gs E(1 {s≤τ } λs |Fs ) 1 E(1 {s≤τ } λs |Gs ) 1 62 = hence Ht − t∧τ 0 Credit Risk, TUNIS 2005 1 {s≤τ } 1 Gs E(Gs λs |Fs ) = 1 {s≤τ } 1 Gs E(fs |Fs ) E(fs |Fs )/Gs ds is a G-martingale, and we are done. 4.2.4 Change of a probability measure Kusuoka [66] shows, by means of a counter-example, that the hypothesis (H) is not invariant with respect to an equivalent change of the underlying probability measure, in general. It is worth noting that his counter-example is based on two filtrations, H1 and H2 , generated by the two random times τ 1 and τ 2 , and he chooses H1 to play the role of the reference filtration F. We shall argue that in the case where F is generated by a Brownian motion (or, more generally, by some martingale orthogonal to M under P), the above-mentioned invariance property is valid under mild technical assumptions. Preliminary lemma Let us first examine a general set-up in which G = F ∨ H, where F is an arbitrary filtration and H is generated by the default process H. We say that Q is locally equivalent to P if Q is equivalent to P on (Ω, Gt ) for every t ∈ I + . Then there exists the Radon-Nikod´m density process η such that R y dQ | Gt = ηt dP | Gt , ∀ t ∈ I +. R (4.5) Part (i) in the next lemma is well known (see Jamshidian [56]). We assume that the hypothesis (H) holds under P. Lemma 4.2.3 (i) Let Q be a probability measure equivalent to P on (Ω, Gt ) for every t ∈ I + , with R the associated Radon-Nikod´m density process η. If the density process η is F-adapted then we have y Q(τ ≤ t | Ft ) = P(τ ≤ t | Ft ) for every t ∈ I + . Hence, the hypothesis (H) is also valid under Q R and the F-intensities of τ under Q and under P coincide. (ii) Assume that Q is equivalent to P on (Ω, G) and dQ = η∞ dP, so that ηt = EP (η∞ | Gt ). Then the hypothesis (H) is valid under Q whenever we have, for every t ∈ I + , R EP (η∞ Ht | F∞ ) EP (ηt Ht | F∞ ) = . EP (η∞ | F∞ ) EP (ηt | F∞ ) (4.6) Proof: To prove (i), assume that the density process η is F-adapted. We have for each t ≤ s ∈ I + R EP (ηt 1 {τ ≤t} | Ft ) 1 Q(τ ≤ t | Ft ) = = P(τ ≤ t | Ft ) = P(τ ≤ t | Fs ) = Q(τ ≤ t | Fs ), EP (ηt | Ft ) where the last equality follows by another application of the Bayes formula. The assertion now follows from part (i) in Lemma 4.2.1. To prove part (ii), it suffices to establish the equality Ft := Q(τ ≤ t | Ft ) = Q(τ ≤ t | F∞ ), ∀ t ∈ I +. R (4.7) Note that since the random variables ηt 1 {τ ≤t} and ηt are P-integrable and Gt -measurable, using 1 the Bayes formula, part (v) in Lemma 4.2.1, and assumed equality (4.6), we obtain the following chain of equalities Q(τ ≤ t | Ft ) = EP (ηt 1 {τ ≤t} | F∞ ) 1 EP (ηt 1 {τ ≤t} | Ft ) 1 = EP (ηt | Ft ) EP (ηt | F∞ ) EP (η∞ 1 {τ ≤t} | F∞ ) 1 = = Q(τ ≤ t | F∞ ). EP (η∞ | F∞ ) M. Jeanblanc 63 We conclude that the hypothesis (H) holds under Q if and only if (4.6) is valid. Unfortunately, straightforward verification of condition (4.6) is rather cumbersome. For this reason, we shall provide alternative sufficient conditions for the preservation of the hypothesis (H) under a locally equivalent probability measure. Case of the Brownian filtration Let W be a Brownian motion under P with respect to its natural filtration F. Since we work under the hypothesis (H), the process W is also a G-martingale, where G = F ∨ H. Hence, W is a Brownian motion with respect to G under P. Our goal is to show that the hypothesis (H) is still valid under Q ∈ Q for a large class Q of (locally) equivalent probability measures on (Ω, G). Let Q be an arbitrary probability measure locally equivalent to P on (Ω, G). Kusuoka [66] (see also Section 5.2.2 in Bielecki and Rutkowski [4]) proved that, under the hypothesis (H), any G-martingale under P can be represented as the sum of stochastic integrals with respect to the Brownian motion W and the jump martingale M . In our set-up, Kusuoka’s representation theorem implies that there exist G-predictable processes θ and ζ > −1, such that the Radon-Nikod´m density η of Q with y respect to P satisfies the following SDE dηt = ηt− θt dWt + ζt dMt with the initial value η0 = 1. More explicitly, the process η equals · · (4.8) ηt = Et 0 θu dWu Et 0 · ζu dMu t = ηt ηt , 1 2 t 0 (1) (2) (4.9) where we write ηt and ηt (2) (1) = Et 0 · θu dWu = exp 0 t θu dWu − 2 θu du , (4.10) t∧τ = Et 0 ζu dMu = exp 0 ln(1 + ζu ) dHu − 0 ζu γu du . (4.11) Moreover, by virtue of a suitable version of Girsanov’s theorem, the following processes W and M are G-martingales under Q t t Wt = Wt − 0 θu du, Mt = Mt − 0 1 {u<τ } γu ζu du. 1 (4.12) Proposition 4.2.3 Assume that the hypothesis (H) holds under P. Let Q be a probability measure locally equivalent to P with the associated Radon-Nikod´m density process η given by formula (4.9) y . If the process θ is F-adapted then the hypothesis (H) is valid under Q and the F-intensity of τ under Q equals γt = (1 + ζt )γt , where ζ is the unique F-predictable process such that the equality ζt 1 {t≤τ } = ζt 1 {t≤τ } holds for every t ∈ I + . 1 1 R Proof: Let P be the probability measure locally equivalent to P on (Ω, G), given by · dP | Gt = Et ζu dMu 0 dP | Gt = ηt dP | Gt . (2) (4.13) We claim that the hypothesis (H) holds under P. From Girsanov’s theorem, the process W follows a Brownian motion under P with respect to both F and G. Moreover, from the predictable representation property of W under P, we deduce that any F-local martingale L under P can be written as a stochastic integral with respect to W . Specifically, there exists an F-predictable process ξ such that t Lt = L0 + 0 ξu dWu . 64 Credit Risk, TUNIS 2005 This shows that L is also a G-local martingale, and thus the hypothesis (H) holds under P. Since · dQ | Gt = Et θu dWu 0 d P | Gt , by virtue of part (i) in Lemma 4.2.3, the hypothesis (H) is valid under Q as well. The last claim in the statement of the lemma can be deduced from the fact that the hypothesis (H) holds under Q and, by Girsanov’s theorem, the process t t Mt = Mt − 0 1 {u<τ } γu ζu du = Ht − 1 0 1 {u<τ } (1 + ζu )γu du 1 is a Q-martingale. We claim that the equality P = P holds on the filtration F. Indeed, we have dP | Ft = ηt dP | Ft , (2) where we write ηt = EP (ηt | Ft ), and EP (ηt | Ft ) = EP Et (2) · ζu dMu 0 F∞ = 1, ∀ t ∈ I +, R (4.14) where the first equality follows from part (v) in Lemma 4.2.1. To establish the second equality in (4.14), we first note that since the process M is stopped at τ , we may assume, without loss of generality, that ζ = ζ where the process ζ is F-predictable (see Lemma ??). Moreover, in view of (??) the conditional cumulative distribution function of τ given F∞ has the form 1 − exp(−Γt (ω)). Hence, for arbitrarily selected sample paths of processes ζ and Γ, the claimed equality can be seen as a consequence of the martingale property of the Dol´ans e exponential. Formally, it can be proved by following elementary calculations, where the first equality is a consequence of (4.11)), · t∧τ EP Et 0 ζu dMu ∞ F∞ = EP 1 + 1 {t≥τ } ζτ exp − 1 t∧u ζu γu du 0 u 0 F∞ = EP 0 t 1 + 1 {t≥u} ζu exp − 1 u ζv γv dv γu e− γv dv du F∞ 0 = EP 0 1 + ζu γu exp − t 0 ∞ (1 + ζv )γv dv du F∞ γu e− u 0 + exp − 0 t ζv γv dv EP t u γv dv du F∞ = 0 1 + ζu γu exp − t 0 ∞ (1 + ζv )γv dv du γu e− u 0 + exp − 0 ζv γv dv t t γv dv du t t = 1 − exp − 0 (1 + ζv )γv dv + exp − 0 ζv γv dv exp − 0 γv dv = 1, where the second last equality follows by an application of the chain rule. Extension to orthogonal martingales Equality (4.14) suggests that Proposition 4.2.3 can be extended to the case of arbitrary orthogonal local martingales. Such a generalization is convenient, if we wish to cover the situation considered in Kusuoka’s counterexample. M. Jeanblanc 65 Let N be a local martingale under P with respect to the filtration F. It is also a G-local martingale, since we maintain the assumption that the hypothesis (H) holds under P. Let Q be an arbitrary probability measure locally equivalent to P on (Ω, G). We assume that the Radon-Nikod´m y density process η of Q with respect to P equals dηt = ηt− θt dNt + ζt dMt (4.15) for some G-predictable processes θ and ζ > −1 (the properties of the process θ depend, of course, on the choice of the local martingale N ). The next result covers the case where N and M are orthogonal G-local martingales under P, so that the product M N follows a G-local martingale. Proposition 4.2.4 Assume that the following conditions hold: (a) N and M are orthogonal G-local martingales under P, (b) N has the predictable representation property under P with respect to F, in the sense that any F-local martingale L under P can be written as t Lt = L0 + 0 ξu dNu , ∀ t ∈ I +, R for some F-predictable process ξ, (c) P is a probability measure on (Ω, G) such that (4.13) holds. Then we have: (i) the hypothesis (H) is valid under P, (ii) if the process θ is F-adapted then the hypothesis (H) is valid under Q. The proof of the proposition hinges on the following simple lemma. Lemma 4.2.4 Under the assumptions of Proposition 4.2.4, we have: (i) N is a G-local martingale under P, (ii) N has the predictable representation property for F-local martingales under P. Proof: In view of (c), we have dP | Gt = ηt dP | Gt , where the density process η (2) is given by (2) (2) (4.11), so that dηt = ηt− ζt dMt . From the assumed orthogonality of N and M , it follows that N and η (2) are orthogonal G-local martingales under P, and thus N η (2) is a G-local martingale under P as well. This means that N is a G-local martingale under P, so that (i) holds. To establish part (ii) in the lemma, we first define the auxiliary process η by setting ηt = (2) EP (ηt | Ft ). Then manifestly dP | Ft = ηt dP | Ft , and thus in order to show that any F-local martingale under P follows an F-local martingale under P, it suffices to check that ηt = 1 for every t ∈ I + , so that P = P on F. To this end, we note that R EP (ηt | Ft ) = EP Et (2) · (2) ζu dMu 0 F∞ = 1, ∀ t ∈ I +, R where the first equality follows from part (v) in Lemma 4.2.1, and the second one can established similarly as the second equality in (4.14). We are in a position to prove (ii). Let L be an F-local martingale under P. Then it follows also an F-local martingale under P and thus, by virtue of (b), it admits an integral representation with respect to N under P and P. This shows that N has the predictable representation property with respect to F under P. We now proceed to the proof of Proposition 4.2.4. Proof of Proposition 4.2.4. We shall argue along the similar lines as in the proof of Proposition 4.2.3. To prove (i), note that by part (ii) in Lemma 4.2.4 we know that any F-local martingale under P admits the integral representation with respect to N . But, by part (i) in Lemma 4.2.4, N is a G-local martingale under P. We conclude that L is a G-local martingale under P, and thus the hypothesis (H) is valid under P. Assertion (ii) now follows from part (i) in Lemma 4.2.3. 66 Credit Risk, TUNIS 2005 Remark 4.2.1 It should be stressed that Proposition 4.2.4 is not directly employed in what follows. We decided to present it here, since it sheds some light on specific technical problems arising in the context of modeling dependent default times through an equivalent change of a probability measure (see Kusuoka [66]). Example 4.2.1 Kusuoka [66] presents a counter-example based on the two independent random t∧τ i times τ1 and τ2 given on some probability space (Ω, G, P). We write Mti = Ht − 0 i γi (u) du, i where Ht = 1 {t≥τi } and γi is the deterministic intensity function of τi under P. Let us set dQ | Gt = 1 (1) (2) ηt dP | Gt , where ηt = ηt ηt and, for i = 1, 2 and every t ∈ I + , R ηt = 1 + (i) t 0 (i) i ηu− ζu dMu = Et (i) · 0 (i) i ζu dMu for some G-predictable processes ζ (i) , i = 1, 2, where G = H1 ∨ H2 . We set F = H1 and H = H2 . Manifestly, the hypothesis (H) holds under P. Moreover, in view of Proposition 4.2.4, it is still valid under the equivalent probability measure P given by · dP | Gt = Et It is clear that P = P on F, since EP (ηt | Ft ) = EP Et (2) · 0 0 (2) 2 ζu dMu dP | Gt . (2) 2 ζu dMu 1 Ht = 1, ∀ t ∈ I +. R However, the hypothesis (H) is not necessarily valid under Q if the process ζ (1) fails to be Fadapted. In Kusuoka’s counter-example, the process ζ (1) was chosen to be explicitly dependent on both random times, and it was shown that the hypothesis (H) does not hold under Q. For an alternative approach to Kusuoka’s example, through an absolutely continuous change of a probability measure, the interested reader may consult Collin-Dufresne et al. [21]. 4.2.5 Stochastic Barrier P (τ ≤ t|F∞ ) = 1 − e−Γt Suppose that where Γ is an arbitrary continuous strictly increasing F-adapted process. Our goal is to show that there exists a random variable Θ, −t. independent of F∞ , with exponential law of parameter 1, such =e −Γ th τQ ’pCy‘Ppty‘RidSpi‘o0’‘ CwydSPpty‘RidS’‘v( P-wy‘uidSU”0 p)‘33000=04 ‘ A jF ∞ ) e ‘‘WIp)‘ ui‘6 Hypx M. Jeanblanc 67 Th´or`me 4.1 Under (H), any G-square integrable martingale admits a representation as the sum e e of a stochastic integral with respect to the Brownian motion and a stochastic integral with respect to the discontinuous martingale M . We assume for simplicity that F is continuous and Ft < 1, ∀t ∈ I + . Since (H) hypothesis holds, R F is an increasing process. Then, dFt = e−Γt dΓt and d(eΓt ) = eΓt dΓt = eΓt dFt . 1 − Ft (4.16) Proposition 4.3.1 Suppose that hypothesis (H) holds under P and that any F-martingale is continuous. Then, the martingale Ht = EP (hτ | Gt ) , where h is an F-predictable process such that E(hτ ) < ∞, admits the following decomposition as the sum of a continuous martingale and a discontinuous martingale Ht = m h + 0 t∧τ 0 eΓu dmh + u (hu − Ju ) dMu , ]0,t∧τ ] (4.17) where mh is the continuous F-martingale mh = EP t t ∞ hu dFu | Ft , 0 Jt = eΓt (mh − t dFu . 1 − Fu hu dFu ) and M is the discontinuous G-martingale Mt = Ht − Λt∧τ where dΛu = 0 Proof: From (3.3) we know that Ht = E(hτ | Gt ) = 1 {τ ≤t} hτ + 1 {τ >t} eΓt E 1 1 ∞ t hu dFu Ft = 1 {τ ≤t} hτ + 1 {τ >t} Jt . 1 1 (4.18) From the facts that Γ is an increasing process and mh a continuous martingale, and using the integration by parts formula, we deduce that dJt = eΓt dmh + Jt e−Γt d(eΓt ) − ht (eΓt dFt ) . t Therefore, from (4.16) dJt = eΓt dmh + (Jt − ht ) t or, in an integrated form, t dFt 1 − Ft Jt = m0 + 0 eΓu dmh + u t (Ju − hu )dΛu . 0 Note that Ju = Hu for u < τ . Therefore, on {t < τ } Ht = mh + 0 t∧τ 0 eΓu dmh + u t∧τ (Ju − hu )dΛu 0 From (4.18), the jump of H at time τ is hτ − Jτ = hτ − Hτ − . Then, (4.17) follows. Remark 4.3.1 Since hypothesis (H) holds, the processes (mt , t ≥ 0) and ( also G-martingales. t∧τ 0 eΓu dmu , t ≥ 0) are 68 Credit Risk, TUNIS 2005 4.4 Partial information As pointed out by Jamshidian [?], “one may wish to apply the general theory perhaps as an intermediate step, to a subfiltration that is not equal to the default-free filtration. In that case, F rarely satisfies hypothesis (H)”. We present here simple cases of such a situation. 4.4.1 Information at discrete times dVt = Vt (µdt + σdWt ), V0 = v Assume that i.e., Vt = ve = ve , with ν = (µ−σ 2 /2)/σ and Xt = Wt +νt. The default time is assumed to be the first hitting time of α with α < v, i.e., τ = inf{t : Vt ≤ α} = inf{t : Xt ≤ a} where a = σ −1 ln(α/v). Here, F is the filtration of the observations of V at discrete times t1 , · · · tn where tn ≤ t < tn+1 , i.e., Ft = σ(Vt1 , · · · , Vtn , ti ≤ t) and we compute Ft = P (τ ≤ t|Ft ). Let us recall that (See Section 9.1.2) P (inf Xs > z) = Φ(ν, t, z) , s≤t σ(Wt +νt) σXt (4.19) where Φ(ν, t, z) = N = 0, Φ(ν, 0, z) = 1, On t < t1 In that case, Ft is the cumulative function of τ . Since a < 0, we obtain Ft = P (τ ≤ t) = P (inf Xs ≤ a) s≤t νt − z z + νt √ √ − e2νz N t t for z ≥ 0, t ≥ 0, for z < 0. , for z < 0, t > 0, = 1 − Φ(ν, t, a) = N On t1 < t < t2 a − νt √ t + e2νa N a + νt √ t . W We denote by Ft = σ(Ws , s ≤ t) the natural filtration of the Brownian motion (this is also the natural filtration of X) Ft = = P (τ ≤ t|Xt1 ) = 1 − P (τ > t|Xt1 ) W E(1 {inf sa} P ( inf Xs > a|Ft1 )|Xt1 ) 1 t1 ≤s a|Ft1 ) = Φ(ν, t − t1 , a − Xt1 ) . t1 ≤s a|Xt1 ) . s a, we obtain (we skip the parameter ν in the definition of Φ) 2a Ft = 1 − Φ(t − t1 , a − Xt1 ) 1 − exp − (a − Xt1 ) . (4.20) t1 The case Xt1 ≤ a corresponds to default and, therefore, for Xt1 ≤ a, Ft = 1. The process F is continuous and increasing in [t1 , t2 [. When t approaches t1 from above, for 2a Xt1 > a, Ft+ = exp − (a − Xt1 ) , because limt→t+ Φ(t − t1 , a − Xt1 ) = 1. 1 1 t1 For Xt1 > a, the jump of F at t1 is ∆Ft2 = exp − 1 2a (a − Xt1 ) − 1 + Φ(t1 , a). t1 For Xt1 ≤ a, Φ(t − t1 , a − Xt1 ) = 0 by the definition of Φ(·) and ∆Ft1 = Φ(t1 , a). General observation times ti < t < ti+1 < T , i ≥ 2 For ti < t < ti+1 , P (τ > t|Xt1 , . . . , Xti ) = = P s≤ti inf Xs > a P ( inf Xs > a|Fti )|Xt1 , . . . , Xti ti ≤s a|Xt1 , . . . , Xti . Write Ki for the second term on the right-hand-side Ki = = Obviously, P( ti−1 ≤s a|Xt1 , . . . , Xti inf Xs > a P ( inf Xs > a|Fti−1 ∨ Xti )|Xt1 , . . . , Xti . s≤ti−1 ti−1 ≤s a|Fti−1 ∨ Xti ) = P( ti−1 ≤s a|Xti−1 , Xti )) = exp − Therefore, Ki = Ki−1 exp − Hence, P (τ ≤ t|Ft ) where = 1 2 (a − Xti−1 )(a − Xti ) . ti − ti−1 (4.21) 2 (a − Xti−1 )(a − Xti ) . ti − ti−1 if Xtj < a for at least one tj , tj < t = 1 − Φ(t − ti , a − Xti )Ki , Ki = k(t1 , Xt1 , 0)k(t2 − t1 , Xt1 , Xt2 ) · · · k(ti − ti−1 , Xti−1 , Xti ) and k(s, x, y) = 1 − exp − 2 (a − x)(a − y) . s Lemma 4.4.1 The process ζ defined by ζt = i,ti ≤t ∆Fti . is an F-martingale. 70 Credit Risk, TUNIS 2005 Proof: Consider first the times ti ≤ s < t ≤ ti+1 . In this case, it is obvious that E(ζt |Hs ) = ζs since ζt = ζs = ζti , which is Hs -measurable. It suffices to show that E(ζt |Fs ) = ζs for ti ≤ s < ti+1 ≤ t < ti+2 . In this case, ζs = ζti and ζt = ζti + ∆Fti+1 . Therefore, E(ζt |Fs ) = E(ζti + ∆Fti+1 |Fs ) = ζti + E(∆Fti+1 |Fs ), which shows that it is necessary to prove that E(∆Fti+1 |Fs ) = 0. Let s < u < ti+1 < v < t. Then, E(Fv − Fu |Fs ) = E(1 u<τ ≤v |Fs ). 1 When v → ti+1 , v > ti+1 u → ti+1 , u < ti+1 , It follows that E(∆Fti+1 |Fs ) = = u→ti+1 ,v→ti+1 and Fv − Fu → ∆Fti+1 . lim E(1 u<τ ≤v |Fs ) 1 E(1 τ =ti+1 |Fs ) = 0. 1 The Doob-Meyer decomposition of F is Ft = ζt + (Ft − ζt ), where ζ is an F-martingale and Ft − ζt is a predictable increasing process. The intensity of the default time would be the process λ defined as λt dt = d(Ft − ζt ) . 1 − (Ft− − ζt− ) Comments 4.4.1 It is also possible, as in Duffie and Lando [34], to assume that the observation at time [t] is only V[t] + where is a noise, modelled as a random variable independent of V . Another example, related with Parisian stopping times is presented in Cetin et al. [18] ¸ 4.4.2 Delayed information In [?] the author suggested to start form a structural model with delayed information. We prove here that (H) hypothesis is not satisfied. Brownian filtration case General case 4.5 Intensity approach In the so-called intensity approach, the starting point is the knowledge of default time τ and some filtration G such that τ is a G-stopping time. The intensity is defined as any non-negative process λ, such that Mt = Dt − 0 def t∧τ λs ds M. Jeanblanc 71 is a G-martingale. The existence of the intensity relies on the fact that D is an increasing process, therefore a sub-martingale and can be written as a martingale M plus a predictable increasing 1 1 process A. The increasing process A is such that At 1 t≥τ = Aτ 1 t≥τ . In the case where τ is a predictable stopping time, obviously A = D. The intensity exists only if τ is a totally inaccessible stopping time. Under some additional properties, Duffie and Singleton [35] establishes formulae similar to (3.2). We emphasize that, in that setting the intensity is not well defined after time τ , i.e., if λ is an intensity, ˜ for any non-negative predictable process g the process λt = λt 1 t≤τ + gt 1 {t>τ } is also an intensity. 1 1 Lemma 4.5.1 The process Lt = 1 {t<τ } exp 1 t 0 λs ds is a martingale. Proof: From Itˆ’s calculus (See Section 9.3.2) o t t dLt = exp 0 λs ds (−dDt + (1 − Dt− )λt dt) = − exp 0 λs ds dMt . T Proposition 4.5.1 If the process Yt = E X exp − t λu du |Gt T is continuous at time τ , then E(X1 {T <τ } |Gt ) = 1 {t<τ } E X exp − 1 1 T λu du |Gt t (4.22) Proof: The process Ut = 1 t<τ exp 1 deed, dUt = Lt− dYt + Yt dLt and t 0 λs ds E(X exp − 0 λu du|Gt ) = Lt Yt is a martingale. In- E(UT |Gt ) = E(X1 {T <τ } |Gt ) = Ut . 1 The result follows. It can be mentioned that the continuity of the process depends on the choice of λ after time τ . If the process Y is not continuous, then dUt = Lt− dYt + Yt− dLt + d[L, Y ]t = Lt− dYt + Yt− dLt − ∆Yt and E(UT |Gt ) = E(X1 {T <τ } |Gt ) = Ut − E(∆Yτ |Gt ) . 1 Nevertheless, in practise, it is difficult to compute the size of the jump. 72 Credit Risk, TUNIS 2005 Chapter 5 Hedging 5.1 Semimartingale Model with a Common Default In what follows, we fix a finite horizon date T > 0. For the purpose of this work, it is enough to formally define a generic defaultable claim through the following definition. Definition 5.1.1 A defaultable claim with maturity date T is represented by a triplet (X, Z, τ ), where: (i) the default time τ specifies the random time of default, and thus also the default events {τ ≤ t} for every t ∈ [0, T ], (ii) the promised payoff X ∈ FT represents the random payoff received by the owner of the claim at time T, provided that there was no default prior to or at time T ; the actual payoff at time T associated with X thus equals X1 {T <τ } , 1 (iii) the F-adapted recovery process Z specifies the recovery payoff Zτ received by the owner of a claim at time of default (or at maturity), provided that the default occurred prior to or at maturity date T . In practice, hedging of a credit derivative after default time is usually of minor interest. Also, in a model with a single default time, hedging after default reduces to replication of a non-defaultable claim. It is thus natural to define the replication of a defaultable claim in the following way. Definition 5.1.2 We say that a self-financing strategy φ replicates a defaultable claim (X, Z, τ ) if its wealth process V (φ) satisfies VT (φ)1 {T <τ } = X1 {T <τ } and Vτ (φ)1 {T ≥τ } = Zτ 1 {T ≥τ } . 1 1 1 1 When dealing with replicating strategies, in the sense of Definition 5.1.2, we will always assume, without loss of generality, that the components of the process φ are F-predictable processes. 5.1.1 Dynamics of asset prices We assume that we are given a probability space (Ω, G, P) endowed with a (possibly multi-dimensional) standard Brownian motion W and a random time τ admitting an F-intensity γ under P, where F is the filtration generated by W . In addition, we assume that τ satisfies (4.4), so that the hypothesis (H) is valid under P for filtrations F and G = F ∨ H. Since the default time admits an F-intensity, it is not an F-stopping time. Indeed, any stopping time with respect to a Brownian filtration is known to be predictable. We interpret τ as the common default time for all defaultable assets in our model. For simplicity, we assume that only three primary assets are traded in the market, and the dynamics under the 73 74 historical probability P of their prices are, for i = 1, 2, 3 and t ∈ [0, T ], i dYti = Yt− µi,t dt + σi,t dWt + κi,t dMt , Credit Risk, TUNIS 2005 (5.1) (5.2) or equivalently, i dYti = Yt− (µi,t − κi,t γt 1 {t≤τ } ) dt + σi,t dWt + κi,t dHt . 1 The processes (µi , σi , κi ) = (µi,t , σi,t , κi,t , t ≥ 0), i = 1, 2, 3, are assumed to be G-adapted, where G = F ∨ H. In addition, we assume that κi ≥ −1 for any i = 1, 2, 3, so that Y i are nonnegative processes, and they are strictly positive prior to τ . Note that, according to Definition 5.1.2, replication refers to the behavior of the wealth process V (φ) on the random interval [[0, τ ∧ T ]] only. Hence, for the purpose of replication of defaultable claims of the form (X, Z, τ ), it is sufficient to consider prices of primary assets stopped at τ ∧T . This implies that instead of dealing with G-adapted coefficients in (5.1), it suffices to focus on F-adapted coefficients of stopped price processes. However, for the sake of completeness, we shall also deal with 2 3 1 T -maturity claims of the form Y = G(YT , YT , YT , HT ) (see Section 5.4 below). Pre-default values As will become clear in what follows, when dealing with defaultable claims of the form (X, Z, τ ), we will be mainly concerned with the so-called pre-default prices. The pre-default price Y i of the ith asset is an F-adapted, continuous process, given by the equation, for i = 1, 2, 3 and t ∈ [0, T ], dYti = Yti (µi,t − κi,t γt ) dt + σi,t dWt (5.3) with Y0i = Y0i . Put another way, Y i is the unique F-predictable process such that (see Lemma ??) Yti 1 {t≤τ } = Yti 1 {t≤τ } for t ∈ I + . When dealing with the pre-default prices, we may and do assume, 1 1 R without loss of generality, that the processes µi , σi and κi are F-predictable. It is worth stressing that the historically observed drift coefficient equals µi,t − κi,t γt , rather than µi,t . The drift coefficient denoted by µi,t is already credit-risk adjusted in the sense of our model, and it is not directly observed. This convention was chosen here for the sake of simplicity of notation. It also lends itself to the following intuitive interpretation: if φi is the number of units of the ith asset held in our portfolio at time t then the gains/losses from trades in this asset, prior to default time, can be represented by the differential φi dYti = φi Yti µi,t dt + σi,t dWt − φi Yti κi,t γt dt. t t t The last term may be here separated, and formally treated as an effect of continuously paid dividends at the dividend rate κi,t γt . However, this interpretation may be misleading, since this quantity is not directly observed. In fact, the mere estimation of the drift coefficient in dynamics (5.3) is not practical. Still, if this formal interpretation is adopted, it is sometimes possible make use of the standard results concerning the valuation of derivatives of dividend-paying assets. It is, of course, a delicate issue how to separate in practice both components of the drift coefficient. We shall argue below that although the dividend-based approach is formally correct, a more pertinent and simpler way of dealing with hedging relies on the assumption that only the effective drift µi,t − κi,t γt is observable. In practical approach to hedging, the values of drift coefficients in dynamics of asset prices play no essential role, so that they are considered as market observables. Market observables To summarize, we assume throughout that the market observables are: the pre-default market prices of primary assets, their volatilities and correlations, as well as the jump coefficients κi,t (the financial M. Jeanblanc 75 interpretation of jump coefficients is examined in the next subsection). To summarize we postulate that under the statistical probability P we have i dYti = Yt− µi,t dt + σi,t dWt + κi,t dHt (5.4) where the drift terms µi,t are not observable, but we can observe the volatilities σi,t (and thus the assets correlations), and we have an a priori assessment of jump coefficients κi,t . In this general set-up, the most natural assumption is that the dimension of a driving Brownian motion W equals the number of tradable assets. However, for the sake of simplicity of presentation, we shall frequently assume that W is one-dimensional. One of our goals will be to derive closed-form solutions for replicating strategies for derivative securities in terms of market observables only (whenever replication of a given claim is actually feasible). To achieve this goal, we shall combine a general theory of hedging defaultable claims within a continuous semimartingale set-up, with a judicious specification of particular models with deterministic volatilities and correlations. Recovery schemes It is clear that the sample paths of price processes Y i are continuous, except for a possible discontinuity at time τ . Specifically, we have that ∆Yτi := Yτi − Yτi− = κi,τ Yτi− , so that Yτi = Yτi− (1 + κi,τ ) = Yτi− (1 + κi,τ ). A primary asset Y i is termed a default-free asset (defaultable asset, respectively) if κi = 0 (κi = 0, respectively). In the special case when κi = −1, we say that a defaultable asset Y i is subject to a total default, since its price drops to zero at time τ and stays there forever. Such an asset ceases to exist after default, in the sense that it is no longer traded after default. This feature makes the case of a total default quite different from other cases, as we shall see in our study below. In market practice, it is common for a credit derivative to deliver a positive recovery (for instance, a protection payment) in case of default. Formally, the value of this recovery at default is determined as the value of some underlying process, that is, it is equal to the value at time τ of some F-adapted recovery process Z. For example, the process Z can be equal to δ, where δ is a constant, or to g(t, δYt ) where g is a deterministic function and (Yt , t ≥ 0) is the price process of some default-free asset. Typically, the recovery is paid at default time, but it may also happen that it is postponed to the maturity date. Let us observe that the case where a defaultable asset Y i pays a pre-determined recovery at default is covered by our set-up defined in (5.1). For instance, the case of a constant recovery payoff i δi ≥ 0 at default time τ corresponds to the process κi,t = δi (Yt− )−1 − 1. Under this convention, the i price Y is governed under P by the SDE i i dYti = Yt− µi,t dt + σi,t dWt + (δi (Yt− )−1 − 1) dMt . (5.5) If the recovery is proportional to the pre-default value Yτi− , and is paid at default time τ (this scheme is known as the fractional recovery of market value), we have κi,t = δi − 1 and i dYti = Yt− µi,t dt + σi,t dWt + (δi − 1) dMt . (5.6) 5.1.2 Risk-neutral valuation To provide a partial justification for the postulated dynamics of the price of a defaultable asset delivering a recovery, let us study a toy example with two assets: a savings account with constant interest rate r and a defaultable asset Y represented by a defaultable claim (X, Z, τ ). In this toy model, the only source of noise is the default time, hence, the only relevant filtration is H (in other 76 Credit Risk, TUNIS 2005 words, the reference filtration F is trivial). We assume that by choosing today’s prices of a large family liquidly traded defaultable assets, the market implicitly specifies a martingale measure Q, equivalent to the historical probability P. More precisely, the probability distribution of τ under an equivalent martingale measure (e.m.m.) Q can be inferred from market data. We are thus interested in the dynamics of the price process of (X, Z, τ ) under Q. It is worth noting that in this subsection we adopt a totally different perspective than in the rest of the present paper. In fact, no attempt to replicate a defaultable claim is done in this section. We assume instead that the risk-neutral default intensity can be uniquely determined from prices of traded assets, and we postulate that the price of (X, Z, τ ) is defined by the standard risk-neutral valuation formula. The argument that formally justifies the use of this pricing rule is that we obtain in this way an arbitrage-free market model in which Q is a martingale measure, and a defaultable claim can be considered to be an additional traded asset. Since we do not assume here that a defaultable claim is attainable, its spot price (that is, the price expressed in units of cash) depends explicitly on the risk-neutral default intensity. As was mentioned above, the arbitrage price of a defaultable claim, when expressed in terms of tradeable assets used for its replication, will be shown to not depend directly on real-world (or risk-neutral) default intensity. To conclude, the rationale for the calculations given below, is that we strive here to justify the dynamics of prices of primary assets in our model. The risk-neutral valuation considered in this subsection is not supported by replication-based arguments, and thus it is not surprising that it exhibits specific features that are not present in the replication-based valuation. We make the standing assumption that τ admits a continuous cumulative distribution function F under Q. Hence, the hazard function Γ is also continuous, and the process Mt = Ht − Γ(t ∧ τ ) is an H-martingale under Q. The following result is standard (see, e.g., Proposition 4.3.2 in Bielecki and Rutkowski [4]). Proposition 5.1.1 Assume that the cumulative distribution function F of τ is continuous. Let M h be an H-martingale given by Mth = EQ (h(τ ) | Ht ) for some Borel measurable function h : I + → I R R such that the random variable h(τ ) is Q-integrable. Then h Mth = M0 + t 0 h (h(u) − g(u)) dMu = M0 + t 0 h (h(u) − Mu− ) dMu , (5.7) where we write g(t) = eΓ(t) EQ 1 {t<τ } h(τ ) . 