Document Sample

CDO Option Market Model A CDO Option Market Model for DORN J. standardized CDS Index Tranches Preliminaries Conference on Numerical Methods in Finance Motivation Recall : CDO Structure CDO Spread Determinants Previous Jochen DORN Research The Model Overview PRISM Research Center Option Pay-Off Finance Department Spread Dynamics Université Paris1 Panthéon-Sorbonne Conclusion Implementation April 16th 2009 CDO Option Economic Motivation Market Model DORN J. Market Context Preliminaries CDO is a OTC Product ⇒ High Transaction Costs Motivation Recall : CDO Structure "Liquidity Gap" costs precious Basis Points CDO Spread Determinants Previous Research ⇒ Initialization of a standardized synthetic CDO Market The Model (CDX/iTraXX) Overview Option Pay-Off Spread Dynamics Conclusion Implementation CDO Option Economic Motivation Market Model DORN J. Market Context Preliminaries CDO is a OTC Product ⇒ High Transaction Costs Motivation Recall : CDO Structure "Liquidity Gap" costs precious Basis Points CDO Spread Determinants Previous Research ⇒ Initialization of a standardized synthetic CDO Market The Model (CDX/iTraXX) Overview Option Pay-Off Spread Dynamics Modeling Constraints Conclusion Implementation Credit Derivatives : "Static" Models ⇒ The investor does not pay for the Véga ! Pricing of CDO tranches with option alike pay-oﬀs (Deal Spread, Cumulative Loss as underlying) "Maturity Trap" → need for Spread Dynamics ! CDO Option Objectives Market Model DORN J. Preliminaries Motivation Recall : CDO Structure CDO Spread Determinants Previous Research We consider a CDO tranche with AP D% and DP E % The Model and tenor [Ta ; Tb ]. Overview Option Pay-Off The aim consists in ﬁnding a recursive formula for Spread Dynamics market-implied Spread Dynamics ! Conclusion Implementation ⇒ need for liquid market data. CDO Option Market Model DORN J. Standardization Assumptions Preliminaries Underlying CDS Portfolio restricted to components of Motivation Recall : CDO CDX / iTraXX index series Structure CDO Spread Determinants Pre-Set Attachment / Detachment Points Previous Research → Success Story ⇒ option trading possible The Model Overview Option Pay-Off Spread Dynamics Conclusion Implementation CDO Option Market Model DORN J. Standardization Assumptions Preliminaries Underlying CDS Portfolio restricted to components of Motivation Recall : CDO CDX / iTraXX index series Structure CDO Spread Determinants Pre-Set Attachment / Detachment Points Previous Research → Success Story ⇒ option trading possible The Model Overview Option Pay-Off Spread Dynamics CDS Index Tranches Conclusion CDS Index Tranches securitize CDS Index Series. Implementation Attachment / Detachment Points are standardized [0%, 3%, 6%, 9%, 12%, 22%, 100%] ⇒ improves liquidity, reduces ramp-up costs for structurers CDO Option Synthetic CDO Structure Market Model Only synthetic CDOs (CDOs on a CDS portfolio) allow DORN J. for product standardization and hence for liquidity Preliminaries CDOs securitize credit spreads and issue tranches → Motivation Recall : CDO Leverage Structure CDO Spread Determinants Previous Research The Model Overview Systemic Risk Option Pay-Off Spread Dynamics Increasing Risk Conclusion Senior EUR 850m Implementation Swap Interest & Premium Principal CDS Reference Pool SPV 100 names Funding X Protection EUR 10m notional Payment = Mezzanine Decreasing Spread EUR 1000m EUR 850m Total otional Equity Idiosyncratic Risk EUR 50m CDO Option Recall CDO Spread Determinants Market Model DORN J. Preliminaries Motivation Recall : CDO Structure CDO Spread Definition "CDO Premium Leg" : Determinants Previous Research Sum of discounted Cash-Flows perceived by the The Model Trancheholder Overview Option Pay-Off Spread Dynamics Conclusion Implementation CDO Option Recall CDO Spread Determinants Market Model DORN J. Preliminaries Motivation Recall : CDO Structure CDO Spread Definition "CDO Premium Leg" : Determinants Previous Research Sum of discounted Cash-Flows perceived by the The Model Trancheholder Overview Option Pay-Off Spread Dynamics Definition "CDO Protection Leg" : Conclusion Sum of the discounted reductions of a tranche’s notional Implementation