1 Remark 5.1.1 Using the above proposition, it can be easily shown that on (Ω, GT ) we have · dP = ET for some H-predictable process ζ. − 0 ζu dMu dQ, 5.2 Trading Strategies in a Semimartingale Set-up We consider trading within the time interval [0, T ] for some finite horizon date T > 0. For the sake of expositional clarity, we restrict our attention to the case where only three primary assets are traded. The general case of k traded assets was examined by Bielecki et al. [?]. We first recall some general properties, which do not depend on the choice of specific dynamics of asset prices. In this section, we consider a fairly general set-up. In particular, processes Y i , i = 1, 2, 3, are assumed to be nonnegative semi-martingales on a probability space (Ω, G, P) endowed with some filtration G. We assume that they represent spot prices of traded assets in our model of the financial market. Neither the existence of a savings account, nor the market completeness are assumed, in general. M. Jeanblanc 77 Our goal is to characterize contingent claims which are hedgeable, in the sense that they can be replicated by continuously rebalanced portfolios consisting of primary assets. Here, by a contingent claim we mean an arbitrary GT -measurable random variable. We work under the standard assumptions of a frictionless market. 5.2.1 Unconstrained strategies Let φ = (φ1 , φ2 , φ3 ) be a trading strategy; in particular, each process φi is predictable with respect to the filtration G. The wealth of φ equals 3 Vt (φ) = i=1 φi Yti , t ∀ t ∈ [0, T ], and a trading strategy φ is said to be self-financing if 3 t 0 Vt (φ) = V0 (φ) + i=1 i φi dYu , u ∀ t ∈ [0, T ]. Let Φ stand for the class of all self-financing trading strategies. We shall first prove that a selffinancing strategy is determined by its initial wealth and the two components φ2 , φ3 . To this end, we postulate that the price of Y 1 follows a strictly positive process, and we choose Y 1 as a num´raire e asset. We shall now analyze the relative values: Vt1 (φ) := Vt (φ)(Yt1 )−1 , Lemma 5.2.1 (i) For any φ ∈ Φ, we have 3 Yti,1 := Yti (Yt1 )−1 . Vt1 (φ) = V01 (φ) + i=2 t 0 i,1 φi dYu , u ∀ t ∈ [0, T ]. (ii) Conversely, let X be a GT -measurable random variable, and let us assume that there exists x ∈ I R and G-predictable processes φi , i = 2, 3 such that 3 1 X = YT T 0 x+ i=2 i,1 φi dYu u . (5.8) Then there exists a G-predictable process φ1 such that the strategy φ = (φ1 , φ2 , φ3 ) is self-financing and replicates X. Moreover, the wealth process of φ (i.e. the time-t price of X) satisfies Vt (φ) = Vt1 Yt1 , where 3 Vt1 = x + i=2 t 0 i,1 φi dYu , u ∀ t ∈ [0, T ]. (5.9) Proof: The proof of part (i) is given, for instance, in Protter [79]. In the case of continuous semimartingales, this is a well-known result; for discontinuous processes, the proof is not much different. We reproduce it here for the reader’s convenience. Let us first introduce some notation. As usual, [X, Y ] stands for the quadratic covariation of the two semi-martingales X and Y , as defined by the integration by parts formula: t t Xt Yt = X0 Y0 + 0 Xu− dYu + 0 Yu− dXu + [X, Y ]t . For any c`dl`g (i.e., RCLL) process Y , we denote by ∆Yt = Yt − Yt− the size of the jump at time a a t. Let V = V (φ) be the value of a self-financing strategy, and let V 1 = V 1 (φ) = V (φ)(Y 1 )−1 be its value relative to the num´raire Y 1 . The integration by parts formula yields e 1 dVt1 = Vt− d(Yt1 )−1 + (Yt− )−1 dVt + d[(Y 1 )−1 , V ]t . 78 3 Credit Risk, TUNIS 2005 From the self-financing condition, we have dVt = i=1 φi dYti . Hence, using elementary rules to t compute the quadratic covariation [X, Y ] of the two semi-martingales X, Y , we obtain dVt1 1 2 3 = φ1 Yt− d(Yt1 )−1 + φ2 Yt− d(Yt1 )−1 + φ3 Yt− d(Yt1 )−1 t t t 1 1 1 + (Yt− )−1 φ1 dYt1 + (Yt− )−1 φ2 dYt1 + (Yt− )−1 φ3 dYt1 t t t 1 1 −1 1 2 1 −1 2 + φt d[(Y ) , Y ]t + φt d[(Y ) , Y ]t + φ3 d[(Y 1 )−1 , Y 1 ]t t 1 1 = φ1 Yt− d(Yt1 )−1 + (Yt− )−1 dYt1 + d[(Y 1 )−1 , Y 1 ]t t 2 1 1 + φ2 Yt− d(Yt1 )−1 + (Yt− )−1 dYt− + d[(Y 1 )−1 , Y 2 ]t t 3 1 1 + φ3 Yt− d(Yt1 )−1 + (Yt− )−1 dYt− + d[(Y 1 )−1 , Y 3 ]t . t We now observe that 1 1 Yt− d(Yt1 )−1 + (Yt− )−1 dYt1 + d[(Y 1 )−1 , Y 1 ]t = d(Yt1 (Yt1 )−1 ) = 0 and Consequently, i 1 Yt− d(Yt1 )−1 + (Yt− )−1 dYti + d[(Y 1 )−1 , Y i ]t = d((Yt1 )−1 Yti ). dVt1 = φ2 dYt2,1 + φ3 dYt3,1 , t t as was claimed in part (i). We now proceed to the proof of part (ii). We assume that (5.8) holds for some constant x and processes φ2 , φ3 , and we define the process V 1 by setting (cf. (5.9)) 3 Vt1 t 0 =x+ i=2 i,1 φi dYu , u ∀ t ∈ [0, T ]. Next, we define the process φ1 as follows: 3 3 φ1 = Vt1 − t i=2 φi Yti,1 = (Yt1 )−1 Vt − t i=2 φi Yti , t where Vt = Vt1 Yt1 . Since dVt1 = dVt = = From the equality 3 i=2 φi dYti,1 , we obtain t 3 1 1 d(Vt1 Yt1 ) = Vt− dYt1 + Yt− dVt1 + d[Y 1 , V 1 ]t 1 Vt− dYt1 + i=2 1 φi Yt− dYti,1 + d[Y 1 , Y i,1 ]t . t i,1 1 dYti = d(Yti,1 Yt1 ) = Yt− dYt1 + Yt− dYti,1 + d[Y 1 , Y i,1 ]t , it follows that 3 1 dVt = Vt− dYt1 + i=2 i,1 1 φi dYti − Yt− dYt1 = Vt− − t i=2 3 i=1 3 i,1 φi Yt− dYt1 + t i=2 3 φi dYti , t and our aim is to prove that dVt = φi dYti . The last equality holds if t 3 3 1 φi Yti,1 = Vt− − t i,1 φi Yt− , t i=2 φ1 = Vt1 − t i=2 3 (5.10) i.e., if ∆Vt1 = i=2 φi ∆Yti,1 , which is the case from the definition (5.9) of V 1 . Note also that from t the second equality in (5.10) it follows that the process φ1 is indeed G-predictable. Finally, the wealth process of φ satisfies Vt (φ) = Vt1 Yt1 for every t ∈ [0, T ], and thus VT (φ) = X. M. Jeanblanc 79 We say that a self-financing strategy φ replicates a claim X ∈ GT if 3 X= i=1 i φi YT = VT (φ), T or equivalently, 3 T 0 X = V0 (φ) + i=1 φi dYti . t Suppose that there exists an e.m.m. for some choice of a num´raire asset, and let us restrict our e attention to the class of all admissible trading strategies, so that our model is arbitrage-free. Assume that a claim X can be replicated by some admissible trading strategy, so that it is attainable (or hedgeable). Then, by definition, the arbitrage price at time t of X, denoted as πt (X), equals Vt (φ) for any admissible trading strategy φ that replicates X. In the context of Lemma 5.2.1, it is natural to choose as an e.m.m. a probability measure Q1 equivalent to P on (Ω, GT ) and such that the prices Y i,1 , i = 2, 3, are G-martingales under Q1 . If a contingent claim X is hedgeable, then its arbitrage price satisfies 1 πt (X) = Yt1 EQ1 (X(YT )−1 | Gt ). We emphasize that even if an e.m.m. Q1 is not unique, the price of any hedgeable claim X is given by this conditional expectation. That is to say, in case of a hedgeable claim these conditional expectations under various equivalent martingale measures coincide. In the special case where Yt1 = B(t, T ) is the price of a default-free zero-coupon bond with maturity T (abbreviated as ZC-bond in what follows), Q1 is called T -forward martingale measure, and it is denoted by QT . Since B(T, T ) = 1, the price of any hedgeable claim X now equals πt (X) = B(t, T ) EQT (X | Gt ). 5.2.2 Constrained strategies In this section, we make an additional assumption that the price process Y 3 is strictly positive. Let φ = (φ1 , φ2 , φ3 ) be a self-financing trading strategy satisfying the following constraint: 2 i φi Yt− = Zt , t i=1 ∀ t ∈ [0, T ], (5.11) for a predetermined, G-predictable process Z. In the financial interpretation, equality (5.11) means that a portfolio φ is rebalanced in such a way that the total wealth invested in assets Y 1 , Y 2 matches a predetermined stochastic process Z. For this reason, the constraint given by (5.11) is referred to as the balance condition. Our first goal is to extend part (i) in Lemma 5.2.1 to the case of constrained strategies. Let Φ(Z) stand for the class of all (admissible) self-financing trading strategies satisfying the balance condition (5.11). They will be sometimes referred to as constrained strategies. Since any strategy 3 φ ∈ Φ(Z) is self-financing, from dVt (φ) = i=1 φi dYti , we obtain t 3 3 ∆Vt (φ) = i=1 φi ∆Yti t = Vt (φ) − i=1 i φi Yt− . t By combining this equality with (5.11), we deduce that 3 Vt− (φ) = i=1 i i φi Yt− = Zt + φ3 Yt− . t t 80 Credit Risk, TUNIS 2005 3 Let us write Yti,3 = Yti (Yt3 )−1 , Zt = Zt (Yt3 )−1 . The following result extends Lemma 1.7 in Bielecki et al. [?] from the case of continuous semi-martingales to the general case (see also [?]). It is apparent from Proposition 5.2.1 that the wealth process V (φ) of a strategy φ ∈ Φ(Z) depends only on a single component of φ, namely, φ2 . Proposition 5.2.1 The relative wealth Vt3 (φ) = Vt (φ)(Yt3 )−1 of any trading strategy φ ∈ Φ(Z) satisfies t t 3 Y 2,3 Zu 2,3 1,3 1,3 φ2 dYu − u− dYu Vt3 (φ) = V03 (φ) + + (5.12) u 1,3 1,3 dYu . Yu− Yu− 0 0 Proof: Let us consider discounted values of price processes Y 1 , Y 2 , Y 3 , with Y 3 taken as a num´raire asset. By virtue of part (i) in Lemma 5.2.1, we thus have e 2 Vt3 (φ) = V03 (φ) + i=1 t 0 i,3 φi dYu . u (5.13) The balance condition (5.11) implies that 2 i,3 3 φi Yt− = Zt , t i=1 and thus 2,3 1,3 3 φ1 = (Yt− )−1 Zt − φ2 Yt− . t t (5.14) By inserting (5.14) into (5.13), we arrive at the desired formula (5.12). The next result will prove particularly useful for deriving replicating strategies for defaultable claims. Proposition 5.2.2 Let a GT -measurable random variable X represent a contingent claim that settles at time T . We set Y 2,3 2,1 dYt∗ = dYt2,3 − t− dYt1,3 = dYt2,3 − Yt− dYt1,3 , (5.15) 1,3 Yt− where, by convention, Y0∗ = 0. Assume that there exists a G-predictable process φ2 , such that 3 X = YT T x+ 0 φ2 dYt∗ + t T 0 3 Zt 1,3 1,3 dYt Yt− . (5.16) Then there exist G-predictable processes φ1 and φ3 such that the strategy φ = (φ1 , φ2 , φ3 ) belongs to Φ(Z) and replicates X. The wealth process of φ equals, for every t ∈ [0, T ], Vt (φ) = Yt3 t x+ 0 ∗ φ2 dYu + u t 0 3 Zu 1,3 1,3 dYu Yu− . (5.17) Proof: As expected, we first set (note that the process φ1 is a G-predictable process) φ1 = t and Vt3 = x + 0 t 1 2 Zt − φ2 Yt− t 1 Yt− ∗ φ2 dYu + u t 0 3 Zu 1,3 1,3 dYu . Yu− (5.18) M. Jeanblanc 81 Arguing along the same lines as in the proof of Proposition 5.2.1, we obtain 2 Vt3 = V03 + i=1 t 0 i,3 φi dYu . u Now, we define 2 2 φ3 = Vt3 − t i=1 φi Yti,3 = (Yt3 )−1 Vt − t i=1 φi Yti , t where Vt = Vt3 Yt3 . As in the proof of Lemma 5.2.1, we check that 2 3 φ3 = Vt− − t i=1 i,3 φi Yt− , t and thus the process φ3 is G-predictable. It is clear that the strategy φ = (φ1 , φ2 , φ3 ) is self-financing and its wealth process satisfies Vt (φ) = Vt for every t ∈ [0, T ]. In particular, VT (φ) = X, so that φ replicates X. Finally, equality (5.18) implies (5.11), and thus φ belongs to the class Φ(Z). Note that equality (5.16) is a necessary (by Lemma 5.2.1) and sufficient (by Proposition 5.2.2) condition for the existence of a constrained strategy that replicates a given contingent claim X. Synthetic asset Let us take Z = 0, so that φ ∈ Φ(0). Then the balance condition becomes formula (5.12) reduces to Y 2,3 dVt3 (φ) = φ2 dYt2,3 − t− dYt1,3 . t 1,3 Yt− 2 i=1 i φi Yt− = 0, and t (5.19) ¯ The process Y 2 = Y 3 Y ∗ , where Y ∗ is defined in (5.15) is called a synthetic asset. It corresponds 2,1 to a particular self-financing portfolio, with the long position in Y 2 and the short position of Yt− 1 number of shares of Y , and suitably re-balanced positions in the third asset so that the portfolio is self-financing, as in Lemma 5.2.1. It can be shown (see Bielecki et al. [?]) that trading in primary assets Y 1 , Y 2 , Y 3 is formally ¯ equivalent to trading in assets Y 1 , Y 2 , Y 3 . This observation supports the name synthetic asset ¯ 2 . Note, however, that the synthetic asset process may take negative attributed to the process Y values. Case of continuous asset prices ¯ In the case of continuous asset prices, the relative price Y ∗ = Y 2 (Y 3 )−1 of the synthetic asset can be given an alternative representation, as the following result shows. Recall that the predictable bracket of the two continuous semi-martingales X and Y , denoted as X, Y , coincides with their quadratic covariation [X, Y ]. Proposition 5.2.3 Assume that the price processes Y 1 and Y 2 are continuous. Then the relative price of the synthetic asset satisfies Yt∗ = where Yt := Yt2,1 e−αt and αt := ln Y 2,1 , ln Y 3,1 t t t 0 3,1 (Yu )−1 eαu dYu , = 0 2,1 3,1 (Yu )−1 (Yu )−1 d Y 2,1 , Y 3,1 u . (5.20) 82 In terms of the auxiliary process Y , formula (5.12) becomes Vt3 (φ) = V03 (φ) + where φt = φ2 (Yt3,1 )−1 eαt . t t t Credit Risk, TUNIS 2005 φu dYu + 0 0 3 Zu 1,3 1,3 dYu , Yu− (5.21) Proof: It suffices to give the proof for Z = 0. The proof relies on the integration by parts formula stating that for any two continuous semi-martingales, say X and Y , we have Yt−1 dXt − Yt−1 d X, Y t = d(Xt Yt−1 ) − Xt dYt−1 , provided that Y is strictly positive. An application of this formula to processes X = Y 2,1 and Y = Y 3,1 leads to (Yt3,1 )−1 dYt2,1 − (Yt3,1 )−1 d Y 2,1 , Y 3,1 t = d(Yt2,1 (Yt3,1 )−1 ) − Yt2,1 d(Y 3,1 )−1 . t The relative wealth Vt3 (φ) = Vt (φ)(Yt3 )−1 of a strategy φ ∈ Φ(0) satisfies Vt3 (φ) = V03 (φ) + = V03 (φ) + = V03 (φ) + where we denote φt = φ2 (Yt3,1 )−1 eαt . t Remark 5.2.1 The financial interpretation of the auxiliary process Y will be studied in Sections 5.3.1 and 5.3.1 below. Let us only observe here that if Y ∗ is a local martingale under some probability Q then Y is a Q-local martingale (and vice versa, if Y is a Q-local martingale under some probability Q then Y ∗ is a Q-local martingale). Nevertheless, for the reader’s convenience, we shall use two symbols Q and Q, since this equivalence holds for continuous processes only. It is thus worth stressing that we will apply Proposition 5.2.3 to pre-default values of assets, rather than directly to asset prices, within the set-up of a semimartingale model with a common default, as described in Section 5.1.1. In this model, the asset prices may have discontinuities, but their pre-default values follow continuous processes. t 0 t 0 t ∗ φ2 dYu u 3,1 φ2 (Yu )−1 eαu dYu , u φu dYu 0 5.3 Martingale Approach to Valuation and Hedging Our goal is to derive quasi-explicit conditions for replicating strategies for a defaultable claim in a fairly general set-up introduced in Section 5.1.1. In this section, we only deal with trading strategies based on the reference filtration F, and the underlying price processes (that is, prices of defaultfree assets and pre-default values of defaultable assets) are assumed to be continuous. Hence, our arguments will hinge on Proposition 5.2.3, rather than on a more general Proposition 5.2.1. We shall also adapt Proposition 5.2.2 to our current purposes. To simplify the presentation, we make a standing assumption that all coefficient processes are such that the SDEs appearing below admit unique strong solutions, and all stochastic exponentials (used as Radon-Nikod´m derivatives) are true martingales under respective probabilities. y M. Jeanblanc 83 5.3.1 Defaultable asset with total default In this section, we shall examine in some detail a particular model where the two assets, Y 1 and Y 2 , are default-free and satisfy dYti = Yti µi,t dt + σi,t dWt , i = 1, 2, where W is a one-dimensional Brownian motion. The third asset is a defaultable asset with total default, so that 3 dYt3 = Yt− µ3,t dt + σ3,t dWt − dMt . Since we will be interested in replicating strategies in the sense of Definition 5.1.2, we may and do assume, without loss of generality, that the coefficients µi,t , σi,t , i = 1, 2, are F-predictable, rather than G-predictable. Recall that, in general, there exist F-predictable processes µ3 and σ3 such that µ3,t 1 {t≤τ } = µ3,t 1 {t≤τ } , 1 1 σ3,t 1 {t≤τ } = σ3,t 1 {t≤τ } . 1 1 (5.22) We assume throughout that Y0i > 0 for every i, so that the price processes Y 1 , Y 2 are strictly positive, and the process Y 3 is nonnegative, and has strictly positive pre-default value. Default-free market It is natural to postulate that the default-free market with the two traded assets, Y 1 and Y 2 , is arbitrage-free. More precisely, we choose Y 1 as a num´raire, and we require that there exists a e probability measure P1 , equivalent to P on (Ω, FT ), and such that the process Y 2,1 is a P1 -martingale. The dynamics of processes (Y 1 )−1 and Y 2,1 are 2 d(Yt1 )−1 = (Yt1 )−1 (σ1,t − µ1,t ) dt − σ1,t dWt , (5.23) and dYt2,1 = Yt2,1 (µ2,t − µ1,t + σ1,t (σ1,t − σ2,t )) dt + (σ2,t − σ1,t ) dWt , respectively. Hence, the necessary condition for the existence of an e.m.m. P1 is the inclusion A ⊆ B, where A = {(t, ω) ∈ [0, T ] × Ω : σ1,t (ω) = σ2,t (ω)} and B = {(t, ω) ∈ [0, T ] × Ω : µ1,t (ω) = µ2,t (ω)}. The necessary and sufficient condition for the existence and uniqueness of an e.m.m. P1 reads · EP ET 0 θu dWu =1 (5.24) where the process θ is given by the formula (by convention, 0/0 = 0) θt = σ1,t − µ1,t − µ2,t , σ1,t − σ2,t ∀ t ∈ [0, T ]. (5.25) Note that in the case of constant coefficients, if σ1 = σ2 then the model is arbitrage-free only in the trivial case when µ2 = µ1 . Remark 5.3.1 Since the martingale measure P1 is unique, the default-free model (Y 1 , Y 2 ) is complete. However, this is not a necessary assumption and thus it can be relaxed. As we shall see in what follows, it is typically more natural to assume that the driving Brownian motion W is multi-dimensional. Arbitrage-free property Let us now consider also a defaultable asset Y 3 . Our goal is now to find a martingale measure Q1 (if it exists) for relative prices Y 2,1 and Y 3,1 . Recall that we postulate that the hypothesis (H) holds under P for filtrations F and G = F ∨ H. The dynamics of Y 3,1 under P are 3,1 dYt3,1 = Yt− µ3,t − µ1,t + σ1,t (σ1,t − σ3,t ) dt + (σ3,t − σ1,t ) dWt − dMt . 84 Credit Risk, TUNIS 2005 Let Q1 be any probability measure equivalent to P on (Ω, GT ), and let η be the associated Radon-Nikod´m density process, so that y dQ1 | Gt = ηt dP | Gt , where the process η satisfies dηt = ηt− (θt dWt + ζt dMt ) for some G-predictable processes θ and ζ, and η is a G-martingale under P. From Girsanov’s theorem, the processes W and M , given by t t (5.26) (5.27) Wt = Wt − 0 θu du, Mt = Mt − 0 1 {u<τ } γu ζu du, 1 (5.28) are G-martingales under Q1 . To ensure that Y 2,1 is a Q1 -martingale, we postulate that (5.24) and (5.25) are valid. Consequently, for the process Y 3,1 to be a Q1 -martingale, it is necessary and sufficient that ζ satisfies γt ζt = µ3,t − µ1,t − µ1,t − µ2,t (σ3,t − σ1,t ). σ1,t − σ2,t ∈∞ T ∪ E y t ζ du, To ensure that Q1 is a probability measure equivalent to P, we require that ζt > −1.Q1,t − unique The 1t 1 T{ ∞ ∈T T{ martingale measure Q is then given by the formula (5.26) where η solbTfn8...lsTf5.15.96134 ,t M. Jeanblanc 85 Remark 5.3.2 Let us stress once again, that the existence of an e.m.m. is a necessary condition for viability of a financial model, but the uniqueness of an e.m.m. is not always a convenient condition to impose on a model. In fact, when constructing a model, we should be mostly concerned with its flexibility and ability to reflect the pertinent risk factors, rather than with its mathematical completeness. In the present context, it is natural to postulate that the dimension of the underlying Brownian motion equals the number of tradeable risky assets. In addition, each particular model should be tailored to provide intuitive and handy solutions for a predetermined family of contingent claims that will be priced and hedged within its framework. Hedging a survival claim We first focus on replication of a survival claim (X, 0, τ ), that is, a defaultable claim represented by the terminal payoff X1 {T <τ } , where X is an FT -measurable random variable. For the moment, we 1 maintain the simplifying assumption that W is one-dimensional. As we shall see in what follows, it may lead to certain pathological features of a model. If, on the contrary, the driving noise is multi-dimensional, most of the analysis remains valid, except that the model completeness is no longer ensured, in general. Recall that Y 3 stands for the pre-default price of Y 3 , defined as (see (5.3)) dYt3 = Yt3 (µ3,t + γt ) dt + σ3,t dWt (5.30) with Y03 = Y03 . This strictly positive, continuous, F-adapted process enjoys the property that Yt3 = e 1 {t<τ } Yt3 . Let us denote the pre-default values in the num´raire Y 3 by Yti,3 = Yti (Yt3 )−1 , i = 1, 2, 1 ¯ and let us introduce the pre-default relative price Y ∗ of the synthetic asset Y 2 by setting dYt∗ := dYt2,3 − Yt2,3 Yt1,3 dYt1,3 = Yt2,3 µ2,t − µ1,t + σ3,t (σ1,t − σ2,t ) dt + (σ2,t − σ1,t ) dWt , and let us assume that σ1,t − σ2,t = 0. It is also useful to note that the process Y , defined in Proposition 5.2.3, satisfies dYt = Yt µ2,t − µ1,t + σ3,t (σ1,t − σ2,t ) dt + (σ2,t − σ1,t ) dWt . In Sections 5.3.1 and 5.3.1, we shall show that in the case, where α given by (5.20) is deterministic, the process Y has a nice financial interpretation as a credit-risk adjusted forward price of Y 2 relative to Y 1 . Therefore, it is more convenient to work with the process Y ∗ when dealing with the general case, but to use the process Y when analyzing a model with deterministic volatilities. φ2 Yt2 t Consider an F-predictable self-financing strategy φ satisfying the balance condition φ1 Yt1 + t = 0, and the corresponding wealth process 3 Vt (φ) := i=1 φi Yti = φ3 Yt3 . t t Let Vt (φ) := φ3 Yt3 . Since the process V (φ) is F-adapted, we see that this is the pre-default price t process of the portfolio φ, that is, we have 1 {τ >t} Vt (φ) = 1 {τ >t} Vt (φ); we shall call this process the 1 1 3 pre-default wealth of φ. Consequently, the process Vt (φ) := Vt (φ)(Yt3 )−1 = φ3 is termed the relative t pre-default wealth. Using Proposition 5.2.1, with suitably modified notation, we find that the F-adapted process V 3 (φ) satisfies, for every t ∈ [0, T ], Vt3 (φ) = V03 (φ) + t 0 ∗ φ2 dYu . u 86 Define a new probability on (Ω, FT ) by setting ∗ dQ∗ = ηT dP, ∗ ∗ ∗ where dηt = ηt θt dWt , and ∗ θt = Credit Risk, TUNIS 2005 µ2,t − µ1,t + σ3,t (σ1,t − σ2,t ) . σ1,t − σ2,t (5.31) The process Yt∗ , t ∈ [0, T ], is a (local) martingale under Q∗ . We shall require that this process is in fact a true martingale; a sufficient condition for this is that T 0 EQ Yt2,3 (σ2,t − σ1,t ) 2 dt < ∞. 3 From the predictable representation theorem, it follows that for any X ∈ FT , such that X(YT )−1 is 2 square-integrable under Q, there exists a constant x and an F-predictable process φ such that 3 X = YT T x+ 0 ∗ φ2 dYu u . (5.32) We now deduce from Proposition 5.2.2 that there exists a self-financing strategy φ with the predefault wealth Vt (φ) = Yt3 Vt3 for every t ∈ [0, T ], where we set Vt3 = x + t 0 ∗ φ2 dYu . u (5.33) Moreover, it satisfies the balance condition φ1 Yt1 + φ2 Yt2 = 0 for every t ∈ [0, T ]. Since clearly t t VT (φ) = X, we have that 3 3 1 1 VT (φ) = φ3 YT = 1 {T <τ } φ3 YT = 1 {T <τ } VT (φ) = 1 {T <τ } X, 1 T T and thus this strategy replicates the survival claim (X, 0, τ ). In fact, we have that Vt (φ) = 0 on the random interval [[τ, T ]]. Definition 5.3.1 We say that a survival claim (X, 0, τ ) is attainable if the process V 3 given by (5.33) is a martingale under Q∗ . The following result is an immediate consequence of (5.32) and (5.33). 3 Corollary 5.3.1 Let X ∈ FT be such that X(YT )−1 is square-integrable under Q∗ . Then the survival claim (X, 0, τ ) is attainable. Moreover, the pre-default price πt (X, 0, τ ) of the claim (X, 0, τ ) is given by the conditional expectation 3 πt (X, 0, τ ) = Yt3 EQ (X(YT )−1 | Ft ), ∀ t ∈ [0, T ]. (5.34) The process π(X, 0, τ )(Y 3 )−1 is an F-martingale under Q. 3 Proof: Since X(YT )−1 is square-integrable under Q, we know from the predictable representa- tion theorem that φ2 in (5.32) is such that EQ T 0 (φ2 )2 d Y ∗ t t < ∞, so that the process V 3 given by (5.33) is a true martingale under Q. We conclude that (X, 0, τ ) is attainable. Now, let us denote by πt (X, 0, τ ) the time-t price of the claim (X, 0, τ ). Since φ is a hedging portfolio for (X, 0, τ ) we thus have Vt (φ) = πt (X, 0, τ ) for each t ∈ [0, T ]. Consequently, 3 1 1 {τ >t} πt (X, 0, τ ) = 1 {τ >t} Vt (φ) = 1 {τ >t} Yt3 EQ (VT | Ft ) 1 1 3 = 1 {τ >t} Yt3 EQ (X(YT )−1 | Ft ) 1 for each t ∈ [0, T ]. This proves equality (5.34). In view of the last result, it is justified to refer to Q as the pricing measure relative to Y 3 for attainable survival claims. M. Jeanblanc 87 Remark 5.3.3 It can be proved that there exists a unique absolutely continuous probability mea¯ sure Q on (Ω, GT ) such that we have Yt3 EQ ¯ 1 {τ >T } X 1 Gt 3 YT = 1 {τ >t} Yt3 EQ 1 X 3 YT Ft . However, this probability measure is not equivalent to Q, since its Radon-Nikod´m density vanishes y after τ (for a related result, see Collin-Dufresne et al. [21]). Example 5.3.2 We provide here an explicit calculation of the pre-default price of a survival claim. For simplicity, we assume that X = 1, so that the claim represents a defaultable zero-coupon bond. Also, we set γt = γ = const, µi,t = 0, and σi,t = σi , i = 1, 2, 3. Straightforward calculations yield the following pricing formula 1 2 π0 (1, 0, τ ) = Y03 e−(γ+ 2 σ3 )T . We see that here the pre-default price π0 (1, 0, τ ) depends explicitly on the intensity γ, or rather, on the drift term in dynamics of pre-default value of defaultable asset. Indeed, from the practical viewpoint, the interpretation of the drift coefficient in dynamics of Y 2 as the real-world default intensity is questionable, since within our set-up the default intensity never appears as an independent variable, but is merely a component of the drift term in dynamics of pre-default value of Y 3 . Note also that we deal here with a model with three tradeable assets driven by a one-dimensional Brownian motion. No wonder that the model enjoys completeness, but as a downside, it has an undesirable property that the pre-default values of all three assets are perfectly correlated. Consequently, the drift terms in dynamics of traded assets are closely linked to each other, in the sense, that their behavior under an equivalent change of a probability measure is quite specific. As we shall see later, if traded primary assets are judiciously chosen then, typically, the predefault price (and hence the price) of a survival claim will not explicitly depend on the intensity process. Remark 5.3.4 Generally speaking, we believe that one can classify a financial model as ‘realistic’ if its implementation does not require estimation of drift parameters in (pre-default) prices, at least for the purpose of hedging and valuation of a sufficiently large class of (defaultable) contingent claims of interest. It is worth recalling that the drift coefficients are not assumed to be market observables. Since the default intensity can formally interpreted as a component of the drift term in dynamics of pre-default prices, in a realistic model there is no need to estimate this quantity. From this perspective, the model considered in Example 5.3.2 may serve as an example of an ‘unrealistic’ model, since its implementation requires the knowledge of the drift parameter in the dynamics of Y 3 . We do not pretend here that it is always possible to hedge derivative assets without using the drift coefficients in dynamics of tradeable assets, but it seems to us that a good idea is to develop models in which this knowledge is not essential. Of course, a generic semimartingale model considered until now provides only a framework for a construction of realistic models for hedging of default risk. A choice of tradeable assets and specification of their dynamics should be examined on a case-by-case basis, rather than in a general semimartingale set-up. We shall address this important issue in the foregoing sections, in which we shall deal with particular examples of practically interesting defaultable claims. Hedging a recovery process Let us now briefly study the situation where the promised payoff equals zero, and the recovery payoff is paid at time τ and equals Zτ for some F-adapted process Z. Put another way, we consider a defaultable claim of the form (0, Z, τ ). Once again, we make use of Propositions 5.2.1 and 5.2.2. In view of (5.16), we need to find a constant x and an F-predictable process φ2 such that T ψT := − 0 Zt dY 1,3 = x + Yt1 t T 0 φ2 dYt∗ . t (5.35) 88 Credit Risk, TUNIS 2005 Similarly as in Section 5.3.1 we conclude that, under suitable integrability conditions on ψT , there exists φ2 such that dψt = φ2 dYt∗ , where ψt = EQ (ψT | Ft ). We now set t Vt3 = x + t 0 ∗ φ2 dYu + u T 0 3 Zu 1,3 Yu 1,3 dYu , 3 so that, in particular, VT = 0. Then it is possible to find processes φ1 and φ3 such that the strategy φ is self-financing and it satisfies: Vt (φ) = Vt3 Yt3 and Vt (φ) = Zt + φ3 Yt3 for every t ∈ [0, T ]. It is t thus clear that Vτ (φ) = Zτ on the set {τ ≤ T } and VT (φ) = 0 on the set {τ > T }. Bond market For the sake of concreteness, we assume that Yt1 = B(t, T ) is the price of a default-free ZC-bond with maturity T , and Yt3 = D(t, T ) is the price of a defaultable ZC-bond with zero recovery, that 3 1 is, an asset with the terminal payoff YT = 1 {T <τ } . We postulate that the dynamics under P of the default-free ZC-bond are dB(t, T ) = B(t, T ) µ(t, T ) dt + b(t, T ) dWt (5.36) for some F-predictable processes µ(t, T ) and b(t, T ). We choose the process Yt1 = B(t, T ) as a num´raire. Since the prices of the other two assets are not given a priori, we may choose any e probability measure Q equivalent to P on (Ω, GT ) to play the role of Q1 . In such a case, an e.m.m. Q1 is referred to as the forward martingale measure for the date T , and is denoted by QT . Hence, the Radon-Nikod´m density of QT with respect to P is given by (5.27) y for some F-predictable processes θ and ζ, and the process WtT = Wt − t θu du, 0 ∀ t ∈ [0, T ], is a Brownian motion under QT . Under QT the default-free ZC-bond is governed by dB(t, T ) = B(t, T ) µ(t, T ) dt + b(t, T ) dWtT where µ(t, T ) = µ(t, T ) + θt b(t, T ). Let Γ stand for the F-hazard process of τ under QT , so that Γt = − ln(1 − Ft ), where Ft = QT (τ ≤ t | Ft ). Assume that the hypothesis (H) holds under QT so that, in particular, the process Γ is increasing. We define the price process of a defaultable ZC-bond with zero recovery by the formula D(t, T ) := B(t, T )EQT (1 {T <τ } | Gt ) = 1 {t<τ } B(t, T ) EQT eΓt −ΓT Ft , 1 1 where the second equality follows from Lemma ??. It is then clear that Yt3,1 = D(t, T )(B(t, T ))−1 is a QT -martingale, and the pre-default price D(t, T ) equals D(t, T ) = B(t, T ) EQT eΓt −ΓT Ft . The next result examines the basic properties of the auxiliary process Γ(t, T ) given as, for every t ∈ [0, T ], Γ(t, T ) = Yt3,1 = D(t, T )(B(t, T ))−1 = EQT eΓt −ΓT Ft . The quantity Γ(t, T ) can be interpreted as the conditional probability (under QT ) that default will not occur prior to the maturity date T , given that we observe Ft and we know that the default has not yet happened. We will be more interested, however, in its volatility process β(t, T ) as defined in the following result. M. Jeanblanc 89 Lemma 5.3.1 Assume that the F-hazard process Γ of τ under QT is continuous. Then the process Γ(t, T ), t ∈ [0, T ], is a continuous F-submartingale and dΓ(t, T ) = Γ(t, T ) dΓt + β(t, T ) dWtT (5.37) for some F-predictable process β(t, T ). The process Γ(t, T ) is of finite variation if and only if the hazard process Γ is deterministic. In this case, we have Γ(t, T ) = eΓt −ΓT . Proof: We have Γ(t, T ) = EQT eΓt −ΓT | Ft = eΓt Lt , where we set Lt = EQT e−ΓT | Ft . Hence, Γ(t, T ) is equal to the product of a strictly positive, increasing, right-continuous, F-adapted process eΓt , and a strictly positive, continuous F-martingale L. Furthermore, there exists an F-predictable process β(t, T ) such that L satisfies dLt = Lt β(t, T ) dWtT o with the initial condition L0 = EQT e−ΓT . Formula (5.37) now follows by an application of Itˆ’s −Γt formula, by setting β(t, T ) = e β(t, T ). To complete the proof, it suffices to recall that a continuous martingale is never of finite variation, unless it is a constant process. Remark 5.3.5 It can be checked that β(t, T ) is also the volatility of the process Γ(t, T ) = EP eΓt −ΓT Ft . Assume that Γt = 0 γu du for some F-predictable, nonnegative process γ. Then we have the following auxiliary result, which gives, in particular, the volatility of the defaultable ZC-bond. Corollary 5.3.2 The dynamics under QT of the pre-default price D(t, T ) equals dD(t, T ) = D(t, T ) µ(t, T ) + b(t, T )β(t, T ) + γt dt + b(t, T ) + β(t, T ) d(t, T ) dWtT . t Equivalently, the price D(t, T ) of the defaultable ZC-bond satisfies under QT dD(t, T ) = D(t, T ) µ(t, T ) + b(t, T )β(t, T ) dt + d(t, T ) dWtT − dMt . where we set d(t, T ) = b(t, T ) + β(t, T ). Note that the process β(t, T ) can be expressed in terms of market observables, since it is simply the difference of volatilities d(t, T ) and b(t, T ) of pre-default prices of tradeable assets. Credit-risk-adjusted forward price Assume that the price Y 2 satisfies under the statistical probability P dYt2 = Yt2 µ2,t dt + σt dWt (5.38) 2 with F-predictable coefficients µ and σ. Let FY 2 (t, T ) = Yt2 (B(t, T ))−1 be the forward price of YT . For an appropriate choice of θ (see 5.31), we shall have that dFY 2 (t, T ) = FY 2 (t, T ) σt − b(t, T ) dWtT . Therefore, the dynamics of the pre-default synthetic asset Yt∗ under QT are dYt∗ = Yt2,3 σt − b(t, T ) dWtT − β(t, T ) dt , 90 and the process Yt = Yt2,1 e−αt satisfies dYt = Yt σt − b(t, T ) dWtT − β(t, T ) dt . Credit Risk, TUNIS 2005 Let Q be an equivalent probability measure on (Ω, GT ) such that Y (or, equivalently, Y ∗ ) is a Q-martingale. By virtue of Girsanov’s theorem, the process W given by the formula Wt = WtT − t β(u, T ) du, 0 ∀ t ∈ [0, T ], is a Brownian motion under Q. Thus, the forward price FY 2 (t, T ) satisfies under Q dFY 2 (t, T ) = FY 2 (t, T ) σt − b(t, T ) dWt + β(t, T ) dt . (5.39) It appears that the valuation results are easier to interpret when they are expressed in terms of forward prices associated with vulnerable forward contracts, rather than in terms of spot prices of primary assets. For this reason, we shall now examine credit-risk-adjusted forward prices of default-free and defaultable assets. Definition 5.3.2 Let Y be a GT -measurable claim. An Ft -measurable random variable K is called the credit-risk-adjusted forward price of Y if the pre-default value at time t of the vulnerable forward contract represented by the claim 1 {T <τ } (Y − K) equals 0. 1 Lemma 5.3.2 The credit-risk-adjusted forward price FY (t, T ) of an attainable survival claim (X, 0, τ ), represented by a GT -measurable claim Y = X1 {T <τ } , equals πt (X, 0, τ )(D(t, T ))−1 , where πt (X, 0, τ ) 1 is the pre-default price of (X, 0, τ ). The process FY (t, T ), t ∈ [0, T ], is an F-martingale under Q. Proof: The forward price is defined as an Ft -measurable random variable K such that the claim 1 {T <τ } (X1 {T <τ } − K) = X1 {T <τ } − KD(T, T ) 1 1 1 is worthless at time t on the set {t < τ }. It is clear that the pre-default value at time t of this claim equals πt (X, 0, τ ) − K D(t, T ). Consequently, we obtain FY (t, T ) = πt (X, 0, τ )(D(t, T ))−1 . Let us now focus on default-free assets. Manifestly, the credit-risk-adjusted forward price of the bond B(t, T ) equals 1. To find the credit-risk-adjusted forward price of Y 2 , let us write FY 2 (t, T ) := FY 2 (t, T ) eαT −αt = Yt2,1 eαT −αt , where α is given by (see (5.20)) t t αt = 0 σu − b(u, T ) β(u, T ) du = 0 σu − b(u, T ) d(u, T ) − b(u, T ) du. (5.40) Lemma 5.3.3 Assume that α given by (5.40) is a deterministic function. Then the credit-riskadjusted forward price of Y 2 equals FY 2 (t, T ) for every t ∈ [0, T ]. Proof: According to Definition 5.3.2, the price FY 2 (t, T ) is an Ft -measurable random variable 2 K, which makes the forward contract represented by the claim D(T, T )(YT − K) worthless on the 2 set {t < τ }. Assume that the claim YT − K is attainable.1 Since D(T, T ) = 1, from equation (5.34) it follows that the pre-default value of this claim is given by the conditional expectation 2 D(t, T ) EQ YT − K Ft . 1 Attainability of this claim can be shown in a similar way as the attainability of a vulnerable call option considered in Section 5.3.1. M. Jeanblanc 91 Consequently, 2 FY 2 (t, T ) = EQ YT Ft = EQ FY 2 (T, T ) Ft = FY 2 (t, T ) eαT −αt , as was claimed. It is worth noting that the process FY 2 (t, T ) is a (local) martingale under the pricing measure Q, since it satisfies dFY 2 (t, T ) = FY 2 (t, T )(σt − b(t, T )) dWt . (5.41) Under the present assumptions, the auxiliary process Y introduced in Proposition 5.2.3 and the credit-risk-adjusted forward price FY 2 (t, T ) are closely related to each other. Indeed, we have FY 2 (t, T ) = Yt eαT , so that the two processes are proportional. Vulnerable option on a default-free asset We shall now analyze a vulnerable call option with the payoff d 2 CT = 1 {T <τ } (YT − K)+ . 1 Here K is a constant. Our goal is to find a replicating strategy for this claim, interpreted as a 2 survival claim (X, 0, τ ) with the promised payoff X = CT = (YT − K)+ , where CT is the payoff of an equivalent non-vulnerable option. The method presented below is quite general, however, so 2 that it can be applied to any survival claim with the promised payoff X = G(YT ) for some function G : I → I satisfying the usual integrability assumptions. R R We assume that Yt1 = B(t, T ), Yt3 = D(t, T ) and the price of a default-free asset Y 2 is governed by (5.38). Then 1 2 2 d 1 1 CT = 1 {T <τ } (YT − K)+ = 1 {T <τ } (YT − KYT )+ . We are going to apply Proposition 5.2.3. In the present set-up, we have Yt2,1 = FY 2 (t, T ) and Yt = FY 2 (t, T )e−αt . Since a vulnerable option is an example of a survival claim, in view of Lemma d 5.3.2, its credit-risk-adjusted forward price satisfies FC d (t, T ) = Ct (D(t, T ))−1 . Proposition 5.3.2 Suppose that the volatilities σ, b and β are deterministic functions. Then the credit-risk-adjusted forward price of a vulnerable call option written on a default-free asset Y 2 equals FC d (t, T ) = FY 2 (t, T )N (d+ (FY 2 (t, T ), t, T )) − KN (d− (FY 2 (t, T ), t, T )) where d± (z, t, T ) = and v 2 (t, T ) = t (5.42) ln z − ln K ± 1 v 2 (t, T ) 2 v(t, T ) T (σu − b(u, T ))2 du. The replicating strategy φ in the spot market satisfies for every t ∈ [0, T ], on the set {t < τ }, d φ1 B(t, T ) = −φ2 Yt2 , φ2 = D(t, T )(B(t, T ))−1 N (d+ (t, T ))eαT −αt , φ3 D(t, T ) = Ct , t t t t where d+ (t, T ) = d+ (FY 2 (t, T ), t, T ). Proof: In the first step, we establish the valuation formula. Assume for the moment that the option is attainable. Then the pre-default value of the option equals, for every t ∈ [0, T ], d Ct = D(t, T ) EQ (FY 2 (T, T ) − K)+ Ft = D(t, T ) EQ (FY 2 (T, T ) − K)+ Ft . (5.43) 92 Credit Risk, TUNIS 2005 In view of (5.41), the conditional expectation above can be computed explicitly, yielding the valuation formula (5.42). To find the replicating strategy, and establish attainability of the option, we consider the Itˆ o differential dFC d (t, T ) and we identify terms in (5.33). It appears that dFC d (t, T ) = N (d+ (t, T )) dFY 2 (t, T ) = N (d+ (t, T ))eαT dYt = N (d+ (t, T ))Yt3,1 eαT −αt dYt∗ , so that the process φ2 in (5.32) equals φ2 = Yt3,1 N (d+ (t, T ))eαT −αt . t d Moreover, φ1 is such that φ1 B(t, T ) + φ2 Yt2 = 0 and φ3 = Ct (D(t, T ))−1 . It is easily seen that this t t t proves also the attainability of the option. (5.44) Let us examine the financial interpretation of the last result. First, equality (5.44) shows that it is easy to replicate the option using vulnerable forward contracts. Indeed, we have FC d (T, T ) = X = d C0 T D(0, T ) + 0 N (d+ (t, T )) dFY 2 (t, T ) d d and thus it is enough to invest the premium C0 = C0 in defaultable ZC-bonds of maturity T , and take at any instant t prior to default N (d+ (t, T )) positions in vulnerable forward contracts. It is understood that if default occurs prior to T , all outstanding vulnerable forward contracts become void. Second, it is worth stressing that neither the arbitrage price, nor the replicating strategy for a vulnerable option, depend explicitly on the default intensity. This remarkable feature is due to the fact that the default risk of the writer of the option can be completely eliminated by trading in defaultable zero-coupon bond with the same exposure to credit risk as a vulnerable option. In fact, since the volatility β is invariant with respect to an equivalent change of a probability measure, and so are the volatilities σ and b(t, T ), the formulae of Proposition 5.3.2 are valid for any choice of a forward measure QT equivalent to P (and, of course, they are valid under P as well). The only way in which the choice of a forward measure QT impacts these results is through the pre-default value of a defaultable ZC-bond. We conclude that we deal here with the volatility based relative pricing a defaultable claim. This should be contrasted with more popular intensity-based risk-neutral pricing, which is commonly used to produce an arbitrage-free model of tradeable defaultable assets. Recall, however, that if tradeable assets are not chosen carefully for a given class of survival claims, then both hedging strategy and pre-default price may depend explicitly on values of drift parameters, which can be linked in our set-up to the default intensity (see Example 5.3.2). 2 Remark 5.3.6 Assume that X = G(YT ) for some function G : I → I R R. Then the credit-riskadjusted forward price of a survival claim satisfies FX (t, T ) = v(t, FY 2 (t, T )), where the pricing function v solves the PDE 1 ∂t v(t, z) + (σt − b(t, T ))2 z 2 ∂zz v(t, z) = 0 2 with the terminal condition v(T, z) = G(z). The PDE approach is studied in Section 5.4 below. Remark 5.3.7 Proposition 5.3.2 is still valid if the driving Brownian motion is two-dimensional, rather than one-dimensional. In an extended model, the volatilities σt , b(t, T ) and β(t, T ) take values in I 2 and the respective products are interpreted as inner products in I 3 . Equivalently, one may R R prefer to deal with real-valued volatilities, but with correlated one-dimensional Brownian motions. M. Jeanblanc 93 Vulnerable swaption In this section, we relax the assumption that Y 1 is the price of a default-free bond. We now let Y 1 and Y 2 to be arbitrary default-free assets, with dynamics dYti = Yti µi,t dt + σi,t dWt , i = 1, 2. We still take D(t, T ) to be the third asset, and we maintain the assumption that the model is arbitrage-free, but we no longer postulate its completeness. In other words, we postulate the existence an e.m.m. Q1 , as defined in Section 5.3.1, but not the uniqueness of Q1 . Yt1,1 We take the first asset as a num´raire, so that all prices are expressed in units of Y 1 . In particular, e = 1 for every t ∈ I + , and the relative prices Y 2,1 and Y 3,1 satisfy under Q1 (cf. Proposition R 5.3.1) dYt2,1 = Yt2,1 (σ2,t − σ1,t ) dWt , 3,1 dYt3,1 = Yt− (σ3,t − σ1,t ) dWt − dMt . It is natural to postulate that the driving Brownian noise is two-dimensional. In such a case, we may represent the joint dynamics of Y 2,1 and Y 3,1 under Q1 as follows dYt2,1 = Yt2,1 (σ2,t − σ1,t ) dWt1 , 3,1 dYt3,1 = Yt− (σ3,t − σ1,t ) dWt2 − dMt , where W 1 , W 2 are one-dimensional Brownian motions under Q1 , such that d W 1 , W 2 a deterministic instantaneous correlation coefficient ρ taking values in [−1, 1]. t t t = ρt dt for We assume from now on that the volatilities σi , i = 1, 2, 3 are deterministic. Let us set αt = ln Y 2,1 , ln Y 3,1 = 0 ρu (σ2,u − σ1,u )(σ3,u − σ1,u ) du, (5.45) and let Q be an equivalent probability measure on (Ω, GT ) such that the process Yt = Yt2,1 e−αt is a Q-martingale. To clarify the financial interpretation of the auxiliary process Y in the present context, we introduce the concept of credit-risk-adjusted forward price relative to the num´raire Y 1 . e Definition 5.3.3 Let Y be a GT -measurable claim. An Ft -measurable random variable K is called the time-t credit-risk-adjusted Y 1 -forward price of Y if the pre-default value at time t of a vulnerable forward contract, represented by the claim 1 1 1 1 {T <τ } (YT )−1 (Y − KYT ) = 1 {T <τ } (Y (YT )−1 − K), 1 1 equals 0. The credit-risk-adjusted Y 1 -forward price of Y is denoted by FY |Y 1 (t, T ), and it is also interpreted as an abstract defaultable swap rate. The following auxiliary results are easy to establish, along the same lines as Lemmas 5.3.2 and 5.3.3. Lemma 5.3.4 The credit-risk-adjusted Y 1 -forward price of a survival claim Y = (X, 0, τ ) equals FY |Y 1 (t, T ) = πt (X 1 , 0, τ )(D(t, T ))−1 1 where X 1 = X(YT )−1 is the price of X in the num´raire Y 1 , and πt (X 1 , 0, τ ) is the pre-default e value of a survival claim with the promised payoff X 1 . Proof: It suffices to note that for Y = 1 {T <τ } X, we have 1 1 1 {T <τ } (Y (YT )−1 − K) = 1 {T <τ } X 1 − KD(T, T ), 1 1 1 where X 1 = X(YT )−1 , and to consider the pre-default values. 94 Credit Risk, TUNIS 2005 Lemma 5.3.5 The credit-risk-adjusted Y 1 -forward price of the asset Y 2 equals FY 2 |Y 1 (t, T ) = Yt2,1 eαT −αt = Yt eαT , where α is given by (5.45). Proof: It suffices to find an Ft -measurable random variable K for which 2 1 D(t, T ) EQ YT (YT )−1 − K Ft = 0. (5.46) Consequently, K = FY 2 |Y 1 (t, T ), where 2,1 FY 2 |Y 1 (t, T ) = EQ YT Ft = Yt2,1 eαT −αt = Yt eαT , where we have used the facts that Yt = Yt2,1 e−αt is a Q-martingale, and α is deterministic. We are in a position to examine a vulnerable option to exchange default-free assets with the payoff 2,1 d 1 2 1 1 CT = 1 {T <τ } (YT )−1 (YT − KYT )+ = 1 {T <τ } (YT − K)+ . 1 (5.47) The last expression shows that the option can be interpreted as a vulnerable swaption associated with the assets Y 1 and Y 2 . It is useful to observe that d 1 {T <τ } 1 CT 1 = 1 YT YT 2 YT 1 −K YT + , so that, when expressed in the num´raire Y 1 , the payoff becomes e 1,d 2,1 CT = D1 (T, T )(YT − K)+ , 1,d d where Ct = Ct (Yt1 )−1 and D1 (t, T ) = D(t, T )(Yt1 )−1 stand for the prices relative to Y 1 . It is clear that we deal here with a model analogous to the model examined in Sections 5.3.1 and 5.3.1 in which, however, all prices are now relative to the num´raire Y 1 . This observation allows us e to directly derive the valuation formula from Proposition 5.3.2. Proposition 5.3.3 The credit-risk-adjusted Y 1 -forward price of a vulnerable call option written with the payoff given by (5.47) equals FC d |Y 1 (t, T ) = FY 2 |Y 1 (t, T )N d+ (FY 2 |Y 1 (t, T ), t, T ) − KN d− (FY 2 |Y 1 (t, T ), t, T ) where d± (z, t, T ) = and v 2 (t, T ) = t ln z − ln K ± 1 v 2 (t, T ) 2 v(t, T ) T (σ2,u − σ1,u )2 du. The replicating strategy φ in the spot market satisfies for every t ∈ [0, T ], on the set {t < τ }, φ1 Yt1 = −φ2 Yt2 , t t φ2 = D(t, T )(Yt1 )−1 N (d+ (t, T ))eαT −αt , t d φ3 D(t, T ) = Ct , t where d+ (t, T ) = d+ FY 2 (t, T ), t, T . Proof: The proof is analogous to that of Proposition 5.3.2, and thus it is omitted. It is worth noting that the payoff (5.47) was judiciously chosen. Suppose instead that the option payoff is not defined by (5.47), but it is given by an apparently simpler expression d 2 1 CT = 1 {T <τ } (YT − KYT )+ . 1 (5.48) M. Jeanblanc d Since the payoff CT can be represented as follows d 1 2 3 3 2 1 CT = G(YT , YT , YT ) = YT (YT − KYT )+ , 95 where G(y1 , y2 , y3 ) = y3 (y2 − Ky1 )+ , the option can be seen an option to exchange the second asset for K units of the first asset, but with the payoff expressed in units of the defaultable asset. When expressed in relative prices, the payoff becomes 1,d 2,1 CT = 1 {T <τ } (YT − K)+ . 1 1 where 1 {T <τ } = D1 (T, T )YT . It is thus rather clear that it is not longer possible to apply the same 1 method as in the proof of Proposition 5.3.2. 5.3.2 Two defaultable assets with total default We shall now assume that we have only two assets, and both are defaultable assets with total default. This case is also examined by Carr [?], who studies some imperfect hedging of digital options. Note that here we present results for perfect hedging. We shall briefly outline the analysis of hedging of a survival claim. Under the present assumptions, we have, for i = 1, 2, i dYti = Yt− µi,t dt + σi,t dWt − dMt , (5.49) where W is a one-dimensional Brownian motion, so that Yt1 = 1 {t<τ } Yt1 , 1 with the pre-default prices governed by the SDEs dYti = Yti (µi,t + γt ) dt + σi,t dWt . (5.50) Yt2 = 1 {t<τ } Yt2 , 1 The wealth process V associated with the self-financing trading strategy (φ1 , φ2 ) satisfies, for every t ∈ [0, T ], Vt = Yt1 V01 + t 0 2,1 φ2 dYu , u where Yt2,1 = Yt2 /Yt1 . Since both primary traded assets are subject to total default, it is clear that the present model is incomplete, in the sense, that not all defaultable claims can be replicated. We shall check in Section 5.3.2 that, under the assumption that the driving Brownian motion W is one-dimensional, all survival claims satisfying natural technical conditions are hedgeable, however. In the more realistic case of a two-dimensional noise, we will still be able to hedge a large class of survival claims, including options on a defaultable asset (see Section 5.3.2) and options to exchange defaultable assets (see Section 5.3.2). Hedging a survival claim For the sake of expositional simplicity, we assume in this section that the driving Brownian motion W is one-dimensional. This is definitely not the right choice, since we deal here with two risky assets, and thus they will be perfectly correlated. However, this assumption is convenient for the expositional purposes, since it will ensure the model completeness with respect to survival claims, and it will be later relaxed anyway. We shall argue that in a model with two defaultable assets governed by (5.49), replication of a survival claim (X, 0, τ ) is in fact equivalent to replication of the promised payoff X using the pre-default processes. 96 Credit Risk, TUNIS 2005 Lemma 5.3.6 If a strategy φi , i = 1, 2, based on pre-default values Y i , i = 1, 2, is a replicating strategy for an FT -measurable claim X, that is, if φ is such that the process Vt (φ) = φ1 Yt1 + φ2 Yt2 t t satisfies, for every t ∈ [0, T ], dVt (φ) = VT (φ) = φ1 dYt1 + φ2 dYt2 , t t X, then for the process Vt (φ) = φ1 Yt1 + φ2 Yt2 we have, for every t ∈ [0, T ], t t dVt (φ) = φ1 dYt1 + φ2 dYt2 , t t VT (φ) = X1 {T <τ } . 1 This means that a strategy φ replicates a survival claim (X, 0, τ ). Proof: It is clear that Vt (φ) = 1 {t<τ } Vt (φ) = 1 {t<τ } Vt (φ). From 1 1 φ1 dYt1 + φ2 dYt2 = −(φ1 Yt1 + φ2 Yt2 ) dHt + (1 − Ht− )(φ1 dYt1 + φ2 dYt2 ), t t t t t t it follows that that is, φ1 dYt1 + φ2 dYt2 = −Vt (φ) dHt + (1 − Ht− )dVt (φ), t t 1 φ1 dYt1 + φ2 dYt2 = d(1 {t<τ } Vt (φ)) = dVt (φ). t t It is also obvious that VT (φ) = X1 {T <τ } . 1 Combining the last result with Lemma 5.2.1, we see that a strategy (φ1 , φ2 ) replicates a survival claim (X, 0, τ ) whenever we have 1 YT x + T 0 φ2 dYt2,1 = X t for some constant x and some F-predictable process φ2 , where, in view of (5.50), dYt2,1 = Yt2,1 µ2,t − µ1,t + σ1,t (σ1,t − σ2,t ) dt + (σ2,t − σ1,t ) dWt . We introduce a probability measure Q, equivalent to P on (Ω, GT ), and such that Y 2,1 is an Fmartingale under Q. It is easily seen that the Radon-Nikod´m density η satisfies, for t ∈ [0, T ], y · dQ | Gt = ηt dP | Gt = Et with θs dWs 0 dP | Gt (5.51) µ2,t − µ1,t + σ1,t (σ1,t − σ2,t ) , σ1,t − σ2,t provided, of course, that the process θ is well defined and satisfies suitable integrability conditions. 1 We shall show that a survival claim is attainable if the random variable X(YT )−1 is Q-integrable. Indeed, the pre-default value Vt at time t of a survival claim equals θt = 1 Vt = Yt1 EQ X(YT )−1 | Ft , and from the predictable representation theorem, we deduce that there exists a process φ2 such that 1 1 EQ X(YT )−1 | Ft = EQ X(YT )−1 + t 0 2,1 φ2 dYu . u The component φ1 of the self-financing trading strategy φ = (φ1 , φ2 ) is then chosen in such a way that φ1 Yt1 + φ2 Yt2 = Vt , ∀ t ∈ [0, T ]. t t TD[/F79.9TBT/F-fre59.96marye To conclude, by f7ETBT/F29.96Tf-7.nTf7./s280(f7ETBT/F29.96T92(b)27(y)-280(f7ETBT/F7.(b)2.96Tfu)]TJ/TD[(-291(e M. Jeanblanc 97 Option on a defaultable asset In order to get a complete model with respect to survival claims, we postulated in the previous section that the driving Brownian motion in dynamics (5.49) is one-dimensional. This assumption is questionable, since it implies the perfect correlation of risky assets. However, we may relax this restriction, and work instead with the two correlated one-dimensional Brownian motions. The model will no longer be complete, but options on a defaultable assets will be still attainable. The payoff of a (non-vulnerable) call option written on the defaultable asset Y 2 equals 2 2 CT = (YT − K)+ = 1 {T <τ } (YT − K)+ , 1 so that it is natural to interpret this contract as a survival claim with the promised payoff X = 2 (YT − K)+ . To deal with this option in an efficient way, we consider a model in which i dYti = Yt− µi,t dt + σi,t dWti − dMt , (5.52) where W 1 and W 2 are two one-dimensional correlated Brownian motions with the instantaneous 1 correlation coefficient ρt . More specifically, we assume that Yt1 = D(t, T ) = 1 {t<τ } D(t, T ) represents a defaultable ZC-bond with zero recovery, and Yt2 = 1 {t<τ } Yt2 is a generic defaultable asset with 1 total default. Within the present set-up, the payoff can also be represented as follows 1 2 2 1 CT = G(YT , YT ) = (YT − KYT )+ , where g(y1 , y2 ) = (y2 − Ky1 )+ , and thus it can also be seen as an option to exchange the second asset for K units of the first asset. The requirement that the process Yt2,1 = Yt2 (Yt1 )−1 follows an F-martingale under Q implies that dYt2,1 = Yt2,1 (σ2,t ρt − σ1,t ) dWt1 + σ2,t 1 − ρ2 dWt2 , (5.53) t 1 where W = (W 1 , W 2 ) follows a two-dimensional Brownian motion under Q. Since YT = 1, replica2 tion of the option reduces to finding a constant x and an F-predictable process φ satisfying T x+ 0 2 φ2 dYt2,1 = (YT − K)+ . t To obtain closed-form expressions for the option price and replicating strategy, we postulate that the volatilities σ1,t , σ2,t and the correlation coefficient ρt are deterministic. Let FY 2 (t, T ) = Yt2 (D(t, T ))−1 (FC (t, T ) = Ct (D(t, T ))−1 , respectively) stand for the credit-risk-adjusted forward price of the second asset (the option, respectively). The proof of the following valuation result is fairly standard, and thus it is omitted. Proposition 5.3.4 The credit-risk-adjusted forward price of the option written on Y 2 equals FC (t, T ) = FY 2 (t, T )N d+ (FY 2 (t, T ), t, T ) − KN d− (FY 2 (t, T ), t, T ) . Equivalently, the pre-default price of the option equals Ct = Yt2 N d+ (FY 2 (t, T ), t, T ) − K D(t, T )N d− (FY 2 (t, T ), t, T ) , where d± (z, t, T ) = and v 2 (t, T ) = t T ln zf − ln K ± 1 v 2 (t, T ) 2 v(t, T ) 2 2 (σ1,u + σ2,u − 2ρu σ1,u σ2,u ) du. Moreover the replicating strategy φ in the spot market satisfies for every t ∈ [0, T ], on the set {t < τ }, φ1 = −KN d− (FY 2 (t, T ), t, T ) , t φ2 = N d+ (FY 2 (t, T ), t, T ) . t 98 Option to exchange defaultable assets Credit Risk, TUNIS 2005 We work here with the two correlated one-dimensional Brownian motions, so that i dYti = Yt− µi,t dt + σi,t dWti − dMt , i = 1, 2, (5.54) where d W 1 , W 2 t = ρt dt for some function ρ with values in [−1, 1]. The model is no longer complete, but it is still not difficult to establish a direct counterpart of Proposition 5.3.4 for the 2 1 exchange option with the payoff (YT − KYT )+ . In fact, the next result shows that the pricing formula expressed in terms of pre-default prices has the same shape as the standard formula for the option to exchange non-defaultable assets with dynamics (5.49). It is notable that we do not need to make any assumption about the behavior of the default intensity. We only assume that the coefficients in (5.54) are such that there exist an e.m.m. for the process Y 2,1 , where dYti = Yti (µi,t + γt ) dt + σi,t dWti , i = 1, 2, (5.55) so that we implicitly impose mild technical conditions on drift coefficients. Proposition 5.3.5 Assume that the volatilities σ1 , σ2 and the instantaneous correlation coefficient ρ are deterministic. Then the pre-default price of the exchange option equals Ct = Yt2 N d+ (Yt2,1 , t, T ) − K Yt1 N d− (Yt2,1 , t, T ) , where d± (z, t, T ) = and v 2 (t, T ) = t T ln z − ln K ± 1 v 2 (t, T ) 2 v(t, T ) 2 2 σ1,u + σ2,u − 2ρu σ1,u σ2,u du. Moreover the replicating strategy φ in the spot market satisfies for every t ∈ [0, T ], on the set {t < τ }, φ1 = −KN d− (Yt2,1 , t, T ) , t φ2 = N d+ (Yt2,1 , t, T ) . t The pricing formula for the option on a defaultable asset (see Proposition 5.3.4) can be seen as a special case of the formula established in Proposition 5.3.5. Similarly as in Sections 5.3.1 and 5.3.1, we conclude that the pricing and hedging of any attainable 1 2 survival claim with the promised payoff X = g(YT , YT ) depends on the choice of a default intensity only through the pre-default prices Yt1 and Yt2 . This property shows that we have correctly specified the hedging instruments for a claim at hand. Of course, the model considered in this section is not complete, even if the concept of completeness is reduced to survival claims. Basically, a survival 2,1 1 claim can be hedged if its promised payoff can be represents as X = YT h(YT ). 5.4 PDE Approach to Valuation and Hedging In the remaining part of the paper, we take a different perspective, and we assume that trading occurs on the time interval [0, T ] and our goal is to replicate a contingent claim of the form 1 2 3 1 2 3 1 2 3 Y = 1 {T ≥τ } g1 (YT , YT , YT ) + 1 {T <τ } g0 (YT , YT , YT ) = G(YT , YT , YT , HT ), 1 1 which settles at time T . We do not need to assume here that the coefficients in dynamics of primary assets are F-predictable. Since our goal is to develop the PDE approach, it will be essential, however, to postulate a Markovian character of a model. For the sake of simplicity, we assume that the coefficients are constant, so that i dYti = Yt− µi dt + σi dWt + κi dMt , i = 1, 2, 3. M. Jeanblanc 99 The assumption of constancy of coefficients is rarely, if ever, satisfied in practically relevant models of credit risk. It is thus important to note that it was postulated here mainly for the sake of notational convenience, and the general results established in this section can be easily extended to a non1 2 3 1 2 3 homogeneous Markov case in which µi,t = µi (t, Yt− , Yt− , Yt− , Ht− ), σi,t = σi (t, Yt− , Yt− , Yt− , Ht− ), etc. 5.4.1 Defaultable asset with total default We first assume that Y 1 and Y 2 are default-free, so that κ1 = κ2 = 0, and the third asset is subject to total default, i.e. κ3 = −1, 3 dYt3 = Yt− µ3 dt + σ3 dWt − dMt . We work throughout under the assumptions of Proposition 5.3.1. This means that any Q1 -integrable 1 2 3 contingent claim Y = G(YT , YT , YT ; HT ) is attainable, and its arbitrage price equals 1 πt (Y ) = Yt1 EQ1 (Y (YT )−1 | Gt ), ∀ t ∈ [0, T ]. (5.56) The following auxiliary result is thus rather obvious. Lemma 5.4.1 The process (Y 1 , Y 2 , Y 3 , H) has the Markov property with respect to the filtration G 1 2 3 under the martingale measure Q1 . For any attainable claim Y = G(YT , YT , YT ; HT ) there exists a 3 1 2 3 function v : [0, T ] × I × {0, 1} → I such that πt (Y ) = v(t, Yt , Yt , Yt ; Ht ). R R We find it convenient to introduce the pre-default pricing function v(· ; 0) = v(t, y1 , y2 , y3 ; 0) and the post-default pricing function v(· ; 1) = v(t, y1 , y2 , y3 ; 1). In fact, since Yt3 = 0 if Ht = 1, it suffices to study the post-default function v(t, y1 , y2 ; 1) = v(t, y1 , y2 , 0; 1). Also, we write αi = µi − σi µ1 − µ2 , σ1 − σ2 b = (µ3 − µ1 )(σ1 − σ2 ) − (µ1 − µ3 )(σ1 − σ3 ). Let γ > 0 be the constant default intensity under P, and let ζ > −1 be given by formula (5.29). Proposition 5.4.1 Assume that the functions v(· ; 0) and v(· ; 1) belong to the class C1,2 ([0, T ] × I +, I R3 R). Then v(t, y1 , y2 , y3 ; 0) satisfies the PDE 2 ∂t v(· ; 0) + i=1 αi yi ∂i v(· ; 0) + (α3 + ζ)y3 ∂3 v(· ; 0) + b σ1 − σ2 1 σi σj yi yj ∂ij v(· ; 0) 2 i,j=1 3 − α1 v(· ; 0) + γ − v(t, y1 , y2 ; 1) − v(t, y1 , y2 , y3 ; 0) = 0 subject to the terminal condition v(T, y1 , y2 , y3 ; 0) = G(y1 , y2 , y3 ; 0), and v(t, y1 , y2 ; 1) satisfies the PDE 2 2 1 σi σj yi yj ∂ij v(· ; 1) − α1 v(· ; 1) = 0 ∂t v(· ; 1) + αi yi ∂i v(· ; 1) + 2 i,j=1 i=1 subject to the terminal condition v(T, y1 , y2 ; 1) = G(y1 , y2 , 0; 1). Proof: For simplicity, we write Ct = πt (Y ). Let us define ∆v(t, y1 , y2 , y3 ) = v(t, y1 , y2 ; 1) − v(t, y1 , y2 , y3 ; 0). Then the jump ∆Ct = Ct − Ct− can be represented as follows: 3 3 ∆Ct = 1 {τ =t} v(t, Yt1 , Yt2 ; 1) − v(t, Yt1 , Yt2 , Yt− ; 0) = 1 {τ =t} ∆v(t, Yt1 , Yt2 , Yt− ). 1 1 100 Credit Risk, TUNIS 2005 We write ∂i to denote the partial derivative with respect to the variable yi , and we typically omit 1 2 3 the variables (t, Yt− , Yt− , Yt− , Ht− ) in expressions ∂t v, ∂i v, ∆v, etc. We shall also make use of the fact that for any Borel measurable function g we have t 0 2 3 g(u, Yu , Yu− ) du = t 0 2 3 g(u, Yu , Yu ) du 3 3 since Yu and Yu− differ only for at most one value of u (for each ω). Let ξt = 1 {t<τ } γ. An 1 application of Itˆ’s formula yields o 3 dCt = ∂t v dt + i=1 ∂i v dYti + 1 j i σi σj Yt− Yt− ∂ij v dt 2 i,j=1 3 3 + ∆v + Yt− ∂3 v dHt 3 = ∂t v dt + i=1 ∂i v dYti + 1 j i σi σj Yt− Yt− ∂ij v dt 2 i,j=1 3 3 + ∆v + Yt− ∂3 v dMt + ξt dt , and this in turn implies that 3 dCt = ∂t v dt + i=1 i Yt− ∂i v µi dt + σi dWt + 1 j i σi σj Yt− Yt− ∂ij v dt 2 i,j=1 3 3 + ∆v dMt + ∆v + Yt− ∂3 v ξt dt   3 3   1 j 3 i i ∂t v + µi Yt− ∂i v + σi σj Yt− Yt− ∂ij v + ∆v + Yt− ∂3 v ξt dt =   2 i=1 i,j=1 3 + i=1 i σi Yt− ∂i v dWt + ∆v dMt . We now use the integration by parts formula together with (5.23) to derive dynamics of the relative price Ct = Ct (Yt1 )−1 . We find that 2 dCt = Ct− (−µ1 + σ1 ) dt − σ1 dWt   3 3   1 j 1 i 3 i + (Yt− )−1 ∂t v + µi Yt− ∂i v + σi σj Yt− Yt− ∂ij v + ∆v + Yt− ∂3 v ξt dt   2 i,j=1 i=1 3 1 + (Yt− )−1 i=1 i 1 1 σi Yt− ∂i v dWt + (Yt− )−1 ∆v dMt − (Yt− )−1 σ1 i=1 3 i σi Yt− ∂i v dt. Hence, using (5.28), we obtain 2 dCt = Ct− − µ1 + σ1 dt + Ct− − σ1 dWt − σ1 θ dt   3 3   1 j i 1 i 3 µi Yt− ∂i v + + (Yt− )−1 ∂t v + σi σj Yt− Yt− ∂ij v + ∆v + Yt− ∂3 v ξt dt   2 i=1 i,j=1 3 1 + (Yt− )−1 i=1 i 1 σi Yt− ∂i v dWt + (Yt− )−1 i=1 3 1 1 1 + (Yt− )−1 ∆v dMt + (Yt− )−1 ζξt ∆v dt − (Yt− )−1 σ1 i=1 i σ i Yt− ∂i v dt. 3 i σi Yt− θ∂i v dt M. Jeanblanc 101 This means that the process C admits the following decomposition under Q1 2 dCt = Ct− − µ1 + σ1 − σ1 θ dt   3 3   1 j 1 i i 3 + (Yt− )−1 ∂t v + µi Yt− ∂i v + σi σj Yt− Yt− ∂ij v + ∆v + Yt− ∂3 v ξt dt   2 i,j=1 i=1 3 1 + (Yt− )−1 i=1 3 1 − (Yt− )−1 σ1 i=1 i σi Yt− ∂i v dt + a Q1 -martingale. i 1 σi Yt− θ∂i v dt + (Yt− )−1 ζξt ∆v dt From (5.56), it follows that the process C is a martingale under Q1 . Therefore, the continuous finite variation part in the above decomposition necessarily vanishes, and thus we get 1 2 0 = Ct− (Yt− )−1 − µ1 + σ1 − σ1 θ   3 3   1 j 1 i i 3 + (Yt− )−1 ∂t v + µi Yt− ∂i v + σi σj Yt− Yt− ∂ij v + ∆v + Yt− ∂3 v ξt   2 i,j=1 i=1 3 1 + (Yt− )−1 i=1 1 1 i σi Yt− θ∂i v + (Yt− )−1 ζξt ∆v − (Yt− )−1 σ1 i=1 3 i σi Yt− ∂i v. Consequently, we have that 2 0 = Ct− − µ1 + σ1 − σ1 θ 3 3 + ∂t v + i=1 3 i µi Yt− ∂i v + 1 j i 3 σi σj Yt− Yt− ∂ij v + ∆v + Yt− ∂3 v ξt 2 i,j=1 3 i σi Yt− ∂i v. i=1 + i=1 i σi Yt− θ∂i v + ζξt ∆v − σ1 Finally, we conclude that 2 ∂t v + i=1 i αi Yt− ∂i v + (α3 + 3 ξt ) Yt− ∂3 v 1 j i + σi σj Yt− Yt− ∂ij v 2 i,j=1 3 − α1 Ct− + (1 + ζ)ξt ∆v = 0. Recall that ξt = 1 {t<τ } γ. It is thus clear that the pricing functions v(·, 0) and v(·; 1) satisfy the 1 PDEs given in the statement of the proposition. The next result deals with a replicating strategy for Y . Proposition 5.4.2 The replicating strategy φ for the claim Y is given by formulae 3 φ3 Yt− t 3 3 = −∆v(t, Yt1 , Yt2 , Yt− ) = v(t, Yt1 , Yt2 , Yt− ; 0) − v(t, Yt1 , Yt2 ; 1), 3 φ2 Yt2 (σ2 − σ1 ) t φ1 Yt1 t = −(σ1 − σ3 )∆v − σ1 v + i=1 i Yt− σi ∂i v, = v − φ2 Yt2 − φ3 Yt3 . t t Proof: As a by-product of our computations, we obtain 3 dCt = −(Yt1 )−1 σ1 v dWt + (Yt1 )−1 i=1 i σi Yt− ∂i v dWt + (Yt1 )−1 ∆v dMt . 