inherent to credit events which lead to a decrease in the Trancheholder’s "spread revenue". CDO Option Market Model Definition "Fair Spread" : DORN J. The t-time Fair Spread is the Spread the investor should Preliminaries have contracted instead of Deal Spread + Euribor/Libor Motivation Recall : CDO at issuing date in order to allow the tranche quote at par Structure CDO Spread at time t. Determinants Previous Research The Model Protection Leg Overview Fair Spread = Option Pay-Off Premium Leg Spread Dynamics Conclusion Notional Erosion → Spread has to be calculated on the Implementation outstanding Tranche Notional CDO Option Market Model Definition "Fair Spread" : DORN J. The t-time Fair Spread is the Spread the investor should Preliminaries have contracted instead of Deal Spread + Euribor/Libor Motivation Recall : CDO at issuing date in order to allow the tranche quote at par Structure CDO Spread at time t. Determinants Previous Research The Model Protection Leg Overview Fair Spread = Option Pay-Off Premium Leg Spread Dynamics Conclusion Notional Erosion → Spread has to be calculated on the Implementation outstanding Tranche Notional Definition Forward Fair Tranche Spread B(t, Ti )Protection Legi Fwd Fair Spread = i B(t, Ti )Premium Legi CDO Option History of Market Models in Market Model Derivatives DORN J. The BGM Model (Brace-Gatarek-Musiela) Preliminaries Motivation Arbitrage-Free model for other than instantaneous, Recall : CDO Structure CDO Spread continuously compounded forward rates Determinants Previous Research The idea is to chose a diﬀerent numeraire other than the The Model risk-free account Overview Option Pay-Off Leads to Black’s formula → we refer to as "market Spread Dynamics Conclusion models" Implementation First attempt to model a market-implied term structure of forward rates CDO Option History of Market Models in Market Model Derivatives DORN J. The BGM Model (Brace-Gatarek-Musiela) Preliminaries Motivation Arbitrage-Free model for other than instantaneous, Recall : CDO Structure CDO Spread continuously compounded forward rates Determinants Previous Research The idea is to chose a diﬀerent numeraire other than the The Model risk-free account Overview Option Pay-Off Leads to Black’s formula → we refer to as "market Spread Dynamics Conclusion models" Implementation First attempt to model a market-implied term structure of forward rates CDS Option Market Model Brigo & Mercurio transferred the idea of a market model into the credit derivatives environment One-Period Spread modeling approach applied to the CDS market, with approximation constraints CDO Option The Model - Central Idea Market Model DORN J. Preliminaries Motivation Recall : CDO Structure Describe a fwd-start option on a synth. CDO Spread CDS Index Tranche (B&S framework) Determinants Previous Research Possibility to select any The Model mtgle dynamics of the Overview fwd spread rate under Option Pay-Off Define forward Fair associated probability The Expected outstanding Spread Dynamics Tranche Spread as a measure Tranche Notional is a t- Conclusion function of the numeraire fwd neutral martingale Implementation Calculate its volatility in Derive forward spread function of the Spread dynamics for different Rate and the associated time horizons observable volatility CDO Option A closed-form Market Formula Market Model DORN J. Lemma Let ΠCallCDO D,E (t, K ) describe the t−time pay-oﬀ of a Preliminaries a,b Motivation forward start call option written on standardized CDO Recall : CDO Structure CDO Spread tranche with boundaries [D%; E %]. The tenor is [Ta ; Tb ]. Determinants Previous Within the Black & Scholes framework the Call option takes Research The Model the value Overview Option Pay-Off ˆ D,E D,E ΠCallCDO D,E (t, K ) = Ca,b (t) × Sa,b (t)N(d1 ) − K × N(d2 ) Spread Dynamics Conclusion a,b Implementation with b ˆ D,E Ca,b (Ta ) =: t δi B(Ta , Ti )EQ Ti [X (Ti )] i=a+1 D,E Sa,b (t) T ln K ± (Ta − t) 1 t a σa,b (s)ds 2 2 d1,2 = √ σa,b (Ta − t) Ta − t CDO Option Step 1- The Fwd Spread Dynamics Market Model DORN J. Definition Fwd-neutral Measure Preliminaries Motivation D,E D,E Protleg (t) dQa,b ≈ Recall : CDO Structure Sa,b (t) = = CDO