102 Credit Risk, TUNIS 2005 The self-financing strategy that replicates Y is determined by two components φ2 , φ3 and the following relationship: 3,1 dCt = φ2 dYt2,1 + φ3 dYt3,1 = φ2 Yt2,1 (σ2 − σ1 ) dWt + φ3 Yt− (σ3 − σ1 ) dWt − dMt . t t t t 3,1 By identification, we obtain φ3 Yt− = (Yt1 )−1 ∆v and t 3 φ2 Yt2 (σ2 − σ1 ) − (σ3 − σ1 )∆v = −σ1 Ct + t i=1 i Yt− σi ∂i v. This yields the claimed formulae. Corollary 5.4.1 In the case of a total default claim, the hedging strategy satisfies the balance condition. Proof: A total default corresponds to the assumption that G(y1 , y2 , y3 , 1) = 0. We now have 3 3 v(t, y1 , y2 ; 1) = 0, and thus φ3 Yt− = v(t, Yt1 , Yt2 , Yt− ; 0) for every t ∈ [0, T ]. Hence, the equality t 1 1 2 2 φt Yt + φt Yt = 0 holds for every t ∈ [0, T ]. The last equality is the balance condition for Z = 0. Recall that it ensures that the wealth of a replicating portfolio jumps to zero at default time. Hedging with the savings account Let us now study the particular case where Y 1 is the savings account, i.e., dYt1 = rYt1 dt, Y01 = 1, which corresponds to µ1 = r and σ1 = 0. Let us write r = r + γ, where γ = γ(1 + ζ) = γ + µ3 − r + σ3 (r − µ2 ) σ2 stands for the intensity of default under Q1 . The quantity r has a natural interpretation as the riskneutral credit-risk adjusted short-term interest rate. Straightforward calculations yield the following corollary to Proposition 5.4.1. Corollary 5.4.2 Assume that σ2 = 0 and dYt1 = rYt1 dt, dYt2 = Yt2 µ2 dt + σ2 dWt , 3 dYt3 = Yt− µ3 dt + σ3 dWt − dMt . Then the function v(· ; 0) satisfies ∂t v(t, y2 , y3 ; 0) + ry2 ∂2 v(t, y2 , y3 ; 0) + ry3 ∂3 v(t, y2 , y3 ; 0) − rv(t, y2 , y3 ; 0) 1 + σi σj yi yj ∂ij v(t, y2 , y3 ; 0) + γv(t, y2 ; 1) = 0 2 i,j=2 with v(T, y2 , y3 ; 0) = G(y2 , y3 ; 0), and the function v(· ; 1) satisfies 1 2 2 ∂t v(t, y2 ; 1) + ry2 ∂2 v(t, y2 ; 1) + σ2 y2 ∂22 v(t, y2 ; 1) − rv(t, y2 ; 1) = 0 2 with v(T, y2 ; 1) = G(y2 , 0; 1). 3 M. Jeanblanc 103 In the special case of a survival claim, the function v(· ; 1) vanishes identically, and thus the following result can be easily established. 2 3 Corollary 5.4.3 The pre-default pricing function v(· ; 0) of a survival claim Y = 1 {T <τ } G(YT , YT ) 1 is a solution of the following PDE: ∂t v(t, y2 , y3 ; 0) + ry2 ∂2 v(t, y2 , y3 ; 0) + ry3 ∂3 v(t, y2 , y3 ; 0) + 1 σi σj yi yj ∂ij v(t, y2 , y3 ; 0) − rv(t, y2 , y3 ; 0) = 0 2 i,j=2 3 with the terminal condition v(T, y2 , y3 ; 0) = G(y2 , y3 ). The components φ2 and φ3 of the replicating strategy satisfy 3 φ2 σ2 Yt2 = t i=2 i 3 3 σi Yt− ∂i v(t, Yt2 , Yt− ; 0) + σ3 v(t, Yt2 , Yt− ; 0), 3 3 φ3 Yt− = v(t, Yt2 , Yt− ; 0). t 2 Example 5.4.1 Consider a survival claim Y = 1 {T <τ } g(YT ), that is, a vulnerable claim with 1 default-free underlying asset. Its pre-default pricing function v(· ; 0) does not depend on y3 , and satisfies the PDE (y stands here for y2 and σ for σ2 ) 1 ∂t v(t, y; 0) + ry∂2 v(t, y; 0) + σ 2 y 2 ∂22 v(t, y; 0) − rv(t, y; 0) = 0 2 with the terminal condition v(T, y; 0) = 1 {t<τ } g(y). The solution to (5.57) is 1 v(t, y) = e(r−r)(t−T ) v r,g,2 (t, y) = eγ(t−T ) v r,g,2 (t, y), (5.57) where the function v r,g,2 is the Black-Scholes price of g(YT ) in a Black-Scholes model for Yt with interest rate r and volatility σ2 . 5.4.2 Defaultable asset with non-zero recovery 3 dYt3 = Yt− (µ3 dt + σ3 dWt + κ3 dMt ) We now assume that with κ3 > −1 and κ3 = 0. We assume that Y03 > 0, so that Yt3 > 0 for every t ∈ I + . We shall R briefly describe the same steps as in the case of a defaultable asset with total default. Arbitrage-free property As usual, we need first to impose specific constraints on model coefficients, so that the model is arbitrage-free. Indeed, an e.m.m. Q1 exists if there exists a pair (θ, ζ) such that θt (σi − σ1 ) + ζt ξt κ1 κi − κ1 = µ1 − µi + σ1 (σi − σ1 ) + ξt (κi − κ1 ) , 1 + κ1 1 + κ1 i = 2, 3. To ensure the existence of a solution (θ, ζ) on the set τ < t, we impose the condition σ1 − that is, µ1 (σ3 − σ2 ) + µ2 (σ1 − σ3 ) + µ3 (σ2 − σ1 ) = 0. µ1 − µ2 µ1 − µ3 = σ1 − , σ1 − σ2 σ1 − σ3 104 Now, on the set τ ≥ t, we have to solve the two equations θt (σ2 − σ1 ) = θt (σ3 − σ1 ) + ζt γκ3 = Credit Risk, TUNIS 2005 µ1 − µ2 + σ1 (σ2 − σ1 ), µ1 − µ3 + σ1 (σ3 − σ1 ). If, in addition, (σ2 − σ1 )κ3 = 0, we obtain the unique solution µ1 − µ2 µ1 − µ3 = σ1 − , σ1 − σ2 σ1 − σ3 ζ = 0 > −1, θ = σ1 − so that the martingale measure Q1 exists and is unique. Pricing PDE and replicating strategy We are in a position to derive the pricing PDEs. For the sake of simplicity, we assume that Y 1 is the savings account, so that Proposition 5.4.3 is a counterpart of Corollary 5.4.2. For the proof of Proposition 5.4.3, the interested reader is referred to Bielecki et al. [?]. Proposition 5.4.3 Let σ2 = 0 and let Y 1 , Y 2 , Y 3 satisfy dYt1 = rYt1 dt, dYt2 = Yt2 µ2 dt + σ2 dWt , 3 dYt3 = Yt− µ3 dt + σ3 dWt + κ3 dMt . Assume, in addition, that σ2 (r − µ3 ) = σ3 (r − µ2 ) and κ3 = 0, κ3 > −1. Then the price of a 2 3 contingent claim Y = G(YT , YT , HT ) can be represented as πt (Y ) = v(t, Yt2 , Yt3 , Ht ), where the pricing functions v(· ; 0) and v(· ; 1) satisfy the following PDEs ∂t v(t, y2 , y3 ; 0) + ry2 ∂2 v(t, y2 , y3 ; 0) + y3 (r − κ3 γ) ∂3 v(t, y2 , y3 ; 0) − rv(t, y2 , y3 ; 0) + and ∂t v(t, y2 , y3 ; 1) + ry2 ∂2 v(t, y2 , y3 ; 1) + ry3 ∂3 v(t, y2 , y3 ; 1) − rv(t, y2 , y3 ; 1) + 1 σi σj yi yj ∂ij v(t, y2 , y3 ; 1) = 0 2 i,j=2 3 1 σi σj yi yj ∂ij v(t, y2 , y3 ; 0) + γ v(t, y2 , y3 (1 + κ3 ); 1) − v(t, y2 , y3 ; 0) = 0 2 i,j=2 3 subject to the terminal conditions v(T, y2 , y3 ; 0) = G(y2 , y3 ; 0), The replicating strategy φ equals φ2 t = 1 σ2 Yt2 3 3 σi yi ∂i v(t, Yt2 , Yt− , Ht− ) i=2 v(T, y2 , y3 ; 1) = G(y2 , y3 ; 1). φ3 t = σ3 3 3 − v(t, Yt2 , Yt− (1 + κ3 ); 1) − v(t, Yt2 , Yt− ; 0) , σ2 κ3 Yt2 1 2 3 2 3 3 v(t, Yt , Yt− (1 + κ3 ); 1) − v(t, Yt , Yt− ; 0) , κ3 Yt− and φ1 is given by φ1 Yt1 + φ2 Yt2 + φ3 Yt3 = Ct . t t t t M. Jeanblanc 105 Hedging of a survival claim We shall illustrate Proposition 5.4.3 by means of examples. First, consider a survival claim of the form 2 3 3 Y = G(YT , YT , HT ) = 1 {T <τ } g(YT ). 1 Then the post-default pricing function v g (· ; 1) vanishes identically, and the pre-default pricing function v g (· ; 0) solves the PDE ∂t v g (· ; 0) + ry2 ∂2 v g (· ; 0) + y3 (r − κ3 γ) ∂3 v g (· ; 0) + 1 σi σj yi yj ∂ij v g (· ; 0) − (r + γ)v g (· ; 0) = 0 2 i,j=2 3 with the terminal condition v g (T, y2 , y3 ; 0) = g(y3 ). Denote α = r − κ3 γ and β = γ(1 + κ3 ). It is not difficult to check that v g (t, y2 , y3 ; 0) = eβ(T −t) v α,g,3 (t, y3 ) is a solution of the above equation, where the function w(t, y) = v α,g,3 (t, y) is the solution of the standard Black-Scholes PDE equation 1 2 ∂t w + yα∂y w + σ3 y 2 ∂yy w − αw = 0 2 with the terminal condition w(T, y) = g(y), that is, the price of the contingent claim g(YT ) in the Black-Scholes framework with the interest rate α and the volatility parameter equal to σ3 . Let Ct be the current value of the contingent claim Y , so that Ct = 1 {t<τ } eβ(T −t) v α,g,3 (t, Yt3 ). 1 The hedging strategy of the survival claim is, on the event {t < τ }, φ3 Yt3 t φ2 Yt2 t = = − 1 −β(T −t) α,g,3 1 e v (t, Yt3 ) = − Ct , κ3 κ3 σ3 Yt3 e−β(T −t) ∂y v α,g,3 (t, Yt3 ) − φ3 Yt3 . t σ2 Hedging of a recovery payoff 2 3 As another illustration of Proposition 5.4.3, we shall now consider the contingent claim G(YT , YT , HT ) = 2 2 1 {T ≥τ } g(YT ), that is, we assume that recovery is paid at maturity and equals g(YT ). Let v g be 1 the pricing function of this claim. The post-default pricing function v g (· ; 1) does not depend on y3 . Indeed, the equation (we write here y2 = y) 1 2 ∂t v g (· ; 1) + ry∂y v g (· ; 1) + σ2 y 2 ∂yy v g (· ; 1) − rv g (· ; 1) = 0, 2 with v g (T, y; 1) = g(y), admits a unique solution v r,g,2 , which is the price of g(YT ) in the BlackScholes model with interest rate r and volatility σ2 . Prior to default, the price of the claim can be found by solving the following PDE ∂t v g (·; 0) + ry2 ∂2 v g (·; 0) + y3 (r − κ3 γ) ∂3 v g (·; 0) + 1 σi σj yi yj ∂ij v g (·; 0) − (r + γ)v g (·; 0) = −γv g (t, y2 ; 1) 2 i,j=2 3 with v g (T, y2 , y3 ; 0) = 0. It is not difficult to check that v g (t, y2 , y3 ; 0) = (1 − eγ(t−T ) )v r,g,2 (t, y2 ). The reader can compare this result with the one of Example 5.4.1. 106 Credit Risk, TUNIS 2005 5.4.3 Two defaultable assets with total default We shall now assume that we have only two assets, and both are defaultable assets with total default. We shall briefly outline the analysis of this case, leaving the details and the study of other relevant cases to the reader. We postulate that i dYti = Yt− µi dt + σi dWt − dMt , i = 1, 2, (5.58) so that 1 Yt1 = 1 {t<τ } Yt1 , with the pre-default prices governed by the SDEs dYti = Yti (µi + γ) dt + σi dWt , i = 1, 2. In the case where the promised payoff X is path-independent, so that 1 2 1 2 X1 {T <τ } = G(YT , YT )1 {T <τ } = G(YT , YT )1 {T <τ } 1 1 1 Yt2 = 1 {t<τ } Yt2 , 1 for some function G, it is possible to use the PDE approach in order to value and replicate survival claims prior to default (needless to say that the valuation and hedging after default are trivial here). We know already from the martingale approach that hedging of a survival claim X1 {T <τ } is 1 formally equivalent to replicating the promised payoff X using the pre-default values of tradeable assets dYti = Yti (µi + γ) dt + σi dWt , i = 1, 2. We need not to worry here about the balance condition, since in case of default the wealth of the portfolio will drop to zero, as it should in view of the equality Z = 0. We shall find the pre-default pricing function v(t, y1 , y2 ), which is required to satisfy the terminal condition v(T, y1 , y2 ) = G(y1 , y2 ), as well as the hedging strategy (φ1 , φ2 ). The replicating strategy φ is such that for the pre-default value C of our claim we have Ct := v(t, Yt1 , Yt2 ) = φ1 Yt1 + φ2 Yt2 , t t and dCt = φ1 dYt1 + φ2 dYt2 . (5.59) t t Proposition 5.4.4 Assume that σ1 = σ2 . Then the pre-default pricing function v satisfies the PDE ∂t v + y1 µ1 + γ − σ1 + µ2 − µ1 σ2 − σ1 ∂1 v + y2 µ2 + γ − σ2 µ2 − µ1 σ2 − σ1 ∂2 v µ2 − µ1 σ2 − σ1 v 1 2 2 2 2 y σ ∂11 v + y2 σ2 ∂22 v + 2y1 y2 σ1 σ2 ∂12 v = 2 1 1 µ1 + γ − σ1 with the terminal condition v(T, y1 , y2 ) = G(y1 , y2 ). Proof: We shall merely sketch the proof. By applying Itˆ’s formula to v(t, Yt1 , Yt2 ), and como paring the diffusion terms in (5.59) and in the Itˆ differential dv(t, Yt1 , Yt2 ), we find that o y1 σ1 ∂1 v + y2 σ2 ∂2 v = φ1 y1 σ1 + φ2 y2 σ2 , where φi = φi (t, y1 , y2 ). Since φ1 y1 = v(t, y1 , y2 ) − φ2 y2 , we deduce from (5.60) that y1 σ1 ∂1 v + y2 σ2 ∂2 v = vσ1 + φ2 y2 (σ2 − σ1 ), and thus φ2 y2 = y1 σ1 ∂1 v + y2 σ2 ∂2 v − vσ1 . σ2 − σ1 (5.60) M. Jeanblanc 107 On the other hand, by identification of drift terms in (5.60), we obtain ∂t v + y1 (µ1 + γ)∂1 v + y2 (µ2 + γ)∂2 v 1 2 2 2 2 + y σ ∂11 v + y2 σ2 ∂22 v + 2y1 y2 σ1 σ2 ∂12 v 2 1 1 φ1 y1 (µ1 + γ) + φ2 y2 (µ2 + γ). = Upon elimination of φ1 and φ2 , we arrive at the stated PDE. Recall that the historically observed drift terms are µi = µi + γ, rather than µi . The pricing PDE can thus be simplified as follows: ∂t v + y1 µ1 − σ1 + µ2 − µ1 σ2 − σ1 ∂1 v + y2 µ2 − σ2 µ2 − µ1 σ2 − σ1 ∂2 v . 1 2 2 µ2 − µ1 2 2 y1 σ1 ∂11 v + y2 σ2 ∂22 v + 2y1 y2 σ1 σ2 ∂12 v = v µ1 − σ1 2 σ2 − σ1 The pre-default pricing function v depends on the market observables (drift coefficients, volatilities, and pre-default prices), but not on the (deterministic) default intensity. To make one more simplifying step, we make an additional assumption about the payoff function. Suppose, in addition, that the payoff function is such that G(y1 , y2 ) = y1 g(y2 /y1 ) for some function g : I + → I (or equivalently, G(y1 , y2 ) = y2 h(y1 /y2 ) for some function h : I + → I R R R R). Then we may focus on relative pre-default prices Ct = Ct (Yt1 )−1 and Y 2,1 = Yt2 (Yt1 )−1 . The corresponding pre-default pricing function v(t, z), such that Ct = v(t, Yt2,1 ) will satisfy the PDE 1 ∂t v + (σ2 − σ1 )2 z 2 ∂zz v = 0 2 with terminal condition v(T, z) = g(z). If the price processes Y 1 and Y 2 in (5.49) are driven by the correlated Brownian motions W and W with the constant instantaneous correlation coefficient ρ, then the PDE becomes 1 2 2 ∂t v + (σ2 + σ1 − 2ρσ1 σ2 )z 2 ∂zz v = 0. 2 Consequently, the pre-default price Ct = Yt1 v(t, Yt2,1 ) will not depend directly on the drift coefficients µ1 and µ2 , and thus, in principle, we should be able to derive an expression the price of the claim in terms of market observables: the prices of the underlying assets, their volatilities and the correlation coefficient. Put another way, neither the default intensity nor the drift coefficients of the underlying assets appear as independent parameters in the pre-default pricing function. Before we conclude this work, let us stress once again that the martingale approach can be used in a fairly general set-up. By contrast, the PDE methodology is only suitable when dealing with a Markovian framework. In a forthcoming paper [?], we analyze a more general situation where a traded defaultable asset is a credit default swap, so that its dynamics involve also a continuous dividend stream. 108 Credit Risk, TUNIS 2005 Chapter 6 Indifference pricing 6.1 Defaultable Claims A defaultable claim (X1 , X2 , τ ) with maturity date T consists of: • The default time τ specifying the random time of default and thus also the default events {τ ≤ t} for every t ∈ [0, T ]. It is always assumed that τ is strictly positive with probability 1. • The promised payoff X1 , which represents the random payoff received by the owner of the claim at time T, if there was no default prior to or at time T . The actual payoff at time T associated with X1 thus equals X1 1 {τ >T } . We assume that X1 is an FT -measurable random 1 variable. • The recovery payoff X2 , where X2 is an FT -measurable random variable which is received by the owner of the claim at maturity, provided that the default occurs prior to or at maturity date T . In what follows, we shall denote by X = X1 1 T <τ + X2 1 τ ≤T the value of the defaultable contingent 1 1 claim at maturity. 6.1.1 Hodges Indifference Price In this section we discuss the concept of Hodges indifference price in our setup. The difference between our approach and the approach of Barrieu and El Karoui (see the corresponding chapter in the present volume) is that we study two different problems, corresponding to the choice of two different filtrations (i.e. the reference filtration and the full filtration). When considering Hodges indifference prices one starts with a given utility function, say u. Typically, u is assumed to be strictly increasing and strictly concave. We shall also apply a similar methodology in the case where u is assumed to be strictly convex (namely u(x) = x2 ) for quadratic hedging. In this case howevere one can not use the term indifference price and one solves a minimization problem. Problem (P): Optimization in the default-free market. The agent invests his initial wealth v > 0 in the default-free financial market using a self-financing strategy. The associated optimization problem is, v (P) : V(v) := sup EP u VT (φ) φ∈Φ(F ) , where the wealth process (Vt = Vtv (φ), t ≤ T ), is solution of dVt = rVt dt + φt (dSt − rSt dt), 109 V0 = v. (6.1) 110 Here Φ(F ) is the class of all F-adapted, self-financing trading strategies. Credit Risk, TUNIS 2005 X Problem (PF ): Optimization in the default-free market using F-adapted strategies and buying the defaultable claim. The agent buys the defaultable claim X at price p, and invests his remaining wealth v − p in the default-free financial market, using a trading strategy φ ∈ Φ(F ). The resulting global terminal wealth will be v−p,X v−p VT (φ) = VT (φ) + X. The associated optimization problem is v−p X F (PF ) : VX (v − p) := sup EP u VT (φ) + X φ∈Φ(F ) , where the process V v−p (φ) is solution of (6.1) with the initial condition V0v−p (φ) = v − p. We emphasize that the class Φ(F ) of admissible strategies is the same as in the problem (P), that is, we restrict here our attention to trading strategies that are adapted to the reference filtration F. X Problem (PG ): Optimization in the default-free market using G-adapted strategies and buying the defaultable claim. The agent buys the defaultable contingent claim X at price p, and invests the remaining wealth v − p in the financial market, using a strategy adapted to the enlarged filtration G. The associated optimization problem is v−p X G (PG ) : VX (v − p) := sup EP u VT (φ) + X φ∈Φ(G) , where Φ(G) is the class of all G-admissible trading strategies. Remark. It is easy to check that the solution of (PG ) : is the same as the solution of (P). Definition 6.1.1 For a given initial endowment v, the F-Hodges buying price of the defaultable claim X is the real number p∗ (v) such that F F V(v) = VX v − p∗ (v) . F G Similarly, the G-Hodges buying price of X is the real number p∗ (v) such that V(v) = VX v−p∗ (v) . G G φ∈Φ(G) v sup EP u VT (φ) , Remark 6.1.1 We can define the F-Hodges selling price pF (v) of X by considering −p, where p is ∗ the buying price of −X, as specified in Definition 6.1.1. If the contingent claim X is FT -measurable, then (See Rouge and ElKaroui (2000)) the F- and the G-Hodges selling and buying prices coincide with the hedging price of X, i.e., p∗ (v) = p∗ (v) = EP (ζT X) = EQ (X) = pG (v) = pF (v) , F G ∗ ∗ where we denote by ζ the deflator process ζt = ηt e−rt . M. Jeanblanc 111 6.2 Hodges prices relative to the reference filtration X In this section, we study the problem (PF ) (i.e., we use strategies adapted to the reference filtraF tion). First, we compute the value function, i.e., VX (v − p). Next, we establish a quasi-explicit representation for the Hodges price of X in the case of exponential utility. Finally, we compare the spread obtained via the risk-neutral valuation with the spread determined by the Hodges price of a defaultable zero-coupon bond. 6.2.1 X Solution of Problem (PF ) In view of the particular form of the defaultable claim X it follows that v−p,X v−p v−p VT (φ) = 1 {τ >T } (VT (φ) + X1 ) + 1 {τ ≤T } (VT (φ) + X2 ). 1 1 v−p Since the trading strategies are F-adapted, the terminal wealth VT (φ) is an FT -measurable random variable. Consequently, it holds that v−p,X EP u VT (φ) = v−p v−p 1 1 = EP u VT (φ) + X1 1 {τ >T } + u VT (φ) + X2 1 {τ ≤T } v−p v−p = EP EP u VT (φ) + X1 1 {τ >T } + u VT (φ) + X2 1 {τ ≤T } |FT 1 1 v−p v−p = EP u VT (φ) + X1 (1 − FT ) + u VT (φ) + X2 FT , X where FT = P {τ ≤ T | FT }. Thus, problem (PF ) is equivalent to the following problem: v−p X F (PF ) : VX (v − p) := sup EP JX VT (φ), · φ∈Φ(F ) , where JX (y, ω) = u(y + X1 (ω))(1 − FT (ω)) + u(y + X2 (ω))FT (ω), for every ω ∈ Ω and y ∈ I The real-valued mapping JX (·, ω) is strictly concave and increasing. ConR. −1 sequently, for any ω ∈ Ω, we can define the mapping IX (z, ω) by setting IX (z, ω) = JX (·, ω) (z) −1 for z ∈ I where (JX (·, ω)) denotes the inverse mapping of the derivative of JX with respect to R, the first variable. To simplify the notation, we shall usually suppress the second variable, and we shall write IX (·) in place of IX (·, ω). X The following lemma provides the form of the optimal solution for the problem (PF ), v−p,∗ X Lemma 6.2.1 The optimal terminal wealth for the problem (PF ) is given by VT = IX (λ∗ ζT ), ∗ P-a.s., for some λ such that v−p,∗ v − p = EP ζT VT . (6.2) v−p,X,∗ v−p,∗ Thus the optimal global wealth equals VT = VT +X = IX (λ∗ ζT )+X and the value function X of the objective criterion for the problem (PF ) is v−p,X,∗ F VX (v − p) = EP (u(VT )) = EP (u(IX (λ∗ ζT ) + X)). (6.3) Proof: It is well known (see, e.g., Karatzas and Shreve (1998)) that, in order to find the optimal wealth it is enough to maximize u(∆) over the set of square-integrable and FT -measurable random variables ∆, subject to the budget constraint, given by EP (ζT ∆) ≤ v − p. 112 Credit Risk, TUNIS 2005 The mapping JX (·) is strictly concave (for all ω). Hence, for every pair of FT -measurable random variables (∆, ∆∗ ) subject to the budget constraint, by tangent inequality, we have EP JX (∆) − JX (∆∗ ) ≤ EP (∆ − ∆∗ )JX (∆∗ ) . v−p,∗ given in the formulation of the Lemma we obtain For ∆∗ = VT v−p,∗ v−p,∗ ) ≤ λ∗ EP ζT (∆ − VT ) ≤ 0, EP JX (∆) − JX (VT where the last inequality follows from the budget constraint and the choice of λ∗ . Hence, for any φ ∈ Φ(F ), v−p v−p,∗ EP JX (VT (φ)) − JX (VT ) ≤ 0. To end the proof, it remains to observe that the first order conditions are also sufficient in the case of a concave criterion. Moreover, by virtue of strict concavity of the function JX , the optimal strategy is unique. 6.2.2 Exponential Utility: Explicit Computation of the Hodges Price For the sake of simplicity, we assume here that r = 0. Proposition 6.2.1 Let u(x) = 1 − exp(− x) for some > 0. Assume that the random variables ζT e− Xi , i = 1, 2 are P-integrable. Then the F-Hodges buying price is given by 1 p∗ (v) = − EP ζT ln (1 − FT )e− F where the FT -measurable random variable Ψ equals 1 Ψ = − ln (1 − FT )e− X1 X1 + F T e− X2 = EP (ζT Ψ), + F T e− X2 . (6.4) Thus, the F-Hodges buying price p∗ (v) is the arbitrage price of the associated claim Ψ. In addition, F the claim Ψ enjoys the following meaningful property EP u X − Ψ FT = 0. (6.5) Proof: In view of the form of the solution to the problem (P), we obtain 1 v,∗ VT = − ln µ∗ ζT . v,∗ The budget constraint EP (ζT VT ) = v implies that the Lagrange multiplier µ∗ satisfies 1 ln µ∗ 1 = − EP ζT ln ζT − v. (6.6) X The solution to the problem (PF ) is obtained in a general setting in Lemma 6.2.1. In the case of an exponential utility, we have (recall that the variable ω is suppressed) JX (y) = (1 − e− so that Thus, setting (y+X1 ) )(1 − FT ) + (1 − e− X1 (y+X2 ) )FT , JX (y) = e− y (e− A = e− X1 (1 − FT ) + e− X2 X2 FT ). Ψ (1 − FT ) + e− FT = e− , M. Jeanblanc 113 we obtain 1 IX (z) = − ln z A 1 = − ln z − Ψ. It follows that the optimal terminal wealth for the initial endowment v − p is 1 v−p,∗ VT = − ln λ∗ ζ T A 1 = − ln λ∗ − 1 ln ζT − Ψ, v−p,∗ where the Lagrange multiplier λ∗ is chosen to satisfy the budget constraint EP (ζT VT ) = v − p, that is, 1 ln λ∗ 1 = − EP ζT ln ζT − EP ζT Ψ − v + p. (6.7) ¿From definition, the F-Hodges buying price is a real number p∗ = p∗ (v) such that F v,∗ v−p EP exp(− VT ) = EP exp(− (VT ∗ ,∗ + X)) , where µ∗ and λ∗ are given by (6.6) and (6.7), respectively. After substitution and simplifications, we arrive at the following equality EP exp − It is easy to check that EP (ζT Ψ) − p∗ + X − Ψ (X−Ψ) = 1. (6.8) (6.9) EP e− FT = 1 so that equality (6.5) holds, and EP e p∗ (v) = EP (ζT Ψ). F − (X−Ψ) = 1. Combining (6.8) and (6.9), we conclude that We briefly provide the analog of (6.4) for the F-Hodges selling price of X . We have pF (v) = EP (ζT Ψ), ∗ where 1 Ψ = ln (1 − FT )e X1 + FT e X2 . (6.10) Remark 6.2.1 It is important to notice that the F-Hodges prices p∗ (v) and pF (v) do not depend ∗ F on the initial endowment v. This is an interesting property of the exponential utility function. In view of (6.5), the random variable Ψ will be called the indifference conditional hedge. From concavity of the logarithm function we obtain ln((1 − FT )e− X1 + FT e− X2 ) ≥ (1 − FT )(− X1 ) + FT (− X2 ). Hence, using that ζT is FT -measurable, p∗ (v) ≤ EP (ζT ((1 − FT )X1 + FT X2 )) = EQ (X). F Comparison with the Davis price. Let us present the results derived from the marginal utility pricing approach. The Davis price (see Davis (1997)) is given by d∗ (v) = In our context, this yields d∗ (v) = EP ζT X1 FT + X2 (1 − FT ) . v,∗ EP u VT X . V (v) In this case, the risk aversion has no influence on the pricing of the contingent claim. In particular, when F is deterministic, the Davis price reduces to the arbitrage price of each (default-free) financial asset X i , i = 1, 2, weighted by the corresponding probabilities FT and 1 − FT . 114 Credit Risk, TUNIS 2005 6.2.3 Risk-Neutral Spread Versus Hodges Spreads In our setting the price process of the T -maturity unit discount Treasury (default-free) bond is B(t, T ) = e−r(T −t) . Let us consider the case of a defaultable bond with zero recovery, i.e., X1 = 1 and X2 = 0. It follows from (6.10) that the F-Hodges buying and selling prices of the bond are (it will be convenient here to indicate the dependence of the Hodges price on maturity T ) 1 ∗ DF (0, T ) = − EP ζT ln(e− (1 − FT ) + FT ) and F D∗ (0, T ) = 1 EP ζT ln(e (1 − FT ) + FT ) , respectively. Let Q be a risk-neutral probability for the filtration G, that is, for the enlarged market. The “market” price at time t = 0 of defaultable bond, denoted as D0 (0, T ), is thus equal to the expectation under Q of its discounted pay-off, that is, D0 (0, T ) = EQ 1 {τ >T } RT = EQ (1 − FT )RT , 1 where Ft = Q {τ ≤ t | Ft } for every t ∈ [0, T ]. Let us emphasize that the risk-neutral probability Q is chosen by the market, via the price of the defaultable asset. The Hodges buying and selling spreads at time t = 0 are defined as S ∗ (0, T ) = − and S∗ (0, T ) = − 1 D∗ (0, T ) ln F T B(0, T ) 1 DF (0, T ) ln ∗ , T B(0, T ) 1 D0 (0, T ) ln . T B(0, T ) respectively. Likewise, the risk-neutral spread at time t = 0 is given as S 0 (0, T ) = − ∗ F Since DF (0, 0) = D∗ (0, 0) = D0 (0, 0) = 1, the respective backward short spreads at time t = 0 are given by the following limits (provided the limits exist) s∗ (0) = lim S ∗ (0, T ) = − T ↓0 ∗ d+ ln DF (0, T ) dT T =0 −r and s∗ (0) = lim S∗ (0, T ) = − T ↓0 F d+ ln D∗ (0, T ) dT T =0 − r, respectively. We also set s0 (0) = lim S 0 (0, T ) = − T ↓0 d+ ln D0 (0, T ) dT T =0 − r. Assuming, as we do, that the processes FT and FT are absolutely continuous with respect to the Lebesgue measure, and using the observation that the restriction of Q to FT is equal to Q, we find out that ∗ DF (0, T ) B(0, T ) = = 1 − EQ ln e− (1 − FT ) + FT 1 − EQ ln e− T T 1− 0 ft dt + 0 ft dt , M. Jeanblanc 115 and F D∗ (0, T ) B(0, T ) = = 1 1 EQ ln e (1 − FT ) + FT T T EQ ln e 1− 0 ft dt + 0 ft dt . Furthermore, D0 (0, T ) = EQ (1 − FT ) = EQ 1 − B(0, T ) Consequently, s∗ (0) = 1 e − 1 f0 , s∗ (0) = 1 T ft dt . 0 1 − e− f0 , and s0 (0) = f0 . Now, if we postulate, for instance, that s∗ (0) = s0 (0) (it would be the case if the market price is the selling Hodges price), then we must have f0 = 1 1 − e− f0 = 1 1 − e− γ0 so that γ0 < γ0 . Similar calculations can be made for any t ∈ [0, T [. It can be noticed that, if the market price is the selling Hodges price, f0 corresponds to the risk-neutral intensity at time 0 whereas γ0 is the historical intensity. The reader may refer to Bernis and Jeanblanc (2002) for other comments. 6.2.4 Recovery paid at time of default 3 Assume now that the recovery payment is made at time τ , if τ ≤ T . More precisely, let (Xt , t ≥ 0) 3 be some F-adapted process. If τ < T , the payoff Xt is paid at time t = τ and re-invested in the riskless asset. The terminal global wealth is now v−p v−p (VT (π) + X1 )1 T <τ + (VT (π) + Zτ )1 τ ≤T 1 1 3 where Zt = Xt er(T −t) , and we are still interested in optimization of wealth at time T . The corresponding optimization problem is v−p v−p Z (PF ) : V(v − p) := sup EP U (VT (φ) + X1 )1 T <τ + U (VT (φ) + Zτ )1 τ ≤T . 1 1 φ∈Φ(F ) The supremum part above can be written as φ∈Φ(F ) v−p sup EP J VT (φ) , where, for P-a.e. ω ∈ Ω, T J(y, ω) = U (y + X1 (ω))(1 − FT (ω)) + 0 U (y + Zt (ω))ft dt. Let us introduce the conditional indifference hedge: 1 Φ := − ln 0 T exp(− Zt )ft dt + exp(− X1 )(1 − FT ) . (6.11) We have the following result, 116 Credit Risk, TUNIS 2005 Th´or`me 6.1 Assume that sup0≤t≤T exp(− Zt ) and exp(− X 1 ) are Q-integrable. The Hodges e e price of (X 1 , X·3 ) is the arbitrage price of the indifference conditional hedgeΦ, the pay-off of which is given by (6.11). Z Proof: Observe first that problem (PF ) can be written as v−p V(x − p) = sup EP exp − [VT (φ) + Φ] φ∈Φ(F ) . Z X Thus, problem (PF ) is the same as problem (PF ) with X = Φ, so that finding the Hodges price 1 3 of (X , X· ) amounts to finding the Hodges price of Φ. But now, the claim Φ is a FT -measurable random variable. Thus, its Hodges price must coincide with its arbitrage price. Observe that Φ is a pay-off at time T . However, at time of default selling the derivative Φ yields enough money to obtain the utility needed. 6.3 Optimization Problems and BSDEs We now consider strategies φ that are predictable with respect to the full filtration G. The dynamics of the risky asset (St , t ≥ 0) are dSt = St (νdt + σdWt ). (6.12) In order to simplify notation, we denote by (ξt , t ≥ 0) the G-predictable process such that dMt = dHt − ξt dt is a G-martingale, i.e., ξt = γt (1 − Ht− ). (See equation (??).) We assume for simplicity that r = 0, so that now θ = ν/σ, and we change the definition of admissible portfolios to one that will be more suitable for problems considered here: instead of using the number of shares φ as before, we set π = φS, so that π represents the value invested in the risky asset. In addition, we adopt here the following relaxed definition of admissibility of trading strategies. Definition 6.3.1 The class Π(F) (Π(G), respectively) of F-admissible (G-admissible, respectively) trading strategies is the set of all F-adapted (G-predictable, respectively) processes π such that T 2 πt dt < ∞, P-a.s. 0 The wealth process of a strategy π satisfies dVt (π) = πt νdt + σdWt . (6.13) Let X be a given contingent claim, represented by a GT -measurable random variable. We shall study the following problem: sup π∈Π(G) v EP u VT (π) + X . in the case of the exponential utility. In a last step, for the determination of Hodges’ price, we shall change v into v − p. 6.3.1 Optimization Problem Our first goal is to solve an optimization problem for an agent who sells a claim X. To this end, it v suffices to find a strategy π ∈ Π(G) that maximizes EP (u(VT (π) + X)), where the wealth process v (Vt = Vt (π), t ≥ 0) (for simplicity, we shall frequently skip v and π from the notation) satisfies dVt = φt dSt = πt (νdt + σdWt ), V0 = v. M. Jeanblanc 117 > 0. Therefore, e . We consider the exponential utility function u(x) = 1 − e− x , with sup π∈Π(G) v EP u(VT (π) + X) = 1 − π∈Π(G) inf EP e v − VT (π) − X We shall give three different methods to solve inf π∈Π(G) EP e− Direct method v VT (π) − X e . We describe the idea of a solution; the idea follows the dynamic programming principle. Suppose that we can find a G-adapted process (Zt , t ≥ 0) with ZT = e− X , which depends v only on the claim X and parameters , σ, ν, and such that the process (e− Vt (π) Zt , t ≥ 0) is a (P, G)-submartingale for any admissible strategy π, and is a martingale under P for some admissible strategy π ∗ ∈ Π(G). Then, we would have EP (e− inf EP e− v VT (π) ZT ) ≥ e − v V0 (π) Z0 = e − v Z0 e = e − v Z0 , (6.14) for any π ∈ Π(G), with equality for some strategy π ∗ ∈ Π(G). Consequently, we would obtain v VT (π) − X π∈Π(G) e = EP e− v VT (π ∗ ) − X and thus we would be in the position to conclude that π ∗ is an optimal strategy. In fact, it will turn out that in order to implement the above idea we shall need to restrict further the class of G-admissible trading strategies to such strategies that the ”martingale part” in (6.16) determines a true martingale rather than a local-martingale. In what follows, we shall use the BSDE framework. We refer the reader to the chapter by ElKaroui and Hamad´ne in this volume and to the papers of Barles (1997), Rong (1997) and the e thesis of Royer (2002) for BSDE with jumps. We shall search the process Z in the class of all processes satisfying the following BSDE dZt = zt dt + zt dWt + zt dMt , t ∈ [0, T [, ZT = e− X , (6.15) where the process z = (zt , t ≥ 0) will be determined later (see equation (6.18) below). By applying Itˆ’s formula, we obtain o d(e− Vt ) = e− Vt 1 118 Credit Risk, TUNIS 2005 Note that with the above choice of the process z the drift term in (6.16) is positive for any admissible strategy π, and it is zero for π = π ∗ . Given the above, it appears that we have reduced our problem to the problem of solving the BSDE (6.15) with the process z given by (6.18), i.e.,  1 2  dZt = ( 1 θ2 Zt + θzt + 2Zt zt )dt + zt dWt + zt dMt , t ∈ [0, T ), 2 (6.19)  ZT = e − X . In fact, assuming that (6.19) admits a solution (Z, z, z), so that with π = π ∗ the ”martingale part” in (6.16) is a true martingale part rather than a local-martingale part, then the process ∗ πt = 1 zt θ+ , σ Zt will be an optimal portfolio, i.e., π∈Π(G) inf EP e− v VT (π) − X e = EP e− v VT (π ∗ ) − X e . However, this BSDE is not of standard. This is a BSDE with jumps, and existence theorems and comparison theorems are known only if the driver is Lipschitz. Hence, we shall establish the existence using another approach, an approach due to Mania and Tevzadze. Mania and Tevzadze approach In a very general setting, when the underlying asset is of the form dSt = dµt + λt d µ t where µ is a continuous local martingale, Mania and Tevzadze (2003a) study the family of processes T Vt (v) = max EP (U (v + φ t φs dSs )|Gt ) where v is a real-valued deterministic parameter. They establish that the process (V(t, v) = Vt (v), t ≥ 0) (which depends on the parameter v) is solution of a BSDE dV(t, v) = 1 1 (ϕv (t, v) + λt Vv (t, v))2 d µ t + ϕ(t, v)dµt + dNt (v), 2 Vvv (t, v) V(T, v) = U (v), (6.20) where N is a martingale orthogonal to µ, and the optimal portfolio is proved to be φ∗ = −St t ϕv (t, Vt∗ ) − λt Vv (t, Vt∗ ) . Vvv (t, Vt∗ ) Analysis of the proof of the equation (1.4) in Mania and Tevzadze (2003a) reveals that their results carry to the case when T Vt (v) = max E(U (v + φ t φs dSs + X)|Gt ) for a claim X satisfying appropriate integrability conditions, in which case the process (Vt (v), t ≥ 0) satisfies the BSDE (6.20) with terminal condition V(T, v) = U (v + X). We note however that there are several technical conditions postulated in Mania and Tevzadze (2003a) that need to be verified before their results can be adopted. M. Jeanblanc 119 In the particular case when the dynamics of the underlying asset follows dSt = St (νdt + σdWt ) we have dµt = St σdWt and λt = ν/(St σ 2 ), and the BSDE (6.20) reads dV(t, v) = = 2 St σ 2 ν (ϕ(t, v) + 2 Vv (t, v))2 dt + ϕ(t, v)St σdWt + dNt 2Vvv (t, v) σ St 1 (ϕ(t, v)σ 2 St + νVv (t, v))2 dt + ϕ(t, v)St σdWt + dNt 2σ 2 Vvv (t, v) t 0 where N is a martingale orthogonal to W (hence, in our setting a martingale of the form The terminal condition is V(T, v) = U (v + X) . and the optimal portfolio is φ∗ = −St t ϕv + Vv ν/(σ 2 St ) . Vvv ψs dMs ). Here, U is an exponential function. Thus, it is convenient to factorize process V as V(t, v) = e− v Zt , and to factorize process ϕ as ϕ(t, v) = ϕ(t)e− v . It follows that Z satisfies (ϕ(t) + dZt = Setting zt = ϕ(t)σSt , we get dZt = 1 ν (zt + Zt )2 dt + zt dWt + dNt , 2Zt σ ZT = e − X ν Zt )2 σ 2 St 2 St σ 2 dt + ϕ(t)St σdWt + dNt , 2Zt ZT = e− X . , which is exactly equation (6.18), where N is a stochastic integral w.r.t. the martingale M , orthogonal to W . Thus, it appears that a solution to equation (6.18) is given as Zt = e v V(t, v), The optimal portfolio is σzt + Zt ν σ 2 Zt which is exactly (6.17). zt = ϕ(t)σSt , and zt = dNt . dMt Remark 6.3.1 Analogous results follow from by Mania and Tevzadze (2003b) where a more general case of utility function is studied. Duality Approach We present now the duality approach (See for example Delbaen et al. (2002), or Mania and Tevzadze (2003b)). In the case dSt = St (νdt + σdWt ), the set of equivalent martingale measure (emm) is the set of probability measures Qψ defined as dQψ |Gt = Lt dP|Gt where dLt = Lt− (−θdWt + ψt dMt ) 120 Credit Risk, TUNIS 2005 where ψ is a G-predictable process, with ψ > −1 and θ is the risk premium θ = ν/σ. Indeed, using Kusuoka representation theorem (1999), we know that any strictly positive martingale can be written of the form dLt = Lt− ( t dWt + ψt dMt ) . The discounted price of the default-free asset is a martingale under the change of probability, hence, it is easy to check that t = −θ. (We have already noticed that the restriction of any emm to the t filtration F is equal to Q.) Let us denote by WtQ = Wt + θt and Mt = Mt − 0 ψs ξs ds. The processes W Q and M are Qψ martingales. Then, Lt 1 = exp −θWt − θ2 t + 2 = exp −θWtQ + θ2 t + 2 t t ln(1 + ψs )dHs − 0 t 0 t ψs ξs ds [(1 + ψs ) ln(1 + ψs ) − ψs ]ξs ds 0 ln(1 + ψs )dMs + 0 Hence, the relative entropy of Qψ with respect to P is H(Qψ |P) = EQψ (ln LT ) = EQψ 1 2 θ T+ 2 T [(1 + ψs ) ln(1 + ψs ) − ψs ]ξs ds 0 . From duality theory, the optimization problem π∈Π(G) inf EP e− v VT (π) − X e reduces to maximization over ψ of 1 EQψ (X − H(Qψ |P)), that is, maximization over ψ of EQψ X− 1 2 1 θ T− 2 T [(1 + ψs ) ln(1 + ψs ) − ψs ]ξs ds 0 . We solve this latter problem by operating dUt UT = 1 [(1 + ψt ) ln(1 + ψt ) − ψt ]ξt dt + ut dWtQ + ut dMt , 1 2 θ T. 2 = X− Setting Yt = Ut we obtain dYt YT = = ([(1 + ψt ) ln(1 + ψt ) − ψt ]ξt ) dt + yt dWtQ + yt dMt , 1 X − θ2 T. 2 In terms of the martingale M , we get dYt = ([(1 + ψt ) ln(1 + ψt ) − ψt (1 + yt )]ξt ) dt + yt dWtQ + yt dMt , The solution is obtained by maximization of the drift in the above equation w.r.t. ψ, which leads to 1 + ψs = ys . Consequently, the BSDE reads dYt = − eyt − 1 − yt ξt dt + yt dWtQ + yt dMt , 1 YT = X − θ2 T, 2 M. Jeanblanc ∗ and setting Zt = exp(−Yt ) we conclude that ∗ dZt = 121 1 ∗ 2 ∗ ∗ Z y dt − Zt yt dWtQ + Zt− (eyt − 1)dMt , 2 t t 1 ∗ ZT = exp(− X + θ2 T ), 2 ∗ ∗ or, denoting zt = −Zt yt , zt = Zt− (eyt − 1) ∗ dZt = 1 2 Q ∗ z dt + zt dWt + zt dMt , 2Zt t 1 2 1 ∗ ZT = exp(− X + θ2 T ), 2 (T −t) ∗ which is equivalent to (6.19). (Note that Zt = Zt e− 2 θ .) 6.3.2 Hodges Buying and Selling Prices Particular case: attainable claims Assume, as before, that r = 0 and let us check that the Hodges buying price is the hedging price in case of attainable claims. Assume that a claim X is FT -measurable. By virtue of the predictable representation theorem, there exists a pair (x, x), where x is a constant and xt is an F-adapted T Q process, such that X = x + 0 xu dWu , where WtQ = Wt + θt. Here x = EQ X is the arbitrage price of X and the replicating portfolio is obtained through x. Hence, the time t value of X is t Q Xt = x + 0 xu dWu . Then dXt = xt dWtQ and the process Zt = e−θ satisfies dZt = Zt = 1 2 1 θ + 2 2 2 2 xt 2 (T −t)/2 − Xt e dt + xt dWtQ ZT 1 (νZt + σ Zt xt )2 dt + Zt xt dWt , 2σ 2 Zt = e− X . X Hence (Zt , Zt xt , 0) is the solution of (6.19) with the terminal condition e− Z0 = e−θ Note that, for X = 0, we get Z0 = e−θ π∈Π(G) 2 2 , and T /2 − x e . T /2 , therefore v VT (π) inf EP (e− ) = e− v e−θ 2 T /2 . The G-Hodges buying price of X is the value of p such that π∈Π(G) inf EP e− v VT (π) = π∈Π(G) inf EP e− v−p (VT (π)+X) , that is, e− v e−θ We conclude easily that General case pG (X) ∗ 2 T /2 = e− (v−p+EQ X) −θ 2 T /2 e . = EQ X. Similar arguments show that p∗ (X) = EQ X. G Assume now that a claim X is GT -measurable. Assuming that the process Z introduced in (6.19) is strictly positive, we can use its logarithm. Let us denote ψt = Zt /zt =, ψt = Zt /zt = and κt = ψt ln(1 + ψt ) ≥ 0. 122 Then we get d(ln Zt ) = and thus d(ln Zt ) = where dMt = dMt + ξt (1 − κt ) dt = dHt − ξt κt dt. 1 2 1 2 Credit Risk, TUNIS 2005 θ2 dt + ψt dWtQ + ln(1 + ψt ) dMt + ξt (1 − κt ) dt , θ2 dt + ψt dWtQ + ln(1 + ψt ) dMt , The process M is a martingale under the probability measure Q defined as dQ|Gt = ηt dP|Gt , where η satisfies dηt = −ηt− θ dWt + ξt (1 − κt ) dMt with η0 = 1. Proposition 6.3.1 The G-Hodges buying price of X with respect to the exponential utility is the X 0 0 X real number p such that e− (v−p) Z0 = e− v Z0 , that is, p∗ (X) = −1 ln(Z0 /Z0 ) or, equivalently, G ∗ pG (X) = EQ X. Our previous study establishes that the dynamic hedging price of a claim X is the process Xt = EQ (X | Gt ). This price is the expectation of the payoff, under some martingale measure, as is any price in the range of no-arbitrage prices. Remark All the results presented in this section remain valid if ν and σ are adapted processes. 6.4 Quadratic Hedging dVtv (π) = πt (νdt + σdWt ), V0v (π) = v . We work under the same hypothesis as before; in particular, the wealth process follows In the last part of this section we shall study a more general case. The objective of this section is to examine the issue of quadratic pricing and hedging. Specifically, for a given P-square-integrable claim X ∈ GT , we study the following problems: • For a given initial endowment v, solve the minimization problem: v min EP ((VT (π) − X)2 ) . π A solution to this problem provides the portfolio which, among the portfolios with a given initial wealth, has the closest terminal wealth to a given claim X, in the sense of L2 -norm under the historical probability P. The solution of this problem exists, since the set of stochastic integrals of T the form 0 φs dSs is closed in L2 . • Solve the minimization problem: v min EP ((VT (π) − X)2 ) . π,v The optimal value of v is called the quadratic hedging price and the optimal π the quadratic hedging strategy. The quadratic hedging problem was examined in a fairly general framework of incomplete markets by means of BSDEs in several papers; see, for example, Mania (2000), Mania and Tevzadze (2003a), Bobrovnytska and Schweizer (2004), Hu and Zhou (2004) or Lim (2004). Since this list is by no means exhaustive, the interested reader is referred to the references quoted in the above-mentioned papers. The reader may refer to Bielecki et al. (2004b) for a study of the same problem under a constraint on the expectation. Also, some additional references can be found in that paper. M. Jeanblanc 123 6.4.1 Quadratic Hedging with F-Adapted Strategies v min EP ((VT (π) − X)2 ), We shall first solve, for a given initial endowment v, the following minimization problem π∈Π(F) where X is given as X = X1 1 {τ >T } + X2 1 {τ ≤T } 1 1 for some FT -measurable, P-square-integrable random variables X1 and X2 . Using the same approach as in Section 6.2.1, we define JX (y) = (y − X1 )2 (1 − FT ) + (y − X2 )2 FT and its derivative JX (y) = 2 [(y − X1 )(1 − FT ) + (y − X2 )FT ] = 2 [y − X1 (1 − FT ) − X2 FT ] . Hence, the inverse of JX (y) is IX (z) = 1 z + X1 (1 − FT ) + X2 FT 2 and thus the optimal terminal wealth equals v,∗ VT = 1 ∗ λ ζT + X1 (1 − FT ) + X2 FT , 2 where λ∗ is specified through the budget constraint: v,∗ EP (ζT VT ) = 1 ∗ 2 λ EP (ζT ) + EP (ζT X1 (1 − FT )) + EP (ζT X2 FT ) = v . 2 The optimal strategy is the one, which hedges the FT -measurable contingent claim λ∗ ζT + X1 (1 − FT ) + X2 FT = 2e−θ2 T (v − EQ (X))ζT + X1 (1 − FT ) + X2 FT . We deduce that v min EP ((VT − X)2 ) π = EP + EP 1 ∗ λ ζT + X1 (1 − FT ) + X2 FT − X1 2 2 (1 − FT ) 2 1 ∗ λ ζT + X1 (1 − FT ) + X2 FT ) − X2 2 2 EP (ζT ) + EP (X1 − X2 )2 FT (1 − FT ) 2 FT = = ∗ 2 1 4 (λ ) 1 v − EP (ζT (X1 + FT (X2 − X1 )) 2 2EP (ζT ) + EP ((X1 − X2 )2 FT (1 − FT )). It remains to minimize over v the right-hand side, which is now simple. Therefore, we obtain the following result. Proposition 6.4.1 If we restrict our attention to F-adapted strategies, the quadratic hedging price of the claim X = X1 1 {τ >T } + X2 1 {τ ≤T } equals 1 1 EP (ζT (X1 + FT (X2 − X1 )) = EQ (X1 (1 − FT ) + FT X2 ) . The optimal quadratic hedging of X is the strategy which replicates the FT -measurable contingent claim X1 (1 − FT ) + FT X2 . 124 Credit Risk, TUNIS 2005 Let us now examine the case of a generic GT -measurable random variable X. Here, we shall only examine the solution of the second problem introduced above, that is, v min EP ((VT (π) − X)2 ) . v,π As explained in Bielecki et al. (2004b), this problem is essentially equivalent to a problem where we restrict our attention to the terminal wealth so that we may reduce the problem to minV ∈FT EP ((V − X)2 ). From the properties of conditional expectations, we have V ∈FT min EP ((V − X)2 ) = EP ((EP (X | FT ) − X)2 ) and the initial value of the strategy with terminal value EP (X | FT ) is EP (ζT EP (X | FT )) = EP (ζT X). In essence, the latter statement is a consequence of the completeness of the default-free market model. Indeed, the fact that the conditional expectation EP (X | FT ) can be written as a stochastic integral w.r.t. S follows directly from the completeness of the default-free market. In conclusion, the quadratic hedging price equals EP (ζT X) = EQ X and the quadratic hedging strategy is the replicating strategy of the attainable claim EP (X | FT ) associated with X. 6.4.2 Quadratic Hedging with G-Adapted Strategies Similarly as in the previous subsection we assume here that the price process of the underlying asset obeys dSt = St (νdt + σdWt ). The wealth process follows dVtv (π) = πt (νdt + σdWt ), V0v (π) = v . We shall first solve, for a given initial endowment v, the following minimization problem π∈Π(G) v min EP ((VT (π) − X)2 ). As discussed in Bielecki et al. (2004b) one way of solving this problem is to project the random T variable X on the closed set of stochastic integrals of the form 0 ϕs dSs . Here, we present an alternative approach. We are looking for G-adapted processes X, Θ and Ψ such that the process Jt (π, v) = Vtv (π) − Xt Θt + Ψt , 2 ∀ t ∈ [0, T ], (6.21) is a G-submartingale for any G-adapted trading strategy π and a G-martingale for some strategy v π ∗ . In addition, we require that XT = X, ΘT = 1, ΦT = 0 so that JT (π, v) = (VT (π) − X)2 . Let us assume that the dynamics of these processes are of the form dXt dΘt dΨt = xt dt + xt dWt + xt dMt , = Θt− ϑt dt + ϑt dWt + ϑt dMt , = ψt dt + ψt dWt + ψt dMt , (6.22) (6.23) (6.24) where the drifts xt , ϑt and ψt are yet to be determined. From Itˆ’s formula, we obtain (recall that o ξt = γt 1 {τ >t} ) 1 d(Vt − Xt )2 = 2(Vt − Xt )(πt σ − xt ) dWt − 2(Vt − Xt− )xt dMt + (Vt − Xt− − xt )2 − (Vt − Xt− )2 dMt + 2(Vt − Xt )(πt ν − xt ) + (πt σ − xt )2 + ξt (Vt − Xt − xt )2 − (Vt − Xt )2 dt, M. Jeanblanc 125 where we denote Vt = Vtv (π). Then, using integration by parts formula, we obtain by straightforward calculations Jt (π) = k(t, πt , ϑt , xt , ψt )dt + martingale where k(t, πt , ϑt , xt , ψt ) = ψt + Θt ϑt (Vt − Xt )2 + 2(Vt − Xt ) (πt ν − xt ) + ϑt (πt σ − xt ) + ξt xt + (πt σ − xt )2 + ξt (ϑt + 1) (Vt − Xt − xt )2 − (Vt − Xt )2 . The process J(π) is a (local) martingale if and only if its drift term k(t, πt , xt , ϑt , ψt ) equals 0 for every t ∈ [0, T ]. ∗ In the first step, for any t ∈ [0, T ] we shall find πt such that the minimum of k(t, πt , xt , ϑt , ψt ) is attained. Subsequently, we shall choose the processes x = x∗ , ϑ = ϑ∗ and ψ = ψ ∗ in such a way that ∗ ∗ ∗ k(t, πt , x∗ , ϑ∗ , ψt ) = 0. This choice will imply that k(t, πt , x∗ , ϑ∗ , ψt ) ≥ 0 for any trading strategy π t t t t and any t ∈ [0, T ]. The strategy π ∗ which minimizes k(t, πt , xt , ϑt , ψt ) is the solution of the following equation: (Vtv (π) − Xt )(ν + ϑt σ) + σ(πt σ − xt ) = 0, Hence, the strategy π ∗ is implicitly given by ∗ πt = σ −1 xt − σ −2 (ν + ϑt σ)(Vtv (π ∗ ) − Xt ) = At − Bt (Vtv (π ∗ ) − Xt ), ∀ t ∈ [0, T ]. where we denote At = σ −1 xt , Bt = σ −2 (ν + ϑt σ). After some computations, we see that the drift term of the process J(π ∗ ) admits the following representation: 2 k(t, πt , ϑt , xt , ψt ) = ψt + Θt (Vt − Xt )2 (ϑt − σ 2 Bt ) + 2Θt (Vt − Xt ) σ 2 At Bt − ϑt xt − ϑt xt ξt − xt + Θt ξt (ϑt + 1)x2 . t From now on, we shall assume that the auxiliary processes ϑ, x and ψ are chosen as follows: 2 ϑt = ϑ∗ = σ 2 Bt , t xt = x∗ = σ 2 At Bt − ϑt xt − ϑt xt ξt , t ∗ ψt = ψt = −Θt ξt (ϑt + 1)x2 . t Straightforward computation verifies that if the drift coefficients ϑ, x, ψ in (6.22)-(6.24) are chosen as above, then the drift term in dynamics of J is always non-negative, and it is equal to 0 for ∗ πt = At − Bt (Vtv (π ∗ ) − Xt ). 2 Our next goal is to solve equations (6.22)-(6.24). Since ϑt = σ 2 Bt , the three-dimensional process (Θ, ϑ, ϑ) is the unique solution to the linear BSDE (6.23) dΘt = Θt σ −2 (ν + ϑt σ)2 dt + ϑt dWt + ϑt dMt , ΘT = 1. It is obvious that a solution is ϑt = 0, ϑt = 0, Θt = exp(−θ2 (T − t)), ∀ t ∈ [0, T ]. (6.25) The three-dimensional process (X, x, x) solves equation (6.22) with xt = x∗ = σ 2 At (ν/σ 2 ) = θxt . t This means that (X, x, x) is the unique solution to the linear BSDE dXt = θxt dt + xt dWt + xt dMt , XT = X. 126 Credit Risk, TUNIS 2005 The unique solution to the last equation is Xt = EQ (X | Gt ). The components x and x are given by the integral representation of the G-martingale (Xt , t ≥ 0) with respect to W Q and M , where WtQ = Wt + θt. Notice also that since ϑ = 0, the optimal portfolio π ∗ is given by the feedback formula ∗ πt = σ −1 xt − θ(Vtv (π ∗ ) − Xt ) . Finally, since ϑ = 0, we have ψt = −ξt x2 Θt . Therefore, we can solve explicitly the BSDE (6.24) for t the process Ψ. Indeed, we are now looking for a three-dimensional process (Ψ, ψ, ψ), which is the unique solution of the BSDE dΨt = −Θt ξt x2 dt + ψt dWt + ψt dMt , ΨT = 0. t Noting that the process Ψt + 0 t Θs ξs x2 ds s T 0 is a G-martingale under P with terminal value form: T Θs ξs x2 ds, we obtain the value of Ψ in a closed s Ψt = = EP t T t T Θs ξs x2 ds Gt s (T −s) e−θ e−θ 2 EP γs x2 1 {τ >s} Gt ds s1 EP γs x2 eΓt −Γs Ft ds s (6.26) = t 2 (T −s) where we have identified the process x with its F-adapted version (recall that any G-predictable process is equal, prior to default, to an F-predictable process). Substituting (6.25) and (6.26) in (6.21), we conclude that for a fixed v the value function for our ∗ problem is Jt (v) = Jt (π ∗ , v), where in turn Jt (π ∗ , v) = (Vtv (π ∗ ) − Xt )2 e−θ In particular, ∗ J0 (v) = e−θ 2 2 (T −t) T + 1 {τ >t} 1 e−θ 2 (T −s) t T 0 EP γs x2 eΓt −Γs Ft ds . s T (v − X0 )2 + EP eθ s γs x2 e−Γs ds s 2 . ∗ The quadratic hedging price, say v ∗ , is obtained by minimizing J0 (v) with respect to v. From the ∗ last formula, it is obvious that the quadratic hedging price is v = X0 = EQ X. We are in the position to formulate the main result of this section. A corresponding theorem for a default-free financial model was established by Kohlmann and Zhou (2000). Proposition 6.4.2 Let a claim X be GT -measurable and square-integrable under P. The optimal trading strategy π ∗ , which solves the quadratic problem π∈Π(G) v min EP ((VT (π) − X)2 ), is given by the feedback formula ∗ πt = σ −1 xt − θ(Vtv (π ∗ ) − Xt ) , where Xt = EQ (X | Gt ) for every t ∈ [0, T ], and the process xt is specified by dXt = xt dWtQ + xt dMt . The quadratic hedging price of X is EQ X. M. Jeanblanc 127 Example: Survival Claim Let us consider a simple survival claim X = 1 {τ >T } , and let us assume that Γ is deterministic, 1 t specifically, Γ(t) = 0 γ(s) ds. In that case, from the representation theorem (see Bielecki and Rutkowski (2002), Page 159), we have dXt = xt dMt with xt = −eΓ(t)−Γ(T ) . Hence T Ψt = = = = EP t T Θs ξs x2 ds Gt s Θs γ(s)1 {τ >s} e2Γ(s)−2Γ(T ) ds Gt 1 T t T t EP t 1 {τ >t} eΓ(t)−2Γ(T ) EP 1 1 {τ >t} eΓ(t)−2Γ(T ) 1 e−θ 2 2 (T −s) γ(s)eΓ(s) ds Ft e−θ (T −s) γ(s)eΓ(s) ds. One can check that, at time 0, the value function is indeed smaller that the one obtained with F-adapted portfolios. Case of an Attainable Claim Assume now that a claim X is FT -measurable. Then Xt = EQ (X | Gt ) is the price of X, and it satisfies dXt = xt dWtQ . The optimal strategy is, in a feedback form, ∗ πt = σ −1 xt − θ(Vt − Xt ) and the associated wealth process satisfies ∗ ∗ dVt = πt (νdt + σdWt ) = πt σ dWtQ = σ −1 σxt − ν(Vt − Xt ) dWtQ . Therefore, d(Vt − Xt ) = −θ(Vt − Xt ) dWtQ . Hence, if we start with an initial wealth equal to the arbitrage price EQ X of X, then we that Vt = Xt for every t ∈ [0, T ], as expected. Hodges Price v Let us emphasize that the Hodges price has no real meaning here, since the problem min EP ((VT )2 ) has no financial interpretation. We have studied in Bielecki et al. (2004b) a more pertinent problem, v with a constraint on the expected value of VT under P. Nevertheless, from a mathematical point of view, the Hodges price would be the value of p such that T 0 2 (v 2 − (v − p)2 ) = eθ s EP (γs x2 e−Γs )1 {τ >t} ds 1 s In the case of the example studied in Section 6.4.2, the Hodges price would be the non-negative value of p such that 2vp − p2 = e−2ΓT 0 T eθ s γs eΓs ds. 2 Let us also mention that our results are different from results of Lim (2004). Indeed, Lim studies a model with Poisson component, and thus in his approach the intensity of this process does not vanish after the first jump. 128 Credit Risk, TUNIS 2005 6.4.3 Jump-Dynamics of Price We assume here that the price process follows dSt = St− (νdt + σdWt + ϕdMt ), S0 > 0 where the constant ϕ satisfy ϕ > −1 so that the price St is strictly positive. Hence, the primary market, where the savings account and the asset S are traded is arbitrage free, but incomplete (in general). It follows that the wealth process follows dVtv (π) = πt (νdt + σdWt + ϕdMt ), V0v (π) = v. As in the previous subsection, our aim is, for a given initial endowment v, solve the minimization problem: v min EP ((VT (π) − X)2 ). π In order to characterize the value function we proceed analogously as before. That is, we are looking for processes X, Θ and Ψ such that the process (for simplicity we write Vt in place of Vtv (π)) J(t, Vt ) = (Vt − Xt )2 Θt + Ψt is a submartingale for any π and a martingale for some π ∗ , and such that ΨT = 0, XT = X, ΘT = 1. (Note that Mania and Tevzadze (2003a) did a similar approach for continuous processes, with a value function of the form Jt = Φ0 (t) + Φ1 (t)Vt + Φ2 (t)Vt2 .) Let us assume that the dynamics of these processes are of the form dXt dΘt dΨt = = = ft dt + xt dWt + xt dMt , Θt (ϑt dt + ϑt dWt + ϑt dMt ) ψt dt + ψt dWt + ψt dMt (6.27) (6.28) (6.29) where the drifts ft , ϑt and ψt have to be determined. From Itˆ’s formula we obtain o d(Vt − Xt )2 = 2(Vt − Xt )(πt σ − xt )dWt + (Vt + πt ϕ − Xt − xt )2 − (Vt − Xt )2 dMt + + 2(Vt − Xt )(πt µ − ft ) + (πt σ − xt )2 ξt (Vt + πt ϕ − Xt − xt )2 − (Vt − Xt )2 − 2(Vt − Xt )(πt ϕ − xt ) dt. Process Θt (Vt − Xt )2 + Ψt is a (local) martingale iff k(πt , ft , ϑt , ψt ) = 0 for all t, where k(π, ϑ, f, ψ) = ψ + Θt ϑt (Vt − Xt )2 + + + 2(Vt − Xt ) (πµ − f ) + ϑt (πσ − xt ) − ξt (πϕ − xt ) (πσ − xt )2 ξt (ϑt + 1) (Vt + πϕ − Xt − xt )2 − (Vt − Xt )2 . In the first step, we find π such that the maximum of k(π) is obtained. Then, one defines (f ∗ , ϑ∗ , ψ ∗ ) such that k(π , f ∗ , ϑ∗ , ψ ∗ ) = 0. This implies that, for any π, k(π, f ∗ , ϑ∗ , ψ ∗ ) ≤ 0, and that k(π , f ∗ , ϑ∗ , ψ ∗ ) = 0. The optimal π is the solution of (Vt − Xt )(µ − ξt ϕ + ϑt σ) + σ(πσ − xt ) + ξt (ϑt + 1)ϕ(Vt + πϕ − Xt − xt ) = 0 . M. Jeanblanc 129 hence πt = = with At Bt ∆t = = σxt + ξt ϕ(ϑt + 1)xt ∆−1 t µ + ϑt σ + ξt ϕϑt ∆−1 t 1 + + 1) At − Bt (Vt − Xt ) σ2 ϕ2 ξt (ϑt (σxt + ξt ϕ(ϑt + 1)xt ) − (µ + ϑt σ + ξt ϕϑt )(Vt − Xt ) = σ 2 + ϕ2 ξt (ϑt + 1) . After some computations the drift term of Θt (Vt − Xt ) + Ψt is found to be 2 Θt (Vt − Xt )2 (ϑt − Bt ∆t ) + 2Θt (Vt − Xt ) At Bt ∆t − ϑt xt − ξt ϑt xt − ft + Then, we choose ϑ∗ t ft∗ ∗ ψt Θt ξt (ϑt + 1)(At ϕ − xt )2 + Θt (At σ − xt )2 + ψt . 2 = B t ∆t = At Bt ∆t − ϕt xt − ξt ϑt xt = −Θt ξt (ϑt + 1)(At ϕ − xt )2 − Θt (At σ − xt )2 . Let us suppose that with this choice of drifts equations (6.28)–(6.29) admit solutions (we shall discuss this issue below). Next, let us denote these solutions as (Θ∗ , ϑ∗ , ϑ∗ ), (X ∗ , x∗ , x∗ ) and (Ψ∗ , ψ ∗ , ψ ∗ ); the corresponding processes A, B and ∆ will be denoted as A∗ , B ∗ and ∆∗ . Consequently, the ∗ drift term of Θ∗ (Vt∗ (π) − Xt ) + Ψ∗ is non-positive for any admissible π and it is equal to 0 for t t ∗ ∗ π ∗ = A∗ − Bt (Vtv,∗ (π ∗ ) − Xt ). t The three dimensional process (Θ∗ , ϑ∗ , ϑ∗ ) is supposed to satisfy the BSDE dΘt ΘT = = Θt 1. (µ + ϑt σ + ξt ϕϑt )2 σ 2 + ϕ2 ξt (ϑt + 1) dt + ϑt dWt + ϑt dMt (6.30) We shall discuss this equation later. The three dimensional process (X ∗ , x∗ , x∗ ) is a solution of the linear BSDE dXt XT where Thus, where dQκ |Gt = Lt dP|Gt and dLt (κ) (κ) = = 1 (κ1,t xt + κ2,t xt ) dt + xt dWt + xt dMt ∆t X κ1,t = σµ + σϕξt ϑt − ϕ2 ϑt ξt (1 + ϑt ), κ2,t = ϕξt (1 + ϑt )(µ + σ ϑt ) − σ 2 ξt ϑt . ∗ Xt = EQκ (X|Gt ), = −Lt− ( (κ) κ1,t κ2,t dWt + dMt ) . ∆t ξ∆t The three dimensional process (Ψ∗ , ψ ∗ , ψ ∗ ) is solution of dΨt ΨT = = −Θt ξt (ϑt + 1)(At ϕ − xt )2 + (At σ − xt )2 dt + ψt dWt + ψt dMt 0. 130 Thus, noting that Ψ∗ + t 0 Credit Risk, TUNIS 2005 t Θs ξs (ϑs + 1)(As ϕ − xs )2 + (As σ − xs )2 ds is a G-martingale, we obtain that Ψ∗ = E t T t Θs ξs (ϑs + 1)(As ϕ − xs )2 + (As σ − xs )2 ds|Gt . (6.31) Discussion of equation (6.30): Duality approach Our aim is here to prove that the BSDE (6.30) has a solution. We take the opportunuity to correct a mistake in Bielecki et al (2004b) where we claim that, in the particular case where the intensity γt is constant, we get a solution of the form θt constant. The solution that appear in Bielecki et al. is valid only in the case P (τ < T ) = 1. We proceed using duality approach. The set of equivalent martingale measure is determined by the set of densities. From Kusuoka (1999) representation theorem, it follows that any strictly positive martingale in the filtration G can be written as dLt = Lt− ( t dWt + χt dMt ) (6.32) for a G-predictable process χ satisfying χt > −1. In order that L corresponds to the Radon-Nikodym density of an emm, a relation between and χ has to be satisfied in order to imply that process Lt St is a P (local) martingale. (Recall that r = 0.) Straightforward application of integration by parts formula proves that the drift term of LS vanishes iff ϕχt ξt + σ t +ν =0 Recall that by definition the variance optimal measure for L is a probability measure Q∗ such that it minimizes EQ∗ (L2 ). At this moment we are unable to verify existence/uniqueness of such a measure T in the context of our model. We thus assume that the measure exists, Hypothesis: We assume that the variance optimal measure exists. In what follows we shall use the same argument as in Bobrovnytska and Schweizer (2004). Towards this end we denote by L∗ the Radon-Nikodym density of the variance optimal martingale measure. Let Z be the martingale Zt = EQ∗ (L∗ |Gt ) and U = L∗ /Z. It is proved in Delbaen and T Shachermayer (Lemma 2.2) that, if the variance optimal martingale measure exists, then there exists a predictable process z such that dZt /Zt− = zt dSt = zt (σdWt + ϕdMt + νdt) where zt = zt St− (in the proof of lemma 2.2, the hypothesis of continuity of the asset is not required). The process L∗ is a (P, G) martingale, hence there exist and χ such that dL∗ = L∗ ( t dWt + χt dMt ) t t− From Itˆ’s calculus, setting U = L∗ /Z, we obtain o dUt = Ut− At dt + ( t − zt σ)dWt + where At = = = 2 zt σ 2 + ξt (1 + χt )(zt ϕ + 2 zt σ 2 + ξt (1 + χt ) 2 zt 1 (χt + 1) − 1) dMt , 1 + zt ϕ 1 − 1) 1 + zt ϕ UT = 1, 2 z t ϕ2 1 + zt ϕ ϕ2 σ 2 + ξt (1 + χt ) 1 + zt ϕ . M. Jeanblanc 131 t We recall that ϕχt ξt + σ + ν = 0 . Hence, letting ut ut = = − zt σ 1 (χt + 1) − 1 , 1 + zt ϕ t we get zt = − It follows that At = = = = so that process U is a solution of dUt = Ut− ν + σut + ϕξt ut . σ 2 + ϕ2 ξt (1 + ut ) ϕ2 1 + zt ϕ ϕ2 1 + zt ϕ 2 zt σ 2 + ξt (1 + χt ) 2 zt σ 2 + ξt (1 + zt ϕ)(1 + ut ) 2 zt σ 2 + ξt (1 + ut )ϕ2 (ν + σut + ϕξt ut )2 σ 2 + ϕ2 ξt (1 + ut ) (ν + σut + ϕξt ut )2 dt + ut dWt + ut dMt , σ 2 + ϕ2 ξt (1 + ut ) UT = 1, which establishes that the BSDE (6.30) has a solution as long as the variance optimal martingale measure exists in our set-up. 6.5 6.6 MeanVariance Hedging Quantile Hedging 132 Credit Risk, TUNIS 2005 Chapter 7 Dependent Defaults and Credit Migrations Arguably, this is the most important and the most difficult research area with regard to credit risk and credit derivatives. We describe the case of conditionally independent default time, the copulabased approach, as well as the Jarrow and Yu [58] approach to the modeling of dependent stochastic intensities. We conclude by summarizing one of the approaches that were recently developed for the purpose of modeling term structure of corporate interest rates. Let us start by providing a tentative classification of issues and techniques related to dependent defaults and credit ratings. Valuation of basket credit derivatives covers, in particular: • Default swaps of type F (Duffie 1998b, Kijima and Muromachi 2000) – they provide a protection against the first default in a basket of defaultable claims. • Default swaps of type D (Kijima and Muromachi 2000) – a protection against the first two defaults in a basket of defaultable claims. • The ith -to-default claims (Bielecki and Rutkowski 2000) – a protection against the first i defaults in a basket of defaultable claims. Technical issues arising in the context of dependent defaults include: • Conditional independence of default times (Kijima and Muromachi [?]). • Simulation of correlated defaults (Duffie and Singleton 1998). • Modeling of infectious defaults (Davis and Lo 1999). • Asymmetric default intensities (Jarrow and Yu 2001). • Copulas (Sch¨nbucher and Schubert[?], Laurent and Gregory 2001). o • Dependent credit ratings (Lando 1998b, Bielecki and Rutkowski 2003). • Simulation of dependent credit migrations (Kijima et al. 2002, Bielecki 2002). 7.1 Basket Credit Derivatives Basket credit derivatives are credit derivatives deriving their cash flows values (and thus their values) from credit risks of several reference entities (or prespecified credit events). Standing assumptions. We assume that: 133 134 Credit Risk, TUNIS 2005 • We are given a collection of default times τ1 , . . . , τn defined on a common probability space (Ω, G, Q). • Q{τi = 0} = 0 and Q{τi > t} > 0 for every i and t. • Q{τi = τj } = 0 for arbitrary i = j (in a continuous time setup). We associate with the collection τ1 , . . . , τn of default times the ordered sequence τ(1) < τ(2) < · · · < τ(n) , where τ(i) stands for the random time of the ith default. Formally, τ(1) = min {τ1 , τ2 , . . . , τn } and for i = 2, . . . , n τ(i) = min τk : k = 1, . . . , n, τk > τ(i−1) . In particular, τ(n) = max {τ1 , τ2 , . . . , τn }. 7.1.1 The ith -to-Default Contingent Claims i We set Ht = 1 {τi ≤t} and we denote e592374(ClaimrT9592:2:2:2:2:2:2:2:2:2:26)-166(:)-167(;)-16J/F174.98Tf3.67-66J/FJ/F7 1 M. Jeanblanc 135 Values of FDC and LDC In general, the value at time t of a defaultable claim (X, Z, τ ) is given by the risk-neutral valuation formula St = Bt EQ ]t,T ] −1 Bu dDu Gt where D is the dividend process, which describes all the cash flows of the claim. Consequently, the value at time t of the FDC equals: St (1) = −1 1 1 Bt EQ Bτ1 Zτ1 1 {τ1 <τ2 , t<τ1 ≤T } Gt −1 2 1 +Bt EQ Bτ2 Zτ2 1 {τ2 <τ1 , t<τ2 ≤T } Gt −1 +Bt EQ BT X1 {T <τ(1) } Gt . 1 The value at time t of the LDC equals: St (2) = −1 1 Bt EQ Bτ1 Zτ1 1 {τ2 <τ1 , t<τ1 ≤T } Gt 1 −1 2 +Bt EQ Bτ2 Zτ2 1 {τ1 <τ2 , t<τ2 ≤T } Gt 1 −1 +Bt EQ BT X1 {T <τ(2) } Gt . 1 Both expressions above are merely special cases of a general formula. The goal is to derive more explicit representations under various assumptions about τ1 and τ2 , or to provide ways of efficient calculation of involved expected values by means of simulation (using perhaps another probability measure). 7.2 Conditionally Independent Defaults Relatively simple representations for prices of basket credit derivatives can be obtained under the assumption of conditional independence of default times. Definition 7.2.1 The random times τi , i = 1, . . . , n are said to be conditionally independent with respect to F under Q if for any T > 0 and any t1 , . . . , tn ∈ [0, T ] we have: n Q{τ1 > t1 , . . . , τn > tn | FT } = i=1 Q{τi > ti | FT }. Let us comment briefly on Definition 7.2.1. • Conditional independence has the following intuitive interpretation: the reference credits (credit names) are subject to common risk factors that may trigger credit (default) events. In addition, each credit name is subject to idiosyncratic risks that are specific for this name. • Conditional independence of default times means that once the common risk factors are fixed then the idiosyncratic risk factors are independent of each other. • The property of conditional independence is not invariant with respect to an equivalent change of a probability measure. • Conditional independence fits into static and dynamic theories of default times. 136 Credit Risk, TUNIS 2005 7.2.1 Canonical Construction Let Γi , i = 1, . . . , n be a given family of F-adapted, increasing, continuous processes, defined on ˜ ˆ ˆ ˆ a probability space (Ω, F, Q). We assume that Γi = 0 and Γi = ∞. Let (Ω, F, P) be an auxil∞ 0 iary probability space with a sequence ξi , i = 1, . . . , n of mutually independent random variables uniformly distributed on [0, 1]. We set τi (˜ , ω ) = inf { t ∈ I + : Γi (˜ ) ≥ − ln ξi (ˆ ) } ω ˆ R ω t ω ˆ ˜ ˆ ˆ on the product probability space (Ω, G, Q) = (Ω × Ω, F∞ ⊗ F, Q ⊗ P). We endow the space (Ω, G, Q) 1 n with the filtration G = F ∨ H ∨ · · · ∨ H . Proposition 7.2.1 The process Γi is the F-hazard process of τi : i 1 Q{τi > s | Ft ∨ Ht } = 1 {τi >t} EQ eΓt −Γs | Ft . i i We have Q{τi = τj } = 0 for every i = j. Moreover, default times τ1 , . . . , τn are conditionally independent with respect to F under Q. Recall that if Γi = t t 0 i γu du then γ i is the F-intensity of τi . Intuitively i i Q{τi ∈ [t, t + dt] | Ft ∨ Ht } ≈ 1 {τi >t} γt dt. 1 7.2.2 Independent Default Times We shall first examine the case of default times τ1 , . . . , τn that are mutually independent under Q. Suppose that for every k = 1, . . . , n we know the cumulative distribution function Fk (t) = Q{τk ≤ t} of the default time of the k th reference entity. The cumulative distribution functions of τ(1) and τ(n) are: n F(1) (t) = Q{τ(1) ≤ t} = 1 − and F(n) (t) = Q{τ(n) ≤ t} = (1 − Fk (t)) k=1 n Fk (t). k=1 More generally, for any i = 1, . . . , n we have n F(i) (t) = Q{τ(i) ≤ t} = m=i π∈Πm j∈π Fkj (t) l∈π (1 − Fkl (t)) where Πm denote the family of all subsets of {1, . . . , n} consisting of m elements. Suppose, in addition, that the default times τ1 , . . . , τn admit intensity functions γ1 (t), . . . , γn (t). It is easily seen that the default time τ(1) has the intensity function γ(1) (t) = γ1 (t) + . . . + γn (t) and for any t ∈ I + R Q{τ(1) > t} = e− t 0 γ(1) (v) dv . By direct calculations, it is also possible to find the intensity function of the ith default time. We do not necessarily need to assume that the reference filtration F is trivial, so that the case of random interest rates is also covered. M. Jeanblanc 137 Example 7.2.1 We shall consider a digital default put of basket type. To be more specific, we postulate that a contract pays a fixed amount (e.g., one unit of cash) at the ith default time τ(i) provided that τ(i) ≤ T. Assume that the interest rates are non-random. Then the value at time 0 of the contract equals −1 S0 = EQ Bτ 1 {τ(i) ≤T } = 1 ]0,T ] −1 Bu dF(i) (u). If τ1 , . . . , τn admit intensities then T S0 = 0 −1 Bu dF(i) (u) = T 0 −1 Bu γ(i) (u)e− u 0 γ(i) (v)dv du. 7.2.3 Signed Intensities Some authors (e.g., Kijima and Muromachi (2000)) examine credit risk models in which the negative values of intensities are not precluded. Negative values of the intensity process clearly contradict the interpretation of the intensity as the conditional probability of survival over an infinitesimal time interval. Nevertheless, the canonical construction of conditionally independent random times also works in this case. For a given collection Γi , i = 1, . . . , n of F-adapted continuous stochastic processes, with Γi = 0, defined on (Ω, F, P). We define τi , i = 1, . . . , n, on the enlarged probability space 0 (Ω, G, Q): τi = inf { t ∈ I + : Γi (ˆ ) ≥ − ln ξi (ˆ ) }. R ω t ω ˆt Let us denote Γi = max u≤t Γi . Observe that if the process Γi is absolutely continuous, than so it u i ˆ ˆ the process Γ ; in this case the intensity of τi is obtained as the derivative of Γi with respect to the time variable. 138 The following result examines the case of signed intensities. Credit Risk, TUNIS 2005 Lemma 7.2.1 Random times τi , i = 1, . . . , n are conditionally independent with respect to F under Q. In particular, for every t1 , . . . , tn ≤ T, n Q{τ1 > t1 , . . . , τn > tn | FT } = i=1 e−Γti = e− ˆi n i=1 ˆ Γi t i . 7.2.4 Valuation of FDC and LDC Valuation of the first-to-default or last-to-default contingent claim in relatively straightforward under the assumption of conditional independence of default times. We have the following result in which, for notational simplicity, we consider only the case of two entities. As usual, we do not state explicitly integrability conditions that should be imposed on recovery processes Z j and the terminal payoff X. Proposition 7.2.2 Let the default times τj , j = 1, 2 be F-conditionally independent with F-intensities γ j . Assume that Z j are F-predictable processes, and that the terminal payoff X is FT -measurable. Then the price at time t = 0 of the first-to-default claim equals S0 where we denote (1) 2 T = i,j=1, i=j EQ 0 −1 j −1 j Bu Zu e−Γu γu e−Γu du + EQ BT XG , i j G = e−(ΓT +ΓT ) = Q{τ1 > T, τ2 > T | FT }. 1 2 The price at time t = 0 of the last-to-default claim equals S0 where we denote H = 1 − (1 − e−ΓT )(1 − e−ΓT ) = 1 − Q{τ1 ≤ T, τ2 ≤ T | FT }. 2 1 (2) 2 T = i,j=1, i=j EQ 0 −1 −1 j j Bu Zu 1 − e−Γu γu e−Γu du + EQ BT XH , i j 7.2.5 General Valuation Formula We shall examine the case of a generic ith -to-default contingent claims. Recall that we have introduced the notation τ(1) < τ(2) < . . . < τ(n) for the ordered sequence of default times. M. Jeanblanc 139 Recall that according to our notational convention: • If the ith default occurs before or at the maturity date and τ(i) = τk for some k ∈ {1, . . . , n}, k k then an immediate recovery cash flow Zτ(i) = Zτk is received at time of the ith default. • The terminal promised payment occurs at the maturity date if the ith default does happen not prior to or at T. We assume that τ1 , . . . , τn are F-conditionally independent with stochastic intensities γ 1 , . . . , γ n . Then we have the following result (recall that, by convention, B0 = 0). Proposition 7.2.3 The price at time t = 0 of the ith -to-default claim equals S0 (i) n T = j=1 EQ 0 n −1 j j Bu Zu gij (u)γu e−Γu du ∞ T j + j=1 −1 EQ BT X j gij (u)γu e−Γu du , j where for every u ∈ I + R gij (u) = π∈Π(i,j) e − l∈π+ Γl u k∈π− 1 − e−Γu , k and where by Πi,j we denote the collection of specific partitions of the set {1, . . . , n}. Specifically, if π ∈ Π(i,j) then π = {π− , {j}, π+ }, where π− = {k1 , k2 , . . . , ki−1 }, and: j ∈ π− , j ∈ π+ , π− ∩ π+ = ∅ and π− ∪ π+ ∪ {j} = {1, . . . , n}. Consider, for instance, n = 2 credit entities. For i = 1 (i.e., in the case of the first-to-default claim) and j = 1, 2 we have Π(1,1) = ∅, {1}, {2} , Π(1,2) = ∅, {2}, {1} . π+ = {ki+1 , ki+2 , . . . , kn }, Likewise, in the case of the second-to-default claim, we have Π(2,1) = {2}, {1}, ∅ , Π(2,2) = {1}, {2}, ∅ . In this example, each set Π(i,j) contains only one partition; for example, the only element of Π(1,1) is the partition π = ∅, {1}, {2} . 7.2.6 Default Swap of Basket Type Let us consider a portfolio of n corporate bonds. The k th bond has the face value Lk and maturity Tk . Its price process is denoted by Dk (t, Tk ), k = 1, . . . , n. By τk we denote the default time of the k th bond, and, as usual, τ(i) stands for the random time of the ith default. We shall examine a default swap, which matures at some future date T < min {T1 , . . . , Tk } and whose covenants are described as follows. If τ(i) ≤ T, the contract holder (i.e., the protection buyer) receives at time τ(i) the recovery payment n Lk − Dk (τ(i) , Tk ) 1 {τ(i) =τk } . 1 k=1 This means that if the ith defaulting bond was issued by the k th reference entity, the recovery payment is based on the value of the k th bond only. A default swap premium in the amount κ is paid by the contract holder at each of prespecified time instants tp ≤ T, p = 1, 2, . . . , m prior to the ith default time or to the maturity T, whichever comes first. 140 Default Swap Premium Credit Risk, TUNIS 2005 We assume that all corporate bonds are subject to the fractional recovery of par value scheme. Specifically, δj is the constant recovery rate of j th bond. We also assume that each default time t j τj , j = 1, . . . , n, admits the F-intensity process γ j so that Γj = 0 γu du. Then, the following result t gives the value κ of the default swap premium, Proposition 7.2.4 The default swap premium κ = J1 /J2 where n T J1 = j=1 EQ Lj (1 − δj ) 0 −1 Bu π∈Π(i,j) k∈π− j j γu e−Γu 1 − e−Γu du k × e m n −1 EQ Btp p=1 j=1 − l l∈π+ Γu T tp − π∈Π(i,j) l l∈π+ Γu J2 = 1 − e−Γu k∈π− j k × e m n −1 EQ Btp p=1 j γu e−Γu du ∞ T π∈Π(i,j) k∈π− + 1 − e−Γu j j γu e−Γu k j=1 × e − l l∈π+ Γu du . 7.3 Copula-Based Approaches The concept of a copula function allows to produce various multidimensional probability distributions with prespecified univariate marginal laws. Definition 7.3.1 A function C : [0, 1]n → [0, 1] is called a copula if the following conditions are satisfied: (i) C(1, . . . , 1, vi , 1, . . . , 1) = vi for any i and any vi ∈ [0, 1], (ii) C is an n-dimensional cumulative distribution function (c.d.f.). Let us give few examples of copulas: • Product copula: Π(v1 , . . . , vn ) = Πn vi , i=1 • Gumbel copula: for θ ∈ [1, ∞) we set  C(v1 , . . . , vn ) = exp − i=1 n 1/θ  , (− ln vi )θ • Gaussian copula: n C(v1 , . . . , vn ) = NΣ N −1 (v1 ), . . . , N −1 (vn ) , n where NΣ is the c.d.f for the n-variate central normal distribution with the linear correlation matrix Σ, and N −1 is the inverse of the c.d.f. for the univariate standard normal distribution. • t-copula: C(v1 , . . . , vn ) = Θn t−1 (v1 ), . . . , t−1 (vn ) , ν,Σ ν ν where Θn is the c.d.f for the n-variate t-distribution with ν degrees of freedom and with the ν,Σ linear correlation matrix Σ, and t−1 is the inverse of the c.d.f. for the univariate t-distribution ν with ν degrees of freedom. M. Jeanblanc 141 The following theorem is the fundamental result underpinning the theory of copulas. Th´or`me 7.1 (Sklar) For any cumulative distribution function F on I n there exists a copula e e R function C such that F (x1 , . . . , xn ) = C(F1 (x1 ), . . . , Fn (xn )) where Fi is the ith marginal cumulative distribution function. If, in addition, F is continuous then C is unique. 7.3.1 Direct Application In a direct application, we first postulate a (univariate marginal) probability distribution for each random variable τi . Let us denote it by Fi for i = 1, 2, . . . , n. Then, a suitable copula function C is chosen in order to introduce an appropriate dependence structure of the random vector (τ1 , τ2 , . . . , τn ). Finally, the joint distribution of the random vector (τ1 , τ2 , . . . , τn ) is derived, specifically, Q{τi ≤ ti , i = 1, 2, . . . , n} = C F1 (t1 ), . . . , Fn (tn ) . In the finance industry, the most commonly used are elliptical copulas (such as the Gaussian copula and the t-copula). The direct approach has an apparent drawback. It is essentially a static approach; it makes no account of changes in credit ratings, and no conditioning on the flow of information is present. Let us mention, however, an interesting theoretical issue, namely, the study of the effect of a change of probability measures on the copula structure. 7.3.2 Indirect Application A less straightforward application of copulas is based on an extension of the canonical construction of conditionally independent default times. This can be considered as the first step towards a dynamic theory, since the techniques of copulas is merged with the flow of available information, in particular, the information regarding the observations of defaults. Assume that the cumulative distribution function of (ξ1 , . . . , ξn ) in the canonical construction (cf. Section 7.2.1) is given by an n-dimensional copula C, and that the univariate marginal laws are uniform on [0, 1]. Similarly as in Section 7.2.1, we postulate that (ξ1 , . . . , ξn ) are independent of F, and we set τi (˜ , ω ) = inf { t ∈ I + : Γi (˜ ) ≥ − ln ξi (ˆ ) }. ω ˆ R ω t ω Then: • The case of default times conditionally independent with respect to F corresponds to the choice of the product copula Π. In this case, for t1 , . . . , tn ≤ T we have 1 n Q{τ1 > t1 , . . . , τn > tn | FT } = Π(Zt1 , . . . , Ztn ), i where we set Zt = e−Γt . i • In general, for t1 , . . . , tn ≤ T we obtain 1 n Q{τ1 > t1 , . . . , τn > tn | FT } = C(Zt1 , . . . , Ztn ), where C is the copula used in the construction of τ1 , . . . , τn . Survival Intensities Sch¨nbucher and Schubert (2001) show that for arbitrary s ≤ t on the set {τ1 > s, . . . , τn > s} we o have 1 i n C(Zs , . . . , Zt , . . . , Zs ) Q{τi > t | Gs } = EQ Fs . 1 n C(Zs , . . . , Zs ) 142 i Consequently, assuming that the derivatives γt = the set {τ1 > t, . . . , τn > t}, ∂ i ∂vi Zt 1 n C(Zt , . . . , Zt ) dΓi t dt Credit Risk, TUNIS 2005 exist, the ith intensity of survival equals, on λi t = i γt 1 n C(Zt , . . . , Zt ) i i = γt Zt ∂ 1 n ln C(Zt , . . . , Zt ), ∂vi where λi is understood as the limit: t λi = lim h−1 Q{t < τi ≤ t + h | Ft , τ1 > t, . . . , τn > t}. t h↓0 It appears that, in general, the ith intensity of survival jumps at time t, if the j th entity defaults at time t for some j = i. In fact, it holds that λi,j t where = i γt ∂2 1 n C(Zt , . . . , Zt ) i ∂vi ∂vj Zt ∂ 1 n ∂vj C(Zt , . . . , Zt ) , λi,j = lim h−1 Q{t < τi ≤ t + h | Ft , τk > t, k = j, τj = t}. t h↓0 Sch¨nbucher and Schubert (2001) also examine the intensities of survival after the default times of o some entities. Let us fix s, and let ti ≤ s for i = 1, 2, . . . , k < n, and Ti ≥ s for i = k + 1, k + 2, . . . , n. Then, Q τi > Ti , i = k + 1, k + 2, . . . , n | Fs , τj = tj , j = 1, 2, . . . , k, τi > s, i = k + 1, k + 2, . . . , n = EQ ∂k ∂v1 ...∂vk k+1 1 k n C(Zt1 , . . . , Ztk , ZTk+1 , . . . , ZTn ) Fs k+1 1 k n C(Zt1 , . . . , Ztk , Zs , . . . , Zs ) ∂k ∂v1 ...∂vk . (7.1) Remark 7.3.1 Jumps of intensities cannot be efficiently controlled, except for the choice of C. In the approach described above, the dependence between the default times is implicitly introduced through Γi s, and explicitly introduced by the choice of a copula C. 7.3.3 Simplified Version Laurent and Gregory (2002) examine a simplified version of the framework of Sch¨nbucher and o Schubert (2001). Namely, they assume that the reference filtration is trivial – that is, Ft = {Ω, ∅} for every t ∈ I + . This implies, in particular, that the default intensities γ i are deterministic functions, R and t i Q{τi > t} = 1 − Fi (t) = e− 0 γu du . They obtain closed-form expressions for certain conditional intensities of default, by making specific assumptions regarding the choice of a copula C. Example 7.3.1 This example describes the use of one-factor Gaussian copula (Bank of International Settlements (BIS) standard). Let Xi = ρi V + ¯ 1 − ρ 2 Vi , i ¯ where V, Vi , i = 1, 2, . . . , n, are independent, standard Gaussian variables under the probability measure Q. Define the copula function C as C(v1 , . . . , vn ) = Q{Xi < N −1 (vi ), i = 1, 2, . . . , n}. M. Jeanblanc 143 Then, a special case of formula (7.1) takes the form (for i > 1) Q τi ≥ Ti | τ1 = s, τj ≥ s, j = 1, 2, . . . , n = ∞ −∞ ∞ −∞ n j=2 n j=2 N N ρj √ 1−ρ2 u+ρj ρ1 x1 −xj 1 √ √ = ρj √ 1−ρ2 j 1−ρ2 j n(u) du n(u) du 1−ρ2 u+ρj ρ1 y1 −yj 1 with xj = yj = N −1 (Fj (s)) for j = i and xi = N −1 (Fi (Ti )), yi = N −1 (Fi (s)), where n is the univariate standard normal density function. 7.4 Jarrow and Yu Model Jarrow and Yu (2001) approach can be considered as another step towards a dynamic theory of dependence between default times. For a given finite family of reference credit names, Jarrow and Yu (2001) propose to make a distinction between the primary firms and the secondary firms. 144 At the intuitive level: Credit Risk, TUNIS 2005 • The class of primary firms encompasses these entities whose probabilities of default are influenced by macroeconomic conditions, but not by the credit risk of counterparties. The pricing of bonds issued by primary firms can be done through the standard intensity-based methodology. • It suffices to focus on securities issued by secondary firms, that is, firms for which the intensity of default depends on the status of some other firms. Formally, the construction is based on the assumption of asymmetric information. Unilateral dependence is not possible in the case of complete (i.e., symmetric) information. 7.4.1 Construction and Properties of the Model Let {1, . . . , n} represent the set of all firms, and let F be the reference filtration. We postulate that: • For any firm from the set {1, . . . , k} of primary firms, the ‘default intensity’ depends only on F. • The ‘default intensity’ of each firm belonging to the set {k + 1, . . . , n} of secondary firms may depend not only on the filtration F, but also on the status (default or no-default) of the primary firms. Construction of Default Times τ1 , . . . , τn First step. We first model default times of primary firms. To this end, we assume that we are given a family of F-adapted ‘intensity processes’ λ1 , . . . , λk and we produce a collection τ1 , . . . , τk of F-conditionally independent random times through the canonical method: t τi = inf t ∈ I + : R 0 λi du ≥ − log ξi u where ξi , i = 1, . . . , k are mutually independent identically distributed random variables with uniform law on [0, 1] under the martingale measure Q. Second step. We now construct default times of secondary firms. We assume that: • The probability space (Ω, G, Q) is large enough to support a family ξi , i = k + 1, . . . , n of mutually independent random variables, with uniform law on [0, 1]. • These random variables are independent not only of the filtration F, but also of the already constructed in the first step default times τ1 , . . . , τk of primary firms. The default times τi , i = k + 1, . . . , n are also defined by means of the standard formula: t τi = inf t ∈ I + : R 0 λi du ≥ − log ξi . u However, the ‘intensity processes’ λi for i = k + 1, . . . , n are now given by the following expression: k λi = µi + t t l=1 i,l νt 1 {τl ≤t} , 1 where µi and ν i,l are F-adapted stochastic processes. If the default of the j th primary firm does not affect the default intensity of the ith secondary firm, we set ν i,j ≡ 0. M. Jeanblanc 145 Main Features ˆ Let G = F ∨ H1 ∨ . . . ∨ Hn stand for the enlarged filtration and let F = F ∨ Hk+1 ∨ . . . ∨ Hn be the filtration generated by the reference filtration F and the observations of defaults of secondary firms. Then: • The default times τ1 , . . . , τk of primary firms are conditionally independent with respect to F. • The default times τ1 , . . . , τk of primary firms are no longer conditionally independent when we ˆ replace the filtration F by F. ˆ • In general, the default intensity of a primary firm with respect to the filtration F differs from i the intensity λ with respect to F. We conclude that defaults of primary firms are also ‘dependent’ of defaults of secondary firms. Case of Two Firms To illustrate the present model, we now consider only two firms, A and B say, and we postulate that A is a primary firm, and B is a secondary firm. Let the constant process λ1 ≡ λ1 represent the t F-intensity of default for firm A, so that t τ1 = inf t ∈ I + : R 0 λ1 du = λ1 t ≥ − log ξ1 , u where ξ1 is a random variable independent of F, with the uniform law on [0, 1]. For the second firm, the ‘intensity’ of default is assumed to satisfy λ2 = λ2 1 {τ1 >t} + α2 1 {τ1 ≤t} 1 1 t for some positive constants λ2 and α2 , and thus t τ2 = inf t ∈ I + : R 0 λ2 du ≥ − log ξ2 u where ξ2 is a random variable with the uniform law, independent of F, and such that ξ1 and ξ2 are mutually independent. Then the following properties hold: • λ1 is the intensity of τ1 with respect to F, • λ2 is the intensity of τ2 with respect to F ∨ H1 , • λ1 is not the intensity of τ1 with respect to F ∨ H2 . 7.4.2 Bond Valuation The following result was established in Jarrow and Yu (2001), who assumed the fractional recovery of Treasury value scheme with the fixed recovery rates δ1 and δ2 . Let λ = λ1 + λ2 . For λ = α2 , we denote 1 cλ1 ,λ2 ,α2 (u) = λ1 e−α2 u + (λ2 − α2 )e−λu . λ − α2 For λ = α2 , we set cλ1 ,λ2 ,α2 (u) = 1 + λ1 u e−λu . Proposition 7.4.1 For the bond issued by the primary firm we have D1 (t, T ) = B(t, T ) δ1 + (1 − δ1 )e−λ1 (T −t) 1 {τ1 >t} . 1 146 Credit Risk, TUNIS 2005 The value of a zero-coupon bond issued by the secondary firm equals, on the set {τ1 > t}, that is, prior to default of the primary firm 1 D2 (t, T ) = B(t, T ) δ2 + (1 − δ2 )cλ1 ,λ2 ,α2 (T − t)1 {τ2 >t} . On the set {τ1 ≤ t}, that is, after default of the primary firm, it equals D2 (t, T ) = B(t, T ) δ2 + (1 − δ2 )e−α2 (T −t) 1 {τ2 >t} . 1 Special Case: Zero Recovery Assume that λ1 + λ2 − α2 = 0 and the bond is subject to the zero recovery scheme. For the sake of brevity, we set r = 0 so that B(t, T ) = 1 for t ≤ T. Under the present assumptions: 1 2 D2 (t, T ) = Q{τ2 > T | Ht ∨ Ht } and the general formula yields D2 (t, T ) = 1 {τ2 >t} 1 1 Q{τ2 > T | Ht } M. Jeanblanc 147 7.5.1 Kusuoka’s Construction We follow here Kusuoka (1999). Under the original probability measure Q the random times τi , i = 1, 2 are assumed to be mutually independent random variables with exponential laws with parameters λ1 and λ2 , respectively. Girsanov’s theorem. For a fixed T > 0, we define a probability measure Q equivalent to Q on (Ω, G) by setting dQ = ηT , dQ Q-a.s. where the Radon-Nikod´m density process ηt , t ∈ [0, T ], satisfies y 2 ηt = 1 + i=1 ]0,t] i ηu− κi dMu u where in turn i Mti = Ht − 0 t∧τi λi du i Here Ht = 1 {τi ≤t} and processes κ1 and κ2 are given by 1 1 κ1 = 1 {τ2 t} + α1 1 {τ2 ≤t} , 1 1 λ2 1 {τ1 >t} + α2 1 {τ1 ≤t} . 1 1 t 0 Main features. We focus on τ1 and we denote Λ1 = t the process λ1 is H2 -predictable, and the process 1 Mt1 = Ht − t∧τ1 0 λ1 du. Let us make few observations. First, u 1 λ1 du = Ht − Λ1 1 u t∧τ is a G-martingale under Q. Next, the process λ1 is not the intensity of the default time τ1 with respect to H2 under Q. Indeed, in general, we have 1 2 2 1 Q{τ1 > s | Ht ∨ Ht } = 1 {τ1 >t} EQ eΛt −Λs | Ht . 1 1 Finally, the process λ1 represents the intensity of the default time τ1 with respect to H2 under a probability measure Q1 equivalent to Q, where dQ1 = ηT , ˜ dQ Q-a.s. and the Radon-Nikod´m density process ηt , t ∈ [0, T ], satisfies y ˜ ηt = 1 + ˜ ]0,t] 2 ηu− κ2 dMu . ˜ u For s > t we have but also 1 2 Q1 {τ1 > s | Ht ∨ Ht } = 1 {τ1 >t} EQ1 eΛt −Λs | Ft 1 1 1 1 2 1 2 Q{τ1 > s | Ht ∨ Ht } = Q1 {τ1 > s | Ht ∨ Ht }. 148 Credit Risk, TUNIS 2005 7.5.2 Interpretation of Intensities Recall that the processes λ1 and λ2 have jumps if αi = λi . The following result shows that the intensities λ1 and λ2 are ‘local intensities’ of default with respect to the information available at time t. It shows also that the model can in fact be reformulated as a two-dimensional Markov chain (see Lando (1998b)). Proposition 7.5.1 For i = 1, 2 and every t ∈ I + we have R λi = lim h−1 Q{t < τi ≤ t + h | τ1 > t, τ2 > t}. h↓0 (7.2) Moreover: and α1 = lim h−1 Q{t < τ1 ≤ t + h | τ1 > t, τ2 ≤ t}. h↓0 α2 = lim h−1 Q{t < τ2 ≤ t + h | τ2 > t, τ1 ≤ t}. h↓0 7.5.3 Bond Valuation Proposition 7.5.2 The price D1 (t, T ) on {τ1 > t} equals D1 (t, T ) = 1 {τ2 ≤t} e−α1 (T −t) 1 1 +1 {τ2 >t} 1 λ2 e−α1 (T −t) + (λ1 − α1 )e−λ(T −t) . λ − α1 Furthermore ˜ D1 (t, T ) = 1 {τ2 ≤t} 1 +1 {τ2 >t} 1 and D1 (t, T ) = 1 {τ1 >t} 1 Observe that: • Formula for D1 (t, T ) coincides with the Jarrow and Yu formula for the bond issued by a secondary firm. • Processes D1 (t, T ) and D1 (t, T ) represent ex-dividend values of the bond, and thus they vanish after default time τ1 . ˜ • The latter remark does not apply to the process D1 (t, T ). (λ − α2 )λ2 e−α1 (T −τ2 ) λ1 α2 e(λ−α2 )τ2 + λ(λ2 − α2 ) λ − α2 (λ1 − α1 )e−λ(T −t) + λ2 e−α1 (T −t) λ − α1 λ1 e−(λ−α2 )t + λ2 − α2 λ2 e−α1 T + (λ1 − α1 )e−λT . λ2 e−α1 t + (λ1 − α1 )e−λt 7.6 Dependent Intensities of Credit Migrations We present here a contribution to the dynamic theory of dependence between credit events. Specifically, we discuss here an approach towards modeling of dependent credit migrations based on the theory of continuous-time conditional Markov chains. We refer to Bielecki and Rutkowski (2002) for information regarding conditional Markov chains. The goal is to extend the previous analysis to the case of multiple credit ratings. Assume that the current financial standing of the ith firm is reflected through the credit ranking process C i with values in a finite set of credit grades Ki = {1, . . . , ki }. M. Jeanblanc 149 For simplicity, we assume that the reference filtration F is trivial, and we consider the case of two firms. Let Fi = FC , i = 1, 2, denote the filtration generated by C i and let G = F1 ∨ F2 . We examine the two following Markovian properties under the martingale measure Q. The Markov property of C = (C 1 , C 2 ): 1 2 1 2 1 2 Q{Cs = k, Cs = l | Gt } = Q{Cs = k, Cs = l | Ct , Ct }. i The Fj -conditional Markov property of C i for i = j: 1 1 1 2 Q{Cs = k | Gt ) = Q{Cs = k | σ(Ct ) ∨ Ft }, 2 Q{Cs = l | Gt } 2 2 1 = Q{Cs = l | σ(Ct ) ∨ Ft }. 7.6.1 Extension of Kusuoka’s Construction Assume that k1 = k2 = 3 (three rating grades). We consider the two independent Markov chains C i , i = 1, 2 defined on (Ω, G, Q) and taking values in K = {1, 2, 3} with generators:   λi λi −λi − λi 13 12 13 12 λi  . −λi − λi λi Λi =  23 23 21 21 0 0 0 2 1 The state k = 3 is the only absorbing state for each chain. We assume that (C0 , C0 ) = (1, 1). In addition, we are given the following matrices:  i|l  i|l i|l i|l λ13 λ12 −λ12 − λ13 i|l i|l i|l i|l Λi|l =  λ21 −λ21 − λ23 λ23  0 0 0 for i = 1, 2 and l = 2, 3. It should be observed that formally Λi = Λi|1 for i = 1, 2. In general, the i|l intensities λi and λkm may follow F-predictable stochastic processes. km Auxiliary Processes and Associated Martingales We define a probability measure Q equivalent to Q. To this end, we introduce auxiliary processes κi , by setting km 3 i|l λkm j i κkm (t) = Hkl (t−) −1 λi km l=2 for i = 1, 2, j = i, k = 1, 2, m = 1, 2, 3, k = m, where for j = 1, 2 and k = 1, 2, 3, j Hj (t) = Hlj (t)Hk (t) kl j with Hk (t) = 1 {C j =k} . We also define, for i = 1, 2 and k = m, the transition counting process 1 t i Hkm (t) = 0 gK−2 (t, T ) > . . . > g1 (t, T ) > f (t, T ) for every t ≤ T. Definition 7.8.1 For every i = 1, 2, . . . , K − 1, the ith forward credit spread equals si (·, T ) = gi (·, T ) − f (·, T ). Martingale Measure Q It is known from the HJM theory that the following condition (M) is sufficient to exclude arbitrage across default-free bonds for all maturities T ≤ T ∗ and the savings account. Condition (M) There exists an F-adapted I d -valued process β such that R T∗ T∗ 0 EP exp 0 βu dWu − 1 2 |βu |2 du =1 and, for any maturity T ≤ T ∗ , we have α∗ (t, T ) = where α∗ (t, T ) = t 1 ∗ |σ (t, T )|2 − σ ∗ (t, T )βt 2 T α(t, u) du T σ ∗ (t, T ) = t σ(t, u) du. Let γ be some process satisfying Condition (M). Then the probability measure Q, given by the formula ∗ T∗ dQ 1 T = exp βu dWu − |βu |2 du , P-a.s., dP 2 0 0 is a martingale measure for the default-free term structure. We will see that for any T the prices B(t, T ) is a martingale under the measure Q, when discounted with the savings account Bt . 154 Zero-Coupon Bonds Credit Risk, TUNIS 2005 The price of the T -maturity default-free zero-coupon bond is given by the equality T B(t, T ) = exp − t f (t, u) du . Formally, such Treasury bond corresponds to credit class K. Similarly, the ‘conditional value’ of T -maturity defaultable zero-coupon bond belonging at time t to the credit class i = 1, 2, . . . , K − 1, equals T Di (t, T ) = exp − t gi (t, u) du . We consider discounted price processes −1 −1 Z(t, T ) = Bt B(t, T ), Zi (t, T ) = Bt Di (t, T ), where B is the savings account Bt = exp 0 t f (u, u) du . Let us define a Brownian motion W ∗ under Q by setting Wt∗ = Wt − t βu du, 0 ∀ t ∈ [0, T ∗ ]. Conditional Dynamics of the Bond Price Lemma 7.8.1 Under the martingale measure Q, for any fixed T ≤ T ∗ , the discounted price processes Z(t, T ) and Zi (t, T ) satisfy dZ(t, T ) = Z(t, T )b(t, T ) dWt∗ , where b(t, T ) = −σ ∗ (t, T ), and dZi (t, T ) = Zi (t, T ) λi (t) dt + bi (t, T ) dWt∗ where λi (t) = ai (t, T ) − f (t, t) + bi (t, T )βt and ∗ ai (t, T ) = gi (t, t) − αi (t, T ) + 1 ∗ |σ (t, T )|2 2 i ∗ bi (t, T ) = −σi (t, T ). Observe that usually the process Zi (t, T ) is not a martingale under the martingale measure Q. This feature is related to the fact that it does not represent the (discounted) price of a tradeable security. 7.8.2 Credit Migration Process Recall that we assumed that the set of rating classes is K = {1, . . . , K}, where the class K corresponds to default. The migration process C is constructed in Bielecki and Rutkowski (2000a) as a (nonhomogeneous) conditionally Markov process on K, with the state K as the unique absorbing state for this process. The process C is constructed on some enlarged probability space (Ω∗ , G, Q), where the probability measure Q is the extended martingale measure. The reference filtration F is contained in the extended filtration G. For simplicity of presentation, we summarize the results for the case K = 3. M. Jeanblanc 155 Given some non-negative and F-adapted processes λ1,2 (t), λ1,3 (t), λ2,1 (t) and λ2,3 (t), a migration process C is constructed as a conditional Markov process with the conditional intensity matrix (infinitesimal generator)   λ1,1 (t) λ1,2 (t) λ1,3 (t) Λ(t) =  λ2,1 (t) λ2,2 (t) λ2,3 (t)  0 0 0 where λi,i (t) = − j=i λi,j (t) for i = 1, 2. 156 Credit Risk, TUNIS 2005 The conditional Markov property (with respect to the reference filtration F) means that if we C denote by Ft the σ-field generated by C up to time t then for arbitrary s ≥ t and i, j ∈ K we have C Q Ct+s = i | Ft ∨ Ft = Q Ct+s = i | Ft ∨ {Ct = j} . The formula above provides the risk-neutral conditional probability that the defaultable bond is in class i at time t + s, given that it was in the credit class Ct at time t. For any date t, we denote by ˆ Ct the previous bond’s rating; we shall need this notation later. Finally, the default time τ is introduced by setting τ = inf {t ∈ I + : Ct = 3 }. R Let Hi (t) = 1 {Ct =i} for i = 1, 2, and let Hi,j (t) represent the number of transitions from i to j 1 by C over the time interval (0, t]. It can be shown that the process t Mi,j (t) = Hi,j (t) − 0 λi,j (s)Hi (s) ds, ∀ t ∈ [0, T ], for i = 1, 2 and j = i, is a martingale on the enlarged probability space (Ω∗ , G, Q). Let us emphasize that due to the judicious construction of the migration process C, appropriate version of the hypotheses (H.1)-(H.3) remain valid here. 7.8.3 Defaultable Term Structure We maintain the simplified framework with K = 3. We assume the fractional recovery of Treasury value scheme. To be more specific, to each credit rating i = 1, . . . , K − 1, we associate the recovery rate δi ∈ [0, 1), where δi is the fraction of par paid at bond’s maturity, if a bond belonging to the ith class defaults prior to its maturity. Thus, the cash flow at maturity is X = 1 {τ >T } + δCτ 1 {τ ≤T } . 1 ˆ 1 In order to provide the model with arbitrage free properties, Bielecki and Rutkowski (2000a) postulate that the risk-neutral intensities of credit migrations λ1,2 (t), λ1,3 (t), λ2,1 (t) and λ2,3 (t) are specified by the no-arbitrage condition (also termed the consistency condition): λ1,2 (t) Z2 (t, T ) − Z1 (t, T ) + λ1,3 (t) δ1 Z(t, T ) − Z1 (t, T ) + λ1 (t)Z1 (t, T ) = 0, ˆ λ2,1 (t) Z1 (t, T ) − Z2 (t, T ) + λ2,3 (t) δ2 Z(t, T ) − Z2 (t, T ) + λ2 (t)Z2 (t, T ) = 0. Martingale Dynamics of a Defaultable Bond First, we introduce the process Z(t, T ) as a solution to the following SDE dZ(t, T ) = Z2 (t, T ) − Z1 (t, T ) dM1,2 (t) + Z1 (t, T ) − Z2 (t, T ) dM2,1 (t) + δ1 Z(t, T ) − Z1 (t, T ) dM1,3 (t) + δ2 Z(t, T ) − Z2 (t, T ) dM2,3 (t) + H1 (t)Z1 (t, T )b1 (t, T ) dWt∗ + H2 (t)Z2 (t, T )b2 (t, T ) dWt∗ + δ1 H1,3 (t) + δ2 H2,3 (t) Z(t, T )b(t, T ) dWt∗ , with the initial condition Z(0, T ) = H1 (0)Z1 (0, T ) + H2 (0)Z2 (0, T ). M. Jeanblanc 157 It appears that the process Z(t, T ) follows a martingale on (Ω∗ , G, Q), so that it is justified to refer to Q as the extended martingale measure). The proof of the next result employs the no-arbitrage condition. Lemma 7.8.2 For any maturity T ≤ T ∗ and for every t ∈ [0, T ] we have Z(t, T ) = 1 {Ct =3} ZCt (t, T ) + 1 {Ct =3} δCt Z(t, T ) 1 1 ˆ Next, we define the price process of a T -maturity defaultable zero-coupon bond by setting DC (t, T ) = Bt Z(t, T ) for any t ∈ [0, T ]. In view of Lemma 7.8.2, we have that DC (t, T ) = 1 {Ct =3} DCt (t, T ) + 1 {Ct =3} δCt B(t, T ). 1 1 ˆ The defaultable bond price DC (t, T ) satisfies the following properties: • The process DC (t, T ) is a G-martingale under Q, when discounted by the savings account. • In contrast to the ‘conditional price’ Di (t, T ), the process DC (t, T ) admits discontinuities. Jumps are directly associated with changes in credit quality (ratings migrations). • The process DC (t, T ) represents the price of a tradeable security: the corporate zero-coupon bond of maturity T. Risk-Neutral Representations Recall that δi ∈ [0, 1) is the recovery rate for a bond which was in the ith rating class just prior to default. Proposition 7.8.1 The price process DC (t, T ) of a T -maturity defaultable zero-coupon bond equals T DC (t, T ) = 1 {Ct =3} B(t, T ) exp − 1 + 1 {Ct =3} δCt B(t, T ) 1 ˆ t sCt (t, u) du where si (t, u) = gi (t, u) − f (t, u) is the ith credit spread. Proposition 7.8.2 The price process DC (t, T ) satisfies the risk-neutral valuation formula −1 −1 DC (t, T ) = Bt EQ δCT BT 1 {τ ≤T } + BT 1 {τ >T } | Gt . 1 1 ˆ It is also clear that DC (t, T ) = B(t, T ) EQT δCT 1 {τ ≤T } + 1 {τ >T } | Gt , 1 ˆ 1 where QT stands for the T -forward measure associated with the extended martingale measure Q. ¨ Let us end this section by mentioning that Eberlein and Ozkan [?] have generalized the model presented above to the case of term structures driven by L´vy processes. e 7.8.4 Premia for Interest Rate and Credit Event Risks We shall now change, using a suitable version of Girsanov’s theorem, the measure Q to the equivalent probability measure Q. In the financial interpretation, the probability measure Q will play the role of the statistical probability (i.e., the real-world probability). It is thus natural to postulate that 158 Credit Risk, TUNIS 2005 the restriction of the probability measure Q to the original probability space Ω necessarily coincides with the statistical probability P for the default-free market. From now on, we shall assume that the following condition holds. Condition (P) We postulate that dQ = ηT ∗ , ˆ dQ Q-a.s., where the positive Q-martingale η is given by the formula ˆ dˆt = −ˆt βt dWt∗ + ηt− dMt , η η ˆ η0 = 1, for some I d -valued F-predictable process β, where the Q-local martingale M equals R dMt = i=j κi,j (t) dMi,j (t) κi,j (t) dHi,j (t) − λi,j (t)Hi (t) dt i=j = for some F-predictable processes κi,j > −1. Assume that for any i = j T∗ κi,j (t) + 1 λi,j (t) dt < ∞, 0 Q-a.s. In addition, we postulate that EQ (ˆT ∗ ) = 1, so that the probability measure Q is indeed well defined η on (Ω∗ , GT ∗ ). The financial interpretation of processes β and κ is similar as in Section ??, namely, • The vector-valued process β corresponds to the premium for the interest rate risk. • The matrix-valued process κ represents the premium for the credit event risk. Statistical Default Intensities Q We define processes li,j by setting, for i = j, λQ (t) = (κi,j (t) + 1)λi,j (t), i,j λQ (t) = − i,i j=i λQ (t). i,j Proposition 7.8.3 Under an equivalent probability Q given by condition (P ), the process C is a conditionally Markov process. The matrix of conditional intensities of C under Q equals Q l1,1 (t) .  ΛQ =  Q t lK−1,1 (t) 0  ... ... ... ...  Q l1,K (t) .  . Q lK−1,K (t) 0 If the market price for credit risk depends only on the current rating i (and not on the rating j after jump), so that κi,j = κi,i for every j = i. Then ΛQ = Φt Λt , where Φt = diag [φi (t)] with t φi (t) = κi,i (t) + 1 is the diagonal matrix (this case was examined, e.g., by Jarrow et al. (1997)). 7.8.5 Defaultable Coupon Bond Consider a defaultable coupon bond with the face value L that matures at time T and promises to pay coupons ci at times T1 < . . . < Tn < T. The coupon payments are only made prior to default, M. Jeanblanc 159 and the recovery payment is made at maturity T , and is proportional to the bond’s face value. Notice that the migration process C introduced in Section 7.8.2 may depend on both the maturity T and on recovery rates. Therefore, it is more appropriate to write Ct = Ct (δ, T ), where δ = (δ1 , . . . , δK ). Similarly, we denote the price of a defaultable zero-coupon bond DC(δ,T ) (t, T ), rather than DC (t, T ). A defaultable coupon bond can be treated as a portfolio consisting of: • Defaultable coupons – that is, defaultable zero-coupon bonds with maturities T1 , . . . , Tn , which are subject to zero recovery. • Defaultable face value – that is, a T -maturity defaultable zero-coupon bond with a constant recovery rate δ. We conclude that the arbitrage price of a defaultable coupon bond equals n Dc (t, T ) = i=1 ci DC(0,Ti ) (t, Ti ) + LDC(δ,T ) (t, T ), where, by convention, we set DC(0,Ti ) (t, Ti ) = 0 for t > Ti . 7.8.6 Examples of Credit Derivatives Credit Default Swap Consider first a basic credit default swap, as described, e.g., in Section 1.3.1 of Bielecki and Rutkowski (2002). In the present setup, the contingent payment is triggered by the event {Ct = K}. The contract is settled at time τ = inf {t < T : Ct = K }, and the payoff equals Zτ = 1 − δCT B(τ, T ) . ˆ Notice the dependence of Zτ on the initial rating C0 through the default time τ and the recovery rate δCT . The following two market conventions are common in practice: ˆ • The buyer pays a lump sum at contract’s inception (default option). • The buyer pays annuities up to default time (default swap). In the first case, the value at time 0 of a default option equals −1 S0 = EQ Bτ 1 − δCT B(τ, T ) 1 {τ ≤T } . 1 ˆ In the second case, the annuity κ can be found from the equation T S0 = κ EQ i=1 −1 Bti 1 {ti <τ } . 1 Notice that both the price S0 and the annuity κ depend on the initial bond’s rating C0 . Total Rate of Return Swap As a reference asset we take the coupon bond with the promised cash flows ci at times Ti . Suppose ˆ the contract maturity is T ≤ T . In addition, suppose that the reference rate payments (the annuity ˆ payments) are made by the investor at fixed scheduled times ti ≤ T , i = 1, 2, . . . , m. The owner of a total rate of return swap is entitled not only to all coupon payments during the life of the contract, but also to the change in the value of the underlying bond. By convention, we assume that the 160 Credit Risk, TUNIS 2005 default event occurs when Ct (δ, T ) = K. According to this convention, the reference rate κ to be paid by the investor satisfies n EQ i=1 −1 −1 ci BTi 1 {Ti ≤T } + EQ Bτ Dc (τ, T ) − Dc (0, T ) 1 ˆ m = κ EQ i=1 −1 Bti 1 {Cti (δ,T )=K} , 1 ˆ where τ = inf {t ≥ 0 : Ct (δ, T ) = K } ∧ T . 7.9 Concluding Remarks It should be acknowledged that we have not discussed in the present text any results or techniques related to hedging of credit risk. Let us conclude, however, by listing the most important issues arising in practical and theoretical approaches to this problem, and giving some references that may be consulted by the interested reader. Simplified approaches. In most practical implementations of credit risk models (see, for instance, Greenfield (2000)), it is common to impose at least some of the following simplifying assumptions: • Only a pure credit risk instrument (e.g., a basic credit default swap) is considered. • One deals with a one-sided counterparty risk with a fixed recovery rate (the same for a derivative product and for a corporate bond). • The mark-to-market value of the contract is assumed to be non-negative to a non-defaultable counterparty (thus, for instance, defaultable loans and bonds or vulnerable options are covered, but defaultable swaps are excluded). • Independence of market and credit risks is frequently postulated. • Existence of a non-defaultable version of the contract and of a liquid market in corporate bonds and other related instruments of various maturities is assumed. Theoretical results. More sophisticated mathematical techniques, which have a potential to be useful in hedging credit risk, have been developed in recent years, in particular: • Suitable versions of a predictable representation theorem with respect to discontinuous martingales associated with the default event, or with credit migrations, were established (see, for instance, B´langer et al. (2001) or Blanchet-Scalliet and Jeanblanc (2003)). Unfortunately, e the general formulae obtained through this technique seem to be very difficult to implement. A more straightforward approach to the replication of credit derivatives was proposed by Vaillant (2001) (see also Jeanblanc and Rutkowski (2003)). • A utility-based approach to hedging of credit risk and valuation of credit derivatives was examined. In this approach, which is based on the idea of indifference pricing, hedging strategies are constructed as solutions to appropriate stochastic control problems (see, for instance, Collin-Dufresne and Hugonnier (2002) or Lukas (2001)). • An alternative approach to hedging of credit risk, in the spirit of Markowitz mean-variance methodology, was recently developed. It involves, in general, constructing of hedging strategies in terms of solutions of certain backward stochastic differential equations (BSDE) as well as in terms of certain orthogonal projections (see Bielecki et al. (2004)). Chapter 8 Portfolio management 8.1 Ratings More generally, let us assume that X is a continuous time Markov chain on a finite state space E = {1, . . . , K, K + 1} with transition probability P (t). In credit rating’s model, the states are the rating classes AAA, AA, . . . as used by Standard and Poors or Moody’s agencies. The state K + 1 corresponds to bankruptcy, and it is assumed that a firm which goes in bankruptcy disappears, that is, remains in that state. Let us call Pi,j (t) the probability that, at time t, the Markov chain starting from state i is in state j. Therefore, PK+1,i = 0, ∀i = K + 1, PK+1,K+1 = 1. This leads to PK+1,i = 0 for all i. The generator matrix of X is the matrix Λ such that P (t) = exp(tΛ) = n (tΛ)n n! Here is an example of rating matrices (Lando’s thesis) Initial Rating AAA AA A BBB BB B D AAA 0,8910 0,0086 0,0009 0,0006 0,0004 0,000 0,000 0,0000 AA 0,0963 0,9010 0,0291 0,0043 0,0022 0,0019 0,0000 0,0000 Rating at the end of the A BBB BB 0,0078 0,0019 0,0030 0,0747 0,0099 0,0029 0,8894 0,0649 0,0101 0,0656 0,8427 0,0644 0,0079 0,0719 0,7764 0,0031 0,0066 0,0517 0,0116 0,0116 0,0203 0,0000 0,0000 0,0000 year B 0,000 0,0029 0,0045 0,0160 0,1043 0,8246 0,0754 0,0000 CCC 0,0000 0,0000 0,0000 0,0018 0,0127 0,0435 0,6493 0,0000 D 0,0000 0,0000 0,0009 0,0045 0,0241 0,0685 0,2319 1,0000 161 162 and Λ is given by Credit Risk, TUNIS 2005 Initial Rating AAA AA A BBB BB B CCC D AAA -0,1154 0,0091 0,0010 0,0007 0,0005 0,0000 0,0000 0,0000 AA 0,1019 -0,1043 0,0309 0,0047 0,0025 0,0021 0,0000 0,0000 Rating at the end of the A BBB BB 0,0083 0,0020 0,0031 0,0787 0,0105 0,0030 -0,1172 0,0688 0,0107 0,0713 -0,1711 0,0701 0,0089 0,0813 -0,2530 0,0034 0,0073 0,0568 0,0142 0,0142 0,0250 0,0000 0,0000 0,0000 year B 0,000 0,0030 0,0048 0,0174 0,1181 -0,1929 0,0818 0,0000 CCC 0,0000 0,0000 0,0000 0,0020 0,0144 0,0479 -0,4318 0,0000 D 0,0000 0,0000 0,0010 0,0049 0,0273 0,0753 0,2756 0,0000 In this model, the ”basic” filtration is the trivial one. Lando generalizes the study to the case where the coefficients of the matrix Λ depend on a random process X. It seems important to underline the fact that the data is a transition matrix conditioned to the information given by the factor at time t. Lando introduces nonnegative functions λi , λi,j such that λi (x) = j,j=i λi,j (x) and the generator matrix Λ(Xt ). 8.1.1 CreditMetrics CreditMetrics (from J.P. Morgan) is based on credit migration analysis and the estimation of the distribution of the changes in value of credit quality at a given horizon. The first step is to provide a transition matrix (Moody’s or Standard & Poor’s). The spreads are evaluated for each of the 7 possible credit qualities. Then, assuming that the firm’s asset value (Vt , t ≥ 0) follows a standard geometric Brownian motion dVt = Vt (µdt + σdWt ), one has to slice the distribution of asset returns into bands in such a way that one reproduces the migration frequencies given in the transition matrix. The terminal value VT has a distribution given by √ σ2 VT = V0 exp (µ − )T + σ T G , 2 where G is a standard Gaussian variable. If p is the probability that a BB-rated obligor defaults, the barrier L is defined as   σ2 ln(V0 /L) + (µ − )T   2 √ p = P (VT ≤ L) = P G ≤ −  = N (−d) σ T where σ2 ln(V0 /L) + (µ − )T 2 √ d= σ T is the distance to default. The barrier L can be translated in a threshold g for the process Zt = ln Vt . 8.1.2 KMV approach KMV (Kealhofer-McQuown, Vacisek) implements an intermediate phase before computing the probabilities of default. From observation, they define Distance to Default (DD) as follows : the default M. Jeanblanc 163 point DPT is equal to STD+(1/2)LTD where STD is the short term debt and LTD the long term debt. The DD is ln(V0 /DP T ) + (µ − σ 2 /2)T √ DD = σ T and can be viewed as a measure for the distance between the expected return on assets and the DPT. KMV estimates the proportion of firms of a given rating which actually defaults after one year, this proportion is called the EDF (Expected Default Frequencies) and deduces the implied rating. 8.1.3 CreditRisk+ CreditRisk+ (Creduit Suisse Financial products) focuses on default. They assume that default for individual bonds or loans follow a Poisson process. 8.1.4 CreditPortfolioView CreditPortfolioView measures default risk a a discrete time multiperiod model where default probabilities are functions of macro-economics variable. See the paper of Crouhy et al. [24] for a comparative study of these approaches. 8.2 Portfolio management In that setting, only the state of the firm et maturity is taken into account. A large portfolio, involving different counterparties is studied. Let Yi be a random variable, valued in {0, 1}, such that Yi = 1 if the i-th counterparty has defaulted before time T . Two approaches are used in order to correlate these variables. a. Latent variable model This method is used by CreditRisk and KMV. The default occurs if a latent variable Xi is smaller than a fixed level Di . b. Mixture model This model is used by CreditRisk+. Default probabilities are made stochastic by the introduction of a r.v. Qi [0, 1]-valued. The conditional law of Yi , given Qi is a Bernoulli law of parameter Qi The correlation between Yi is made using copulas function. 8.2.1 Copulas A recent approach for modeling dependent credit risks is the use of copulas. The reader is refereed to various works, e.g. Durrleman et al [38], Nelsen [77] for definitions and properties of Copulas and to Frey and McNeil [43], Embrechts [41], for finance purpose. We recall the main obvious lemma: Lemma 8.2.1 If X is a r.v. with continuous distribution function F , the r.v. F (X) has a uniform distribution on [0, 1]. 164 Credit Risk, TUNIS 2005 Definition 8.2.1 A copula C is a joint cumulative distribution of n random variables uniformly distributed on [0, 1]. A mapping C defined on [0, 1]d is a copula if and only if it satisfies : (i) C(u1 , . . . , un ) is increasing with respect to each component ui , (ii) C(1, . . . , ui , . . . , 1) = ui , for any i, for any ui ∈ [0, 1], (iii) For any a, b ∈ [0, 1]d with a ≤ b (i.e., ai ≤ bi , ∀i) 2 2 ... i1 =1 (−1)i1 +...+in C(u1,i1 , . . . , un,in ) ≥ 0 , in =1 where uj,1 = aj , uj,2 = bj . The survival copula is C(u1 , . . . , un ) = P (U1 > u1 , . . . , Un > un ). Th´or`me 8.1 Sklar Theorem. Let F be an n-dimensional cumulative distribution with margin e e Fi . Then there exists a copula C such that F (x) = C(F1 (x1 ), · · · , Fn (xn )) A particular copula is the independent copula C(u1 , . . . , un ) = n i=1 ui . 8.2.2 Davis and Lo Davis et Lo [27, 26] assume that a counterparty can default for two reasons: - either directly, i.e. Yi = 1 - or may be infected by default of company j The random variable Zi is equal to 1 if the firm i has defaulted, where Zi = Yi + (1 − Yi )(1 − j (1 − Yj Cji )) Let N be the number of firms in default at date T . If (Yi , Cji ) are independent and P (Yi = 1) = p, P (Cij = 1) = q . n we get P (N = k) = Ck αp,q,n (k) where k−1 αp,q,n (k) = pk (1 − p)n−k q k(n−k) + ¯ i=1 k Ci pi (1 − p)n−i (1 − q i )k−i q i(n−k) ¯ ¯ where q = 1 − q. ¯ 8.2.3 Portfolio management A portfolio is made of positions on different counterparties. Let n V = i=1 Zi be the global position, where Zi = ρZ + i M. Jeanblanc 165 the r.v. i are independent and independent of the common factor Z. The default for the i-th component occurs if Zi < d. KMV assumes that Z and are Gaussian variables. The coefficient ρ stands for the correlation between the Zi . They study P( This can be approximated, as follows P( 1 N N i i=1 1 N N Zi ≤ d) i=1 1 N N Zi ≤ d|Z = z) = ΦN (z) i=1 where ΦN (z) = P ( ≤ d − ρz) is a non standard Gaussian cumulative function. 166 Credit Risk, TUNIS 2005 Chapter 9 Appendix 9.1 Hitting times In this chapter, a Brownian motion (Wt , t ≥ 0) starting from 0 is given on a probability space x 2 1 (Ω, F, P ), and F = (Ft , t ≥ 0) is its natural filtration. The function N (x) = √ e−u /2 du is 2π −∞ the cumulative function of the standard Gaussian law. 9.1.1 Hitting times of a level and law of the maximum for Brownian motion def Let us study the law of the pair of random variables (Wt , Mt ) where M is the maximum process of the Brownian motion, i.e., Mt = sups≤t Ws . The law of hitting times of a given level by the Brownian motion will be obtained. Law of the pair of the random variables (Wt , Mt ) Let us remark that the process M is an increasing process, with non negative values. Th´or`me 9.1 Let W be a Brownian motion starting from 0 and Mt = sup (Ws , 0 ≤ s ≤ t). e e   for         for        for x − 2y x N ( √ ) − N ( √ ), t t y −y P (Mt ≤ y) = N ( √ ) − N ( √ ), t t 0. . (9.2) (9.1) y ≥ 0, x ≤ y y ≥ 0, x ≥ y y≤0 P (Wt ≤ x, Mt ≤ y) = P (Wt ≤ x, Mt ≤ y) = P (Wt ≤ x, Mt ≤ y) = P (Wt ∈ dx, Mt ∈ dy) = 1 y≥0 1 x≤y 1 1 Proof: • Reflexion Principle Let us show that for 0 ≤ y, x ≤ y : 2 (x − 2y)2 (2y − x) exp − πt3 2t P (Wt ≤ x , Mt ≥ y) = P (Wt ≥ 2y − x) . 167 (9.3) 168 It will follow that Credit Risk, TUNIS 2005 P (Wt ≤ x , Mt ≤ y) = P (Wt ≤ x) − P (Wt ≤ x , Mt ≥ y) = P (Wt ≤ x) − P (Wt ≥ 2y − x) , hence the first equality. (9.4) Let Ty = inf{t : Wt ≥ y} be the first time where the Brownian motion is greater than y. This is a stopping time in the filtration F and (Ty ≤ t) = (Mt ≥ y) for y ≥ 0. Furthermore, from y ≥ 0 and by relying on the continuity of Brownian motion paths, the stopping time Ty is also the hitting time of y, i.e., Ty = inf{t : Wt = y} and WTy = y. Therefore P (Wt ≤ x , Mt ≥ y) = P (Wt ≤ x , Ty ≤ t) = P (Wt − WTy ≤ x − y , Ty ≤ t) . For the sake of simplicity, we denote EP (1 A |Ty ) = P (A|Ty ). By relying on the strong Markov 1 property, we obtain P (Wt − WTy ≤ x − y , Ty ≤ t) = E(1 Ty ≤t P (Wt − WTy ≤ x − y |Ty )) = E(1 Ty ≤t Φ(Ty )) 1 1 ˜ ˜ def with Φ(u) = P (Wt−u ≤ x−y ) where (Wu = WTy +u −WTy , u ≥ 0) is a Brownian motion independent ˜ ˜ of (Wt , t ≤ Ty ), and with the same law as −W . Therefore Φ(u) = P (Wt−u ≥ y − x ) and by proceeding backward: E(1 Ty ≤t Φ(Ty )) = E[1 Ty ≤t P (Wt − WTy ≥ y − x |Ty )] = P (Wt ≥ 2y − x , Ty ≤ t). 1 1 Hence P (Wt ≤ x, Mt ≥ y) = P (Wt ≥ 2y − x, Mt ≥ y) (9.5) The right-hand side of (9.5) is equal to P (Wt ≥ 2y − x) since 2y − x ≥ y which implies that, on the set {Wt ≥ 2y − x}, the hitting time Ty is smaller than t (or Mt ≥ y.) • For {0 ≤ y ≤ x}, P (Wt ≤ x, Mt ≤ y) = P (Wt ≤ y, Mt ≤ y) since Mt ≥ Wt . Furthermore, by setting x = y in (9.4) P (Mt ≤ y) = P (Wt ≤ y, Mt ≤ y) = N • Finally, for y ≤ 0, y √ −N −y √ t . (9.6) t P (Wt ≤ x, Mt ≤ y) = 0 since Mt ≥ M0 = 0. Law of the supremum Proposition 9.1.1 The random variable Mt has the same law as |Wt |. Proof: This follows from (9.6). In a different way, for x ≥ 0 : P (Mt ≥ x) = P (Mt ≥ x, Wt ≥ x) + P (Mt ≥ x, Wt ≤ x) , therefore, from (9.3) (with x = y ) we obtain, using again that Wt = −Wt , P (Mt ≥ x) = P (Wt ≥ x) + P (Wt ≥ x) = P (Wt ≥ x) + P (Wt ≤ −x) = P (|Wt | ≥ x) . law M. Jeanblanc 169 Law of the hitting time For x > 0, the law of Tx = inf{s : Ws ≥ x} can be now easily deduced from √ x2 P (Tx ≤ t) = P (x ≤ Mt ) = P (x ≤ |Wt |) = P (x ≤ |G| t) = P ( 2 ≤ t) , G (9.7) where, as usual G stands for a Gaussian random variable, with zero expectation and unit variance. 2 law x Hence, Tx = 2 and the density of Tx follows: G P (Tx ∈ dt) = √ x 2πt3 exp(− x2 )1 t≥0 dt . 1 2t For x < 0, we have, using the symmetry of the BM Tx = inf{t : Wt ≤ x} = inf{t : Wt = x} = T−x and |x| x2 exp(− )1 t≥0 dt . 1 P (Tx ∈ dt) = √ 2t 2πt3 law (9.8) Comments 9.1.1 The process M does not have the same law as the process |W |. Indeed, the process M is an increasing process, this is not the case for |W |. Nevertheless, there are some further equalities in law, as M − W = |W | between these processes. Law of the infimum The law of the infimum of a Brownian motion is obtained by relying on the same procedure. It can also be deduced by observing that mt = inf Ws = − sup(−Ws ) = − sup(Bs ) s≤t s≤t s≤t def law where B = −W is a Brownian motion. Hence for y ≤ 0, x ≥ y for y ≤ 0, x ≤ y −x 2y − x P (Wt ≥ x, mt ≥ y) = N ( √ ) − N ( √ ), t t −y y P (Wt ≥ x, mt ≥ y) = N ( √ ) − N ( √ ) , t t P (Wt ≥ x, mt ≥ y) = 0 . (9.9) for y ≥ 0 −y y In particular, P (mt ≥ y) = N ( √ ) − N ( √ ). As an immediate consequence, we obtain that, for t t x > 0 and y > 0, Px (Wt ∈ dy, T0 > t) = = P0 (Wt + x ∈ dy, T−x > t) = P0 (Wt + x ∈ dy, mt > −x) (x − y)2 (x + y)2 1 √ exp − − exp − dx dy . 2t 2t 2πt (9.10) Laplace transform of the hitting time λ2 We have recalled that, for any λ > 0 the process (exp(λWt − t), t ≥ 0) is a martingale. Let y ≥ 0, 2 λ ≥ 0 and Ty be the hitting time of y. The martingale (exp(λWt∧Ty − λ2 (t ∧ Ty )), t ≥ 0) 2 170 is bounded by eλy . Doob’s optional sampling theorem yields E[exp(λWTy − λ2 Ty )] = 1 . 2 Credit Risk, TUNIS 2005 The case where y < 0 is obtained by studying the Brownian motion −W . Warning 1 In order to apply Doob’s optional sampling theorem, we have to check carefully that λ2 (t ∧ Ty )) is uniformely integrable. In the case λ > 0 and y < 0, a the martingale exp(λWt∧Ty − 2 wrong use of this theorem would lead to the equality between 1 and E[exp(λWTy − that is between E[exp(− λ2 λ2 Ty )] = eλy E[exp(− Ty )] 2 2 λ2 λ2 Ty )] and exp(−yλ) This is obvioulsly wrong since the quantity E[exp(− Ty )] 2 2 is smaller than 1 whereas exp(−yλ) is strictly greater than 1. Proposition 9.1.2 Let Ty be the hitting time of y ∈ I for a standard Brownian motion. Then, for R λ>0 λ2 E[exp(− Ty )] = exp(−|y|λ) . 2 9.1.2 Hitting times for a Drifted Brownian motion We study now the case where Xt = νt+Wt , where W is a Brownian motion. Let MtX = sup (Xs , s ≤ t), mX = inf (Xs , s ≤ t) and Ty (X) = inf{t ≥ 0 | Xt = y}. t Laws of the pairs M, X and m, X at time t Proposition 9.1.3 For y ≥ 0, y ≥ x x − 2y − νt x − νt √ ) P (Xt ≤ x, MtX ≤ y) = N ( √ ) − e2νy N ( t t and for y ≤ 0, y ≤ x P (Xt ≥ x, mX ≥ y) = N ( t Proof: From Cameron-Martin’s theorem P (ν) (Xt ≤ x, MtX ≥ y) = E exp[νWt − From the reflection principle (9.5) for y ≥ 0, x ≤ y, P (Wt ≤ x, Mt ≥ y) = P (x ≥ 2y − Wt , Mt ≥ y) hence E exp[νWt − ν2 t] 1 1 {Wt ≤ x, MtW ≥ y} 2 = = ν2 t] 1 1 {2y − Wt ≤ x, MtW ≥ y} 2 ν2 e2νy E exp[−νWt − t] 1 {W ≥ 2y − x} . 1 t 2 E exp[ν(2y − Wt ) − ν2 t] 1 1 . {Wt ≤ x, MtW ≥ y} 2 −x + 2y + νt −x + νt √ √ ) − e2νy N ( ). t t M. Jeanblanc 171 We apply again Cameron-Martin’s theorem E exp[−νWt − ν2 t] 1 {W ≥ 2y − x} 1 t 2 = = It follows that for y ≥ 0, y ≥ x P (Xt ≤ x, MtX ≥ y) = = P (Xt ≤ x, MtX ≤ y) = = = e2νy P (Xt ≥ 2y − x + 2νt) −2y + x − νt √ e2νy N ( ) t P (Xt ≤ x) − P (Xt ≤ x, MtX ≥ y) P (Xt ≤ x) − e2νy P (Xt ≥ 2y − x + 2νt) x − νt −2y + x − νt √ N ( √ ) − e2νy N ( ). t t P (Wt − νt ≥ 2y − x) = P (Xt ≥ 2y − x + 2νt) P (Wt ≤ −2y + x − 2νt) . Laws of maximum, minimum and hitting times In particular, the laws of the maximum and of the minimum are deduced : P (MtX ≤ y) P (MtX ≥ y) and for y > 0 P (Ty (X) ≥ t) = P (MtX ≤ y) . The law of the variable Ty (X) has density P (Ty (X) ∈ dt) = √ dt 2πt3 yeνy exp − 1 2 y2 + ν2t t =√ dt 2πt3 y exp − 1 2 (y − νt) 2t , y − νt −y − νt = N ( √ ) − e2νy N ( √ ), y ≥ 0 t t −y + νt −y − νt = N( √ ) + e2νy N ( √ ), y ≥ 0 t t named inverse Gaussian law with parameter (y, ν). In particular, when t → ∞ in y − νt −y − νt P (Ty ≥ t) = N ( √ ) − e2νy N ( √ ), t t we obtain P (Ty = ∞) = 1 − e2νy , for ν ≤ 0 and y > 0. −y + νt y + νt √ ) − e2νy N ( √ ), y ≤ 0 t t y − νt y + νt ≤ y) = N ( √ ) + e2νy N ( √ ), y ≤ 0 . t t P (mX ≥ y) = N ( t P (mX t Laplace transforms From Cameron-Martin’s theorem E(exp − (9.11) (9.12) λ2 ν 2 + λ2 Ty (X)) = E exp(νWTy − Ty (W )) 2 2 . 172 From Proposition 9.1.2, the right hand side equals Credit Risk, TUNIS 2005 1 eνy E[exp(− (ν 2 + λ2 )Ty (W ))] = eνy exp[−|y| ν 2 + λ2 ] . 2 Therefore λ2 Ty (X)) = eνy exp[−|y| ν 2 + λ2 ] . 2 Therefore, setting λ = 0 in (9.13), in the case ν > 0, y < 0, or ν < 0, y > 0 E(exp − P (Ty < ∞) = e2νy , which proves again that, in that case the probability that a Brownian motion with positive drift hits a negative level is not equal to one. In the case νy > 0, obviously P (Ty < ∞) = 1 . This is explained by the fact that (Wt + νt)/t goes to ν when t goes to infinity, hence the drift drives the process to infinity. In the case νy > 0, taking the derivative of (9.13) for λ = 0, we obtain E(Ty (X)) = y/ν. When νy < 0, the differentiation does not work, and obvioussly the expectation of the stopping time is equal to infinity. (9.13) 9.1.3 Hitting Times for Geometric Brownian Motion Let us assume that the dynamics of the risky asset are, under the risk neutral probability Q, given by dSt = St (µdt + σdWt ) , S0 = x (9.14) with σ > 0, i.e., St = x exp (µ − σ 2 /2)t + σWt = xeσXt , µ σ where µ = r − δ, Xt = νt + Wt , ν = − . We denote the first hitting time of a by σ 2 Ta (S) = inf{t ≥ 0 : St = a} = inf{t ≥ 0 : Xt = 1 ln(a/x)} σ 1 Then Ta (S) = Tα (X) where α = ln(a/x). When a level b is used for the geometric Brownian σ 1 motion S, we shall denote β = ln(b/x). σ Law of the pair (maximum, minimum) We deduce from Proposition 9.1.3 that for b > a, b > x α − νt α − 2β − νt √ P (St ≤ a, MtS ≤ b) = P (Xt ≤ α, MtX ≤ β) = N ( √ ) − e2νβ N ( ) t t whereas, for a > b, b < x P (St ≥ a, mS ≥ b) = P (Xt ≥ α, mX ≥ β) = N ( t t It follows that, for a > x (or α > 0) P (Ta (S) < t) = P (Tα (X) < t) = 1 − P (MtX ≤ α) = 1 − P (Xt ≤ α, MtX ≤ α) α − νt −νt − α = 1 − N ( √ ) + e2να N ( √ ) t t −α + νt −νt − α = N( √ ) + e2να N ( √ ) t t −α + νt −α + 2β + νt √ √ ) − e2νβ N ( ) t t (9.15) (9.16) (9.17) M. Jeanblanc 173 and, for a < x (or α < 0) P (Ta (S) < t) = = P (Tα (X) < t) = 1 − P (mX ≥ α) t α − νt νt + α 2να N( √ ) + e N( √ ). t t Laplace transforms From the previous remarks E(exp − Therefore, from (9.13) E(exp − λ2 Ta (S)) = exp να − |α| ν 2 + λ2 . 2 (9.18) λ2 λ2 Ta (S)) = E (ν) (exp − Tα (X)) . 2 2 9.1.4 Other processes OU Process Let (rt , t ≥ 0) be defined as drt = k(θ − rt ) dt + σ dWt , r0 = 0, and τρ = inf {t ≥ 0 : rt ≥ ρ}. For any ρ > r0 = 0, the density function of τρ equals f (t) = ρ √ σ 2π k sinh kt 3/2 ekt/2 exp − k 2σ2 ρ−θ 2 − θ2 + ρ2 coth kt . For the derivation of the last formula, the reader is referred to G¨ing and Yor [45]. The formula in o Leblanc and Scaillet [69] is only valid for r0 = 0. The Laplace transform of the stopping time τρ is known (see Borodin and Salminen [10]): Er (exp (−δτρ )) = where Υ(r) = exp (r − θ) 4σ 2 2 Υ(r) Υ(ρ) D−k − r−k σ where D is the parabolic cylinder function : D−ν (z) = exp − z2 4 √ 2−ν/2 π +∞ k 1 ν (ν + 2) . . . (ν + 2k − 2) z 2 1+ Γ ((ν + 1) /2) 3.5 . . . (2k − 1) k! 2 k=1 √ +∞ z 2 (ν + 1) (ν + 3) . . . (ν + 2k − 1) z 2 − 1+ Γ (ν/2) 3.5 . . . (2k + 1) k! 2 k=1 k . 174 Credit Risk, TUNIS 2005 9.1.5 Non-constant Barrier The case of non-constant barrier would be of great interest. For example, the process X is a geometric Brownian motion with deterministic volatility dSt = St (rdt + σ(t)dBt ), S0 = x and 1 Ta (S) = inf{t : St = a} = inf{t : rt − [ 2 0 t t t σ(s)]2 ds + 0 σs dBs = α} , where α = ln(a/x) As we shall see below, the process Ut = 0 σs dBs is a changed time Brownian t 0 motion and can be written as ZA(t) where Z is a brownian motion and A(t) = introducing the inverse C of the function A [σ(s)]2 ds. Hence, 1 1 Ta (S) = inf{t : rt − A(t) + ZA(t) = α} = inf{C(u) : rC(u) − u + Zu = α} , 2 2 and we are reduced to the study of the hitting time of the non-constant boundary C(u) by the drifted Brownian motion Zt − 1 t. 2 Bibliography More generally, let τf (V ) = inf {t ≥ 0 : Vt = f (t)}, where f is a deterministic function and V a diffusion process. There are only few cases for which the law of τf (V ) is explicitly known; for instance, the previous case when V is a Brownian motion and f is an affine function. This problem is studied in a general framework in Alili’s thesis [1], Barndorff-Nielsen et al. [3], Daniels [25], Durbin [37], Ferebee [42], Hobson et al. [51], Jennen and Lerche [62][63], Lerche [72], Salminen [82] and Siegmund and Yuh [86]. √ Breiman [11] studies the case of a square root boundary, i.e. T = inf{t : x + Bt = α t} . Groeneboom [47] studies the case T = inf{t : x + Bt = αt2 }. For any x > 0 and α < 0, 2 Ai(λn − 2αcx) Px (T ∈ dt) = 2(αc)2 Σ∞ exp(−µn − α2 t3 ) n=0 3 Ai (λn ) here λn are the zeros on the negative half-line of the Airy function Ai, the unique bounded solution of u − xu = 0, u(0) = 1, and µn = −λn /c. This last expression was obtained by Salminen [82]. The Airy function is defined as (Ai)(x) = def 1 π x 3 1/2 K1/3 2 3/2 x 3 . 9.1.6 Let Fokker-Planck equation dXt = b(t, xt )dt + σ(t, X − t)dWt be a diffusion. M. Jeanblanc 175 Proposition 9.1.4 Let h be a deterministic function, τ = inf{t ≥ 0 : Xt ≤ h(t)} and g(t, x)dx = P (Xt ∈ dx, τ > t) . The measure g(t, x)dx satisfies d ∂ 1 ∂2 2 g(t, x) = − (b(t, x)g(t, x)) + σ (t, x)g(t, x) ; x > h(t) dt ∂x 2 ∂x2 and the boundary conditions g(t, x)dx|t=0 g(t, x)|x=h(t) = δ(x − X0 ) = 0 Proof: The proof is done as for the backward Kolmogorov equation. If ϕ is a C 2 function with compact support, for s < t t∧τ ϕ(Xt∧τ ) = ϕ(Xs∧τ ) + s∧τ ϕ (Xu )dXu + 1 2 t∧τ s∧τ ϕ (Xu )σ 2 (u, Xu )du . (9.19) Hence, on the one hand E(ϕ(Xt∧τ )) = = = E(ϕ(Xt )1 {t≤τ } ) + E(ϕ(Xτ )1 {τ 0. Hence, the explosion time is infinite and P (Nt = n) = e−λt (λt)n . n! The standard Poisson process can be redefined as follows (See e.g., Cinlar [?]): it is a counting process without explosion (i.e., T = ∞) such that N - for every s, t, Nt+s − Nt is independent of Ft , - for every s, t, the r.v. Nt+s − Nt has the same law as Ns . or, in an equivalent way, its increments are independent and stationary. Definition 9.3.1 Let F be a given filtration and λ a positive constant. The process N is an FPoisson process with intensity λ if N is an F-adapted process, such that for any (t, s), the random variable Nt+s − Nt is independent of Ft and follows the Poisson law with parameter λs. Exercise 9.3.1 Let N be a Poisson process and Tn its n-th jump time. Prove that ∞ P (Tn ≥ x|Ft ) = 1 x≤Tn ≤t + 1 t0} dt, 1 (n − 1)! . From the properties of the Poisson distribution, it follows that for every t > 0, E(Nt ) = λt, and for every x > 0, t > 0, u, α ∈ I R E(xNt ) = eλt(x−1) ; E(eiuNt ) = eλt(e iu Var (Nt ) = λt −1) ; E(eαNt ) = eλt(e α −1) . (9.20) 178 Credit Risk, TUNIS 2005 • Conditionally on (Nt = n), the law of (T1 , T2 , . . . , Tn ) is a multinomial distribution . (t) (t) • Let Ti be equal to the time of the i-th jump which occurs after t minus t, i.e., Ti = TNt +i − t, (t) for i ≥ 1. It can be proved that, for any t, the sequence of times (Ti , 1 ≤ i) has the same law as (Ti , 1 ≤ i). This property is called the lack of memory of the Poisson process. We can associate a random measure to any Poisson process as follows. For any Borel set Λ, for any ω, let µ(ω, Λ) = #{n ≥ 1 : Tn (ω) ∈ Λ} . The quantity µ(Λ) is almost surely finite for any bounded set Λ, and E(µ(Λ)) = λ|Λ| where |Λ| is the Lebesgue measure of the set Λ. The random variable Nt can be written as Nt (ω) = ]0,t] µ(ω, ds). Martingale Properties From the independence of the increments of the Poisson process, we derive the following martingale properties: Proposition 9.3.1 Let N be an F-Poisson process. For each α ∈ I for each bounded Borel R, function h, the following processes are F-martingales: (i) (ii) (iii) (iv) Mt = Nt − λt, Mt2 − λt = (Nt − λt)2 − λt, exp(αNt − λt(eα − 1)), t t (9.21) (9.22) exp 0 h(s)dNs − λ 0 (eh(s) − 1)ds . Proof: Let s < t. From the independence of increments of the Poisson process, we obtain (i) E(Mt − Ms |Fs ) = E(Nt − Ns ) − λ(t − s) = 0, hence M is a martingale. (ii) The martingale property of M implies 2 E(Mt2 − Ms |Fs ) = E[(Mt − Ms )2 |Fs ] = E[(Nt − Ns − λ(t − s))2 |Fs ] = E[(Nt − Ns )2 ] − λ2 (t − s)2 2 = E[Nt−s ] − λ2 (t − s)2 = VarNt−s , 2 E(Mt2 − Ms |Fs ) = λ(t − s) , hence, and the process (Mt2 − λt, t ≥ 0) is a martingale. (iii) E[exp[α(Nt − Ns ) − λ(t − s)(eα − 1)]|Fs ] = 1, hence the martingale property of the process 1 in (iii). Assertion (iv) can be proved for elementary functions h = i ai 1 {]ti ,ti+1 ]} and passing to the limit for general bounded Borel functions h. Exercise 9.3.2 Prove that, for any β > −1, any bounded Borel function h, and any bounded Borel function ϕ valued in ] − 1, ∞[, the processes exp[ln(1 + β)Nt − λβt] = (1 + β)Nt e−λβt , t t exp 0 h(s)dNs + λ 0 (1 − eh(s) )ds t t = exp 0 t h(s)dMs + λ 0 t (1 + h(s) − eh(s) )ds , exp 0 ln(1 + ϕ(s))dNs − λ 0 t ϕ(s)ds t = exp 0 ln(1 + ϕ(s))dMs + λ 0 (ln(1 + ϕ(s)) − ϕ(s))ds , M. Jeanblanc 179 are martingales. Hint: these formulae are “avatars” of those of Proposition 9.3.1. Definition 9.3.2 The martingale (Mt = Nt − λt, t ≥ 0) is called the compensated process of N , and λ is the intensity of the process N . Remark 9.3.1 Note that M is a discontinuous martingale with bounded variation. t 1 We give an example of martingale which is not square integrable. let Xt = 0 √s d(Ns − λs). The process X is a martingale, however, this is not a square integrable martingale. The previous Proposition 9.3.1 can be generalized to predictable integrands: Proposition 9.3.2 Let N be an F-Poisson process and H be an F-predictable bounded process, then the following processes are martingales t t t (H ((H exp M )t = 0 Hs dMs = t 0 0 2 Hs ds t 0 Hs dNs − λ 0 Hs ds (9.23) M )t )2 − λ t 0 Hs dNs + λ (1 − eHs )ds Proof: One establishes (9.23) for predictable processes of the form Ht = KS 1 ]S,T ] (t) where S and 1 T are two stopping times and KS is FS -measurable. In that case, t Hs dMs = KS (MT ∧t − MS∧t ) 0 and the martingale property follows. Then, one passes to the limit. The same procedure can be applied to prove that the two last processes of (9.23) are martingales. Warning 2 The martingale property (9.23) does not extend to adapted processes H. Indeed, from the definition of the stochastic integral w.r.t. N , and the fact that ∀s, Ns − Ns− = 0, P a.s., t t t (Ns − Ns− )dMs 0 = 0 (Ns − Ns− )dNs − λ 0 t (Ns − Ns− )ds = Nt − λ 0 (Ns − Ns− )ds = Nt . Hence, the left-hand side, where one integrates the adapted process Ns − Ns− with respect to the martingale M is not a martingale. Equivalently, the process t t Ns dMs = 0 0 Ns− dMs + Nt , is not a martingale. Comments 9.3.1 Property (i) of Proposition 9.3.1 enables us to prove that the jump times (Ti , i ≥ 1) are not predictable. Indeed, T1 were a predictable stopping time, the process (1 {t −1 and Lt = exp(log(1 + φ)Nt − λφt), then dLt = Lt− φdMt . More generally, for any a ∈ I prove that Lt = (1 + a)Nt e−λat satisfies R, dLt = Lt− adMt . Not that, for a < −1, Lt takes values in I The process L is the Dol´ans-Dade R. e exponential of the martingale aM . M. Jeanblanc 181 Watanabe’s Characterization of the Poisson Process Let N be a counting process and assume that there exists a constant λ > 0 such that Mt = Nt − λt is a martingale. Then N is a Poisson process with intensity λ. This result will be proved in a more general setting in Proposition 9.3.7. Change of Probability If N is a Poisson process with constant intensity λ, then, from Exercises 9.3.2 and 9.3.3, for β > −1, Lt = (1 + β)Nt e−λβt is a strictly positive martingale with expectation equal to 1. Let Q be the probability defined as Q|Ft = Lt P |Ft . From EQ (xNt ) = EP (Lt xNt ) = e−λβt EP ([(1 + β)x]Nt ) = exp((1 + β)λt(x − 1)) we deduce that the r.v. Nt follows the Poisson law of parameter (1 + β)λt under Q. The equalities n n EQ i=1 xi Nti+1 −Nti = EP n ((1 + β)xi )Nti+1 −Nti e−λβt i=1 = i=1 n e−λ(ti+1 −ti ) eλ(ti+1 −ti )(1+β)xi e−λβt e(1+β)λ(ti+1 −ti )(xi −1) , i=1 law = establish that N has independent increments under Q and that Nti+1 − Nti = Nti+1 −ti . Therefore, the process N is a Q-Poisson process with intensity equal to (1 + β)λ. Let us state this result as a proposition: Proposition 9.3.4 Let Πλ be the probability on the canonical space such that the canonical process is a Poisson process with intensity λ. Then, the following absolute continuity relationship holds Π(1+β)λ |Ft = (1 + β)Nt e−λβt Πλ |Ft . Comments 9.3.2 One should note the analogy between the change of intensity of Poisson processes and the change of drift of a BM under a change of probability. The process N remains a Poisson process under the above change of probability. However, let us point out a main difference. If Q is equivalent to P , we know that if B is a P -BM and B the martingale part of B under Q, then 2 2 Bt − t is a P -martingale and Bt − t is a Q-martingale (in other terms the brackets are the same, i.e., B = B ). If Q is equivalent to P , and Mt = Nt − λt the compensated martingale associated with a Poisson process, the process Mt2 − λt is a P -martingale and the P -(predictable) bracket of M is λt. We have proved above that the Q-(predictable) bracket of Mt = Nt − (1 + β)λt is (1 + β)λt. Hence, the predictable bracket is no longer the same under a change of probability. See Section ?? for a general Girsanov theorem and Section ?? for the case of L´vy processes. e Hitting Times Let Tx = inf{t, Nt ≥ x}. Then, for n − 1 < x ≤ n, Tx is equal to the time of the nth -jump of N , hence has a Gamma (n, λ) law. Exercise 9.3.4 Let Xt = Nt + ct. Compute P (inf s≤t Xs ≤ a). One should distinguish the cases c > 0 and c < 0. 182 Credit Risk, TUNIS 2005 9.3.2 Inhomogeneous Poisson Processes Definition Instead of considering a constant intensity λ as before, now (λ(t), t ≥ 0) is an I + -valued function R t satisfying 0 λ(u)du < ∞, ∀t. An inhomogeneous Poisson process N with intensity λ is a (Λ(s, t))n n! counting process with independent increments which satisfies for t > s P (Nt − Ns = n) = e−Λ(s,t) t t (9.25) where Λ(s, t) = Λ(t) − Λ(s) = s λ(u)du, and Λ(t) = 0 t 0 λ(u)du. If (Tn , n ≥ 1) is the sequence of successive jump times associated with N , the law of Tn is: P (Tn ≤ t) = 1 n! exp(−Λ(s)) (Λ(s))n−1 dΛ(s) . It can easily be shown that an inhomogeneous Poisson process with deterministic intensity is an inhomogeneous Markov process. Moreover, E(Nt ) = Λ(t), Var(Nt ) = Λ(t). An inhomogeneous Poisson process can be constructed as a deterministic changed time Poisson process. Martingale Properties Proposition 9.3.5 Let N be an inhomogeneous Poisson process with deterministic intensity λ and FN its natural filtration. The process t (Mt = Nt − 0 λ(s)ds, t ≥ 0) t 0 is an FN -martingale, and the increasing function Λ(t) = pensator of N . t λ(s)ds is called the (deterministic) com- Let φ be an FN -predictable process such that E( 0 |φs |λ(s)ds) < ∞ for every t. Then, the process t ( 0 φs dMs , t ≥ 0) is an FN -martingale. In particular, t t E 0 φs dNs =E 0 φs λ(s)ds . (9.26) As in the constant intensity case, for any bounded predictable process H, the following processes are martingales t t t a) b) c) (H ((H exp M )t = 0 Hs dMs = t 0 0 2 λ(s)Hs ds t 0 Hs dNs − 0 λ(s)Hs ds M )t )2 − t 0 Hs dNs − λ(s)(eHs − 1)ds . As a consequence of the martingale property process in (c), we obtain the important result which generalizes the constant intensity case: Proposition 9.3.6 (Compensation formula.) Let N be an inhomogeneous Poisson process with deterministic intensity λ, then, for any real numbers u and α, for any t ≥ 0 E(eiuNt ) = E(e αNt exp((eiu − 1)Λ(t)) exp((eα − 1)Λ(t)) . ) = M. Jeanblanc 183 Watanabe’s Characterization of Inhomogeneous Poisson Processes The study of inhomogeneous Poisson processes can be generalized to the case where the intensity is not absolutely continuous with respect to the Lebesgue measure. In this case, Λ is an increasing, right continuous deterministic function with zero value at time zero, and it satisfies Λ(∞) = ∞. If N is a counting process with independent increments and if (9.25) holds, the process (Nt −Λ(t), t ≥ 0) is t t a martingale and for any bounded predictable process φ, the equality E( 0 φs dNs ) = E( 0 φs dΛ(s)) is satisfied for any t. This result admits a converse. Proposition 9.3.7 (Watanabe’s characterization.) Let N be a counting process and Λ an increasing, continuous function with zero value at time zero. Let us assume that the process Mt = Nt − Λ(t) is a martingale. Then N is an inhomogeneous Poisson process with compensator Λ. Proof: Let s < t and θ > 0. eθNt − eθNs = s s, φ(t) = φ(s) + (eθ − 1) s t φ(u)dΛ(u) . Solving this equation leads to φ(t) = eθNs exp (eθ − 1) s t dΛ(u) . This shows that the process N has independent increments and that, for s < t, the r.v. Nt − Ns has a Poisson law with parameter Λ(t) − Λ(s) . Stochastic Calculus In this section, M is the compensated martingale of an inhomogeneous Poisson process N with deterministic intensity (λ(s), s ≥ 0). From now on, we restrict our attention to integrals of predictable processes, even if the stochastic integral is defined in a more general setting. Integration by parts formula t Let g and g be two predictable processes and define two processes X and Y as Xt = x + ˜ t 0 gs dNs and Yt = y + 0 gs dNs . The jumps of X (resp. of Y ) occur at the same times as the jumps of N 184 Credit Risk, TUNIS 2005 and ∆Xs = gs ∆Ns , ∆Ys = gs ∆Ns . The processes X and Y are of finite variation and are constant ˜ between two jumps. Then Xt Yt = xy + s≤t t t ∆(XY )s = xy + s≤t Xs− ∆Ys + s≤t Ys− ∆Xs + s≤t ∆Xs ∆Ys = xy + 0 Ys− dXs + 0 Xs− dYs + [X, Y ]t where (note that (∆Nt )2 = ∆Nt ) t [X, Y ]t = s≤t ∆Xs ∆Ys = s≤t gs gs ∆Ns = 0 gs gs dNs . More generally, if dXt = ht dt + gt dNt with X0 = x and dYt = ht dt + gt dNt with Y0 = y, one gets t t Xt Yt = xy + 0 Ys− dXs + 0 t Xs− dYs + [X, Y ]t where [X, Y ]t = 0 gs gs dNs . In particular, if dXt = gt dMt and dYt = gt dMt , the process Xt Yt − [X, Y ]t is a martingale. Itˆ’s Formula o For Poisson processes, Itˆ’s formula is obvious as we now explain. We shall give an extension of this o formula for more general processes in the following Chapter ??. Let N be a Poisson process and f a bounded Borel function. The decomposition f (Nt ) = f (N0 ) + 0 0, the process (Lt = exp(Nt ln(1 + a) − λat), t ≥ 0) is a martingale and that, if Q|Ft = Lt P |Ft , the process N is a Q-Poisson process with intensity λ(1 + a). Note the progression made from Exercise 9.3.3. Predictable Representation Property Proposition 9.3.8 Let FN be the completion of the canonical filtration of the Poisson process N N and H ∈ L2 (F∞ ), a square integrable random variable. Then, there exists a unique predictable process (ht , t ≥ 0) such that ∞ H = E(H) + 0 hs dMs and E( ∞ 0 h2 ds) < ∞. s Proof: The family of exponential random variables ∞ ∞ Y = exp 0 ϕ(s)dNs − λ 0 (eϕ(s) − 1)ds , N where ϕ is a bounded deterministic function with compact support, is total in L2 (F∞ ). Any Y in this family can be written as a stochastic integral with respect to dM . Indeed, from Exercise 9.3.2 the process t t Yt = exp 0 ϕ(s)dNs − λ 0 (eϕ(s) − 1)ds N = E(Y |Ft ) is a martingale, and is the solution of dYt = Yt− (eϕ(t) − 1)dMt , so that, Y =1+ 0 ∞ Ys− (eϕ(s) − 1)dMs . Hence, with the notation of the statement, hs = Ys− (eϕ(s) − 1). For more general random variables, the result follows by passing to the limit. Comments 9.3.3 This result goes back to Br´maud and Jacod [?], Chou and Meyer [19], Davis e [?]. M. Jeanblanc 187 9.4 9.4.1 General theory Semimartingales Recall that if X is a semimartingale, then the process Z = E(X) is the unique solution to the SDE Zt = 1 + ]0,t] Zu− dXu . It is known that Et (X) = exp Xt − X0 − c 1 c X 2 t u≤t (1 + ∆Xu )e−∆Xu , where X is the continuous martingale component of X. 9.4.2 9.4.3 Itˆ’s formula o Stopping times Predictable stopping time, Totally inacessible 9.5 Markov Chains Let X be a right-continuous process with values in a finite set E. Le G be some filtration larger than the natural filtration of X. Definition 9.5.1 A process X is a continuous time G-Markov-chain if for any function h : E → I R and any t, t E(h(Xt+s )|Gt ) = E(h(Xt+s )|Xt ) = Ψ(t, Xt , t + s) A continuous time G-Markov-chain is time homogeneous if Ψ(t, x, t + s) = Ψ(t + u, x, t + s + u) Definition 9.5.2 A family pi,j (t, s) is called a transition probability matrix if P (Xs = j|Xt = i) = pi,j (t, s) In the case of time-homogeneous Markov chain, the one-parameter family pi,j (t) is the family of transition probability if P (Xs+t = j|Xt = i) = pi,j (s) Observe that for all i, pi,j ≥ 0 for all i, Then P (Xs+t ∈ A|Xt = i) = j∈A j∈E pi,j = 1 pi,j (s) The Chapman-Kolmogorov equation pi,j (t + s) = k∈E pi,k (t)pk,j (s) = k∈E pi,k (s)pk,j (s) 188 is satisfied and can be written in a matrix form P (t + s) = P (t)P (s) = P (s)P (t) The following limit λi,j = lim exists. Observe that for i = j, λi,j ≥ 0 λi,i = − j=i Credit Risk, TUNIS 2005 pi,j (t) − pi,j (0) pi,j (t) − δi,j = lim t t λi, j The matrix Λ = (λi,j ) is called the infinitesimal generator matrix. The backward Kologorov equation is dP (t) = ΛP (t), P (0) = Id dt The forward Kolmogorov equation is dP (t) = P (t)Λ, P (0) = Id dt These equation have the unique solution P (t) = etΛ Note that, for any function h, the process t h(Xt ) − 0 (Λh)(Xu )du is a martingale. Elementary case Let us study a continuous time Markov chain with two states 0 and 1. T If P0,0 (t) = P (τ > t) = e−λt , P0,1 (t) = 1 − e−λt , P1,0 (t) = 0, P1,1 (t) = 1 The transition matrix is P (t) = and can be written in the form P (t) = eΛt = n e−λt 0 1 − e−λt 1 (tΛ)n n! −λ λ . The matrix Λ is called the generator of the Markov chain. The probability 0 0 for going from state 0 to state 1 between the date t and t + dt is λdt. (See Karlin and Taylor [64]) with Λ = 9.6 9.6.1 Ornstein-Uhlenbeck processes Vacisek model Proposition 9.6.1 Let k, θ and σ be bounded Borel functions, and W a Brownian motion. The solution of drt = k(t)(θ(t) − rt )dt + σ(t)dWt (9.31) M. Jeanblanc 189 is rt = e−K(t) r0 + 0 t t eK(s) k(s)θ(s)ds + 0 t eK(s) σ(s)dWs where K(t) = 0 k(s)ds. The process (rt , t ≥ 0) is a Gaussian process with mean E(rt ) = e−K(t) r0 + 0 t eK(s) k(s)θ(s)ds and covariance e−K(t)−K(s) 0 t∧s e2K(u) σ 2 (u)du . Proof: The solution of (9.31) is a particular case of Example ??. The values of the mean and of the covariance follow from Exercise ??. The Hull and White model correspond to the dynamics (9.31) where k is a positive function. In the particular case where θ and k are constant, we obtain Corollary 9.6.1 The solution of drt = k(θ − rt )dt + σdWt is rt = (r0 − θ)e−kt + θ + σ 0 (9.32) t e−k(t−u) dWu . The process (rt , t ≥ 0) is a Gaussian process with mean (r0 − θ)e−kt + θ and covariance Cov(rs , rt ) = for s ≤ t. In finance, the solution of (9.32) is called a Vasicek process. In general, k is chosen to be positive, so that E(rt ) → θ as t → ∞. The process (9.31) is called a Generalized Vasicek process (GV). Since r is a Gaussian process, it can take negative values. This is one of the reasons why this process is no longer used for modelling interest rates. When θ = 0, the process r is called an OrnsteinUhlenbeck (OU) process. Consequently, for a general θ, the process (rt − θ, t ≥ 0) is a OU process with parameter k. More formally, here is a Definition 9.6.1 An Ornstein-Uhlenbeck (OU) process driven by a BM follows the dynamics drt = −krt dt + σdWt . An OU process can be constructed in terms of time-changed BM: Proposition 9.6.2 i) If W is a BM starting from x and A(t) = σ 2 e2kt − 1 , the process Zt = 2k σ 2 −k(s+t) 2ks σ 2 −kt e (e − 1) = e sinh(ks) 2k k e−kt WA(t) is an OU process starting from x. ii) Conversely, if U is an OU process starting from x, then there exists a BM W starting from x such that Ut = e−kt WA(t) . Proof: Indeed, the process Z is a Gaussian process, with mean xe−kt and covariance e−k(t+s) (A(t)∧ A(s)). From the Markov property of the process r it follows, in the case of constant coefficients: 190 Credit Risk, TUNIS 2005 Proposition 9.6.3 Let r be the solution of (9.32) and F the natural filtration of the Brownian motion W . For s < t, the conditional expectation and the conditional variance of rt with respect to Fs are given by E(rt |rs ) = Vars (rt ) = (rs − θ)e−k(t−s) + θ σ2 (1 − e−2k(t−s) ) . 2k Note that the filtration generated by the process r is equal to the Brownian filtration. Due to the t Gaussian property of the process r, the integrated process 0 rs ds can be characterized as follows: Proposition 9.6.4 Let r be a solution of (9.32). The process mean E( t r ds) 0 s t r ds, 0 s t≥0 is Gaussian with = θt + (r0 − Var( 0 −kt θ) 1−e , k variance σ2 1 − e−kt σ2 (1 − e−kt )2 + 2 (t − ) 3 2k k k t rs ds) = − and covariance (for s < t) σ2 k2 s − e−kt eks − 1 1 − e−ks e2ks − 1 − + e−k(t+s) k k 2k t . Proof: From the definition, rt = r0 + kθt − k 0 t rs ds + σWt , hence rs ds = 0 = 1 [−rt + r0 + kθt + σWt ] k 1 [kθt + (r0 − θ)(1 − e−kt ) − σ k t 0 e−k(t−u) dWu + σWt ]. Obviously, from the properties of the Wiener integral, the right-hand side defines a Gaussian process. It remains to compute the expectation and the variance of the Gaussian variable on the right-hand side. More generally, for t ≥ s t E s t ru du|Fs ru du =− = θ(t − s) + (rs − θ) 1 − e−k(t−s) def = M (s, t) k Var s s σ2 σ2 1 − e−k(t−s) def (1 − e−k(t−s) )2 + 2 (t − s − ) = V (s, t) 3 2k k k where Vars is the conditional variance with respect to Fs . Exercise 9.6.1 Let B be a Brownian motion, and P b |FT = exp −b T Bs dBs − 0 b2 2 T 0 2 Bs ds P |FT . Prove that the process (Bt , t ≥ 0) is a P b -Ornstein-Uhlenbeck process and that EP 2 exp −αBt − b2 2 t 0 2 Bs ds b 2 2 = E b exp(−αBt + (Bt − t)) 2 . Deduce the L´vy’s area formula e EP 2 exp(−αBt − b2 2 t 0 2 Bs ds) = (cosh bt + 2 1 α sinh bt)− 2 . b M. Jeanblanc 191 t 0 Hint : Note that starting from a Bs dBs = 1 2 2 (Bt − t). One can also prove that if B is an n-dimensional BM b2 2 t 0 Ea exp(−α|Bt |2 − = Zero-coupon Bond cosh bt + 2α sinh bt b 2 |Bs |2 ds) exp − a2 b 1 + 2a cosh bt b 2 coth bt + 2α/b Suppose that the dynamics of the interest rate under the risk-neutral probability are given by (9.32). The value P (t, T ) of a zero-coupon bond maturing at date T is given as the conditional expectation of the discounted payoff. Using the Laplace transform of a Gaussian law (see Proposition ??), and using Proposition 9.6.4, we obtain T P (t, T ) = E i.e., exp − t ru du Ft 1 = exp(−M (t, T ) + V (t, T )) , 2 Proposition 9.6.5 In a Vasicek model, the price of a zero-coupon with maturity T is P (t, T ) = exp −θ(T − t) − (rt − θ) + σ2 1 − e−k(T −t) − 3 (1 − e−k(T −t) )2 k 4k σ2 1 − e−k(T −t) T −t− 2k 2 k = exp(a(t, T ) − b(t, T )rt ) , with b(t, T ) = 1−e−k(T −t) . k It is not difficult to check that the risk-neutral dynamics of the zero-coupon bond is dP (t, T ) = P (t, T )(rt dt − b(t, T )dWt ) . Note that we know in advance, without any computation that dP (t, T ) = P (t, T )(rt dt − σt dWt ) , since the discounted value of the zero-coupon bond is a martingale. It suffices to identify the volatility term. 9.7 9.7.1 Cox-Ingersoll-Ross Processes CIR Processes and BESQ From general Theorem on the existence of solutions to one dimensional SDE, the equation drt = k(θ − rt ) dt + σ |rt |dWt , (9.33) admits a unique solution which is strong. For θ = 0 and r0 = 0, the solution is rt = 0, and from the comparison Theorem ??, we deduce that, in the case kθ > 0, rt ≥ 0 for r0 ≥ 0. In that case, we omit the absolute value and consider the positive solution of √ drt = k(θ − rt ) dt + σ rt dWt . (9.34) 192 Credit Risk, TUNIS 2005 This solution is called a Cox-Ingersoll-Ross (CIR) process or a square-root process (See Feller [?]). For σ = 2, this process is the square of the norm of a δ-dimensional OU process, with parameter kθ. ( Section ?? and Section ??). We shall denote by k Qkθ,σ the law of the CIR process solution of the equation (9.33). In the case σ = 2, we simply note k Qkθ,2 = k Qkθ . The elementary change of time A(t) = 4t/σ 2 reduces the study of the solution of (9.34) to the case σ = 2: indeed, if Zt = r(4t/σ 2 ), then dZt = k (θ − Zt ) dt + 2 with k = 4k/σ 2 and B a Brownian motion. Many authors prefer to write the dynamics of a square root process as drt = (ν − λrt ) dt + σ |rt |dWt (9.35) Zt dBt allowing to consider the interesting case ν = 0. In the case ν = 0, when a CIR process hits 0, it remains at 0. Proposition 9.7.1 The CIR process (9.34) is a space-time changed BESQ process: more precisely, rt = e−kt ρ σ 2 kt 4k (e − 1) 4kθ . σ2 where (ρ(s), s ≥ 0) is a BESQδ process, with dimension δ = Proof: See the following Theorem ??. It follows that for 2kθ ≥ σ 2 , a CIR process starting from a positive initial point stays always positive. For 0 ≤ 2kθ < σ 2 , a CIR process starting from a positive initial point hits 0 with probability x x p ∈]0, 1[ if k < 0 (P (T0 < ∞) = p) and almost surely if k ≥ 0 (P (T0 < ∞) = 1). In the case 0 < 2kθ, the boundary 0 is instantaneously reflecting, whereas in the case 2kθ < 0, the process r starting from a positive initial point remains positive until T0 = inf{t : rt = 0}. Setting Zt = −rT0 +t , we obtain that dZt = (−δ + λZt )dt + σ |Zt |dBt where B is a BM. We know that Zt ≥ 0, thus rT0 +t takes values in I − . R Absolute Continuity Relationship A routine application of Girsanov’s theorem (See Example ?? or Proposition ??) leads to k Qkθ |Ft = exp x k2 k [x + kθt − ρt ] − 4 8 t 0 ρs ds Qkθ |Ft x (9.36) Comments 9.7.1 From an elementary point of view, if the process r reaches 0 at time t, the formal equality between drt and kθdt explains that the increment of rt is positive if kθ > 0. Again formally, for k > 0, if at time t, the inequality rt > θ holds (resp. rt < θ), then the drift k(θ − rt ) is negative (resp. positive) and, at least in mean, r M. Jeanblanc 193 9.7.2 Transition Probabilities for a CIR Process From the expression of a CIR process as a squared Bessel process time-changed, using the transition density of the squared Bessel process given in (??), we obtain its transition density. Proposition 9.7.2 Let r be a CIR process following (9.34). The transition density k Qkθ,σ (rt+s ∈ dy|rs = x) = ft (x, y)dy is given by ft (x, y) = where c(t) = ekt 2c(t) yekt x ν/2 exp − x + yekt 2c(t) Iν 1 c(t) xyekt 1 {y≥0} , 1 σ 2 kt 2kθ (e − 1) and ν = 2 − 1. The cumulative distribution function is 4k σ k Qkθ,σ (rt < y) = χ2 x 4kθ x yekt , ; σ 2 c(t) c(t) , where the function χ2 is defined in Exercise ??. Proof: From the relation rt = e−kt ρc(t) , where ρ is a BES(ν) , we obtain k Qkθ,σ (rt+s ∈ dy|rs = x) = ekt qc(t) (x, yekt )dy . (ν) Denoting by (rt (x); t ≥ 0) the CIR process with initial value r0 = x, the random variable Yt = rt (x)ekt [c(t)]−1 has density P (Yt ∈ dy)/dy = = c(t)e−kt ft (x, yc(t)e−kt )1 {y>0} 1 e−α/2 −y/2 ν/2 √ 1 e y Iν ( yα)1 {y≥0} 2αν/2 where α = x/c(t). This law is a non-central chi-square χ2 (δ, α) with δ = 2(ν + 1) degrees of freedom, and α the parameter of non-centrality. Using the notation of Exercise ??, we obtain k Qkθ,σ (rt < y) = χ2 x 4kθ x yekt , ; σ 2 c(t) c(t) . 9.7.3 CIR Processes as Spot Rate Models The Cox-Ingersoll-Ross model for the short interest rate is the object of many studies since the seminal paper of Cox et al. [?] where the authors assume that the riskless rate r follows a square root process under the historical probability given by √ ˜ ˜ ˜ drt = k(θ − rt ) dt + σ rt dWt . ˜ ˜ ˜ Here k(θ − r) defines a mean reverting drift pulling the interest rate towards its long term value θ ˜ In the risk adjusted economy, the dynamics are supposed to with a speed of adjustment equal to k. be given by: √ ˜ ˜ drt = (k(θ − rt ) − λrt )dt + σ rt dWt t λ√ where (Wt = Wt + 0 σ rs ds, t ≥ 0) is a Brownian motion under the risk adjusted probability Q ˜ ˜ ˜ where λ denotes the market price of risk. Setting k = k + λ, θ = k(θ/k), the Q -dynamics of r are √ drt = k(θ − rt )dt + σ rt dWt . Therefore, we shall establish formulae under general dynamics of the form (9.34). Even if no closed-form expression as a functional of W can be written for rt , it is remarkable that the Laplace transform of the process, i.e. k Qkθ,σ exp − x t du φ(u)ru 0 is known (See Exercise ??). 194 Th´or`me 9.3 Let r be a CIR process, the solution of e e √ drt = k(θ − rt )dt + σ rt dWt . Credit Risk, TUNIS 2005 (9.37) The conditional expectation and the conditional variance of the r.v. rt are given by, for s < t, k Qkθ,σ (rt |Fs ) = rs e−k(t−s) + θ(1 − e−k(t−s) ), x σ 2 (e−k(t−s) − e−2k(t−s) ) θσ 2 (1 − e−k(t−s) )2 + . k 2k t t Var(rt |Fs ) = rs Proof: From the definition, for s ≤ t, one has rt = rs + k s (θ − ru )du + σ s t √ ru dWu . t Itˆ’s formula leads to o 2 rt = = 2 rs + 2k t (θ − ru )ru du + 2σ s t s t (ru )3/2 dWu + σ 2 s 2 ru du + 2σ t s ru du 2 rs + (2kθ + σ 2 ) ru du − 2k s s (ru )3/2 dWu . It can be checked that the stochastic integrals involved in both formulae are martingales: indeed, r is a time changed Bessel process, hence it admits moments of any order. Therefore, the expectation of rt is given by E(rt ) =k Qkθ,σ (rt ) = r0 + k θt − x t E(ru )du 0 . t We now introduce Φ(t) = E(rt ). The integral equation Φ(t) = r0 + k(θt − 0 Φ(u)du) can be written in differential form Φ (t) = k(θ − Φ(t)) where Φ satisfies the initial condition Φ(0) = r0 . Hence E[r(t)] = θ + (r0 − θ)e−kt . In the same way, from 2 2 E(rt ) = r0 + (2kθ + σ 2 ) t t E(ru )du − 2k 0 0 2 E(ru )du 2 setting Ψ(t) = E(rt ) leads to Ψ (t) = (2kθ + σ 2 )Φ(t) − 2kΨ(t), hence σ2 θ (1 − e−kt )[r0 e−kt + (1 − e−kt )] . k 2 Thanks to the Markovian character of r, the conditional expectation can also be computed: Var [rt ] = E(rt |Fs ) Var(rt |Fs ) = θ + (rs − θ)e−k(t−s) = rs e−k(t−s) + θ(1 − e−k(t−s) ), = rs σ 2 (e−k(t−s) − e−2k(t−s) ) θσ 2 (1 − e−k(t−s) )2 + . k 2k Note that, if k > 0, E(rt ) → θ as t goes to infinity. Comments 9.7.2 Using an induction procedure, or using computations done for squared Bessel processes, all the moments of rt can be computed. See Dufresne [?]. Exercise 9.7.1 If r is a CIR process and Z = rα , prove that dZt = αZt 1−1/α (kθ + (α − 1)σ 2 /2) − Zt αk dt + αZt 3/2 1−1/(2α) σdWt . In particular, for α = −1, dZt = Zt (k − Zt (kθ − σ 2 ))dt − Zt Section ?? on CEV processes and Lewis [?]). σdWt is the so-called 3/2 model (see M. Jeanblanc 195 9.7.4 Zero-coupon Bond We now address the problem of the valuation of a zero-coupon bond, i.e., we assume that the dynamics of the interest rate are given by a CIR process under the risk neutral probability and we T compute E exp − t ru du |Ft . Proposition 9.7.3 Let r be a CIR process defined as in (9.34) by √ drt = k(θ − rt ) dt + σ rt dWt , and k Qkθ,σ its law. Then, for any pair (λ, µ) of positive numbers k Qkθ,σ x T exp −λrT − µ 0 ru du = exp[−Aλ,µ (T ) − xGλ,µ (T )] with Gλ,µ (s) Aλ,µ (s) where γ = k 2 + 2σ 2 µ . = λ(γ + k + eγs (γ − k)) + 2µ(eγs − 1) σ 2 λ(eγs − 1) + γ(eγs + 1) + k(eγs − 1) 2kθ ln σ2 σ 2 λ(eγs 2γe(γ+k)s/2 − 1) + γ(eγs + 1) + k(eγs − 1) = − Proof: Using Itˆ’s formula, we obtain that the process o t ϕ(rt , t) exp −µ 0 rs ds is a martingale if ϕ satisfies the equation − ∂ϕ ∂ϕ 1 2 ∂ 2 ϕ = −xµϕ + k(θ − x) + σ x 2. ∂t ∂x 2 ∂x (9.38) Furthermore, if ϕ(x, T ) = e−λx , we obtain k Qkθ,σ x T exp −λrT − µ 0 ru du = ϕ(x, 0) It remains to prove that there exists two functions A and B such that exp(−A(T − t) − xG(T − t)) is a solution of the PDE (9.38, where A and B satisfy A(0) = 0, G(0) = λ. Some involved calculation leads to the proposition. Corollary 9.7.1 In particular, k Qkθ,σ exp(−µ x = e k2 θt/σ 2 t rs ds) 0 −2kθ/σ 2 γt γt k + sinh cosh 2 γ 2 exp −2µx k + γ coth γt 2 where γ 2 = k 2 + 2µσ 2 . These formulae may be considered as extensions of L´vy’s area formula for planar Brownian motion. e See Pitman and Yor [?]. 196 Credit Risk, TUNIS 2005 Corollary 9.7.2 Let r be a CIR process defined as in (9.34) under the risk-neutral probability. The-4 4(tet)]TJ F69.964531 .560Tt[(r)]TJ F129.963. .560T-timee-4pric(pr)ee-4ofe- 4 3(4ze2(pr)5-51(c)oup1(c)on3(4b( etrr et Index Absolute continuity Square CIR and BESQ, 192 Airy function, 174 Change of time for CIR, 192 Compensation formula, 182 Compensator, 27 Corporate discount bond, 7 Credit spread, 10 Default probability, 22 Default time, 6, 11 Defaultable zero-coupon bond, 7 Defaultable zero-coupon bond, 24 Defaultable zero-coupon, 24 Dividend process, 6 Dol´ans-Dade exponential, 180 e Expected discounted loss given default, 22 First passage time, 11 First passage structural models, 6 Fokker-Planck equation, 175 Forward martingale measure, 15 price of a defaultable bond, 16 value, 15 Hazard function, 23, 27 Infinitesimal generator of a Poisson process, 180 Integration by parts for Poisson processes, 183 Intensity of a Poisson process, 179 of a random time, 27 of an inhomogeneous Poisson process, 182 Intensity approach, 70 Itˆ’s formula o for Poisson processes, 184 Laplace transform 197 Law Gamma -, 177 Inverse Gaussian -, 171 Markov process, 180 Martingale measure, 6 Measure random -, 178 Model Hull and White -, 189 Vasicek -, 189 Partial information, 36 Predictable representation theorem, 66 for Poisson process, 186 Probability risk-neutral, 6 statistical, 6 Process CIR -, 192, 194 compensated -, 179 generalized Vasicek, 189 inhomogeneous Poisson -, 182 Ornstein-Uhlenbeck -, 189 Poisson -, 177 square-root -, 192 Vasicek -, 189 Range of prices, 34 Recovery, 6 Reflection principle, 167 Representation theorem in default setting, 28 Risk-neutral probability, 6 for DZC, 34 Spread, 35 credit -, 10, 23 forward short -, 10 Spreads, 25 Transition density hitting time for a GBM, 173 hitting time for OU, 173 of hitting time for a BM, 169 of hitting time for a drifted BM, 171 198 for a CIR, 193 Watanabe’s characterization, 183 Zero-coupon bond defaultable -, 21 deterministic case, 21 in a CIR framework, 195 in an OU framework, 191 Credit Risk, TUNIS 2005 Bibliography [1] L. 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