Spread Premleg(t) dQ Premleg(t) Determinants Previous Research The Model Overview Option Pay-Off Spread Dynamics Conclusion Implementation CDO Option Step 1- The Fwd Spread Dynamics Market Model DORN J. Definition Fwd-neutral Measure Preliminaries Motivation D,E D,E Protleg (t) dQa,b ≈ Recall : CDO Structure Sa,b (t) = = CDO Spread Premleg(t) dQ Premleg(t) Determinants Previous Research The Model Overview Option Pay-Off Spread Dynamics Conclusion Implementation CDO Option Step 1- The Fwd Spread Dynamics Market Model DORN J. Definition Fwd-neutral Measure Preliminaries Motivation D,E D,E Protleg (t) dQa,b ≈ Recall : CDO Structure Sa,b (t) = = CDO Spread Premleg(t) dQ Premleg(t) Determinants Previous Research The Model Overview Option Pay-Off Spread Dynamics Conclusion Implementation D,E D,E Hence Sa,b (t) is a Qa,b - martingale. D,E dSa,b (t) D,E = σa,b (t)dWta,b Sa,b (t) Yi −1 (t) Yi (t) introduces recursion CDO Option Step 2 - Shortfall Dynamics Market Model DORN J. Corollary The expected outstanding tranche notional Yi (t) is a Preliminaries Motivation Q t -martingale. Its dynamics under the forward-neutral Recall : CDO Structure probability Q t follows : CDO Spread Determinants Previous Research dYi (t) The Model = γi (t)dZt Yi (t) Overview Option Pay-Off Spread Dynamics Conclusion Implementation CDO Option Step 2 - Shortfall Dynamics Market Model DORN J. Corollary The expected outstanding tranche notional Yi (t) is a Preliminaries Motivation Q t -martingale. Its dynamics under the forward-neutral Recall : CDO Structure probability Q t follows : CDO Spread Determinants Previous Research dYi (t) The Model = γi (t)dZt Yi (t) Overview Option Pay-Off Spread Dynamics Conclusion Step 3 - Deriving the Volatility Implementation Lemma ∀k ∈ [a + 1; b] the volatility of the process Yk related to tenor [Ta , Tb ] is given by k D,E δj Sj−1,j (t) γk (t) = − D,E σj−1,j (t) j=a+1 1 + δj Sj−1,j (t) CDO Option Step 4 - The Fwd one-period Spread Market Model Dynamics DORN J. Corollary Preliminaries Motivation Consider a deal with tenor [Ta , Tb ] and tranche [D, E ]. The Recall : CDO Structure dynamics of the forward one-period Fair Tranche Spread on CDO Spread Determinants Previous tenor [Ti−1 , Ti ] is given by : Research The Model D,E i D,E Overview dSi−1,i (t) δj Sj−1,j (t) Option Pay-Off D,E = σi−1,i (t)ρ D,E (σj−1,j (t)) dt Spread Dynamics Si−1,i (t) j=a+1 1 + δj Sj−1,j (t) Conclusion Implementation + σi−1,i (t)dZt More precisely, for a deal with tenor [Ti−1 , Ti ], the forward one-period Fair Tranche Spread dynamics for the same tenor amounts to : D,E D,E dSi−1,i (t) δi Si−1,i (t) D,E = D,E |σi−1,i (t)|2 dt + σi−1,i (t)dWt Si−1,i (t) 1 + δi Si−1,i (t) CDO Option Step 5 - The Multi-Period Extension Market Model DORN J. Lemma Preliminaries Again consider a deal with tenor [Ta , Tb ] and tranche [D, E ]. Motivation Recall : CDO Structure The forward multi-period spread dynamics with the same CDO Spread Determinants D,E tenor, note Sa,b , can be written as Previous Research The Model D,E dSa,b (t) Overview Option Pay-Off D,E = (Λ(t) + ς(t)) ρ (Λ(t)) dt − (Λ(t) + ς(t)) dZt Spread Dynamics Sa,b (t) Conclusion Implementation with b δi A(t, Ti )Yi (t) Λ(t) = γi (t) i=a+1 C D,E (t) ˆ a,b A(t, Tb )Yb (t) ς(t) = γb (t) A(t, Ta )Ya (t) − A(t, Tb )Yb (t) CDO Option Conclusion Market Model Market Model allows for calibration of options with DORN J. bespoke exercise periods to options with more liquid Preliminaries tenors thanks to multi-period fwd Tranche Spread Motivation Recall : CDO Dynamics ⇒ More realistic prices. Structure CDO Spread Determinants Previous Research The Model Overview Option Pay-Off Spread Dynamics T_a T_b Illiquid Tenor Conclusion Implementation T_c T_d Liquid Tenor CDO Option Conclusion Market Model DORN J. Preliminaries Motivation Recall : CDO Structure CDO Spread Determinants Possibility of pricing options on tranches with future Previous Research ramp-up dates ⇒ Fwd spread is no longer a martingale The Model Overview ⇒Calculate expectations of the fwd spread dynamics ! Option Pay-Off Spread Dynamics Fwd spread dynamics allow for modeling of deals with Conclusion complicated pay-oﬀs ! Implementation The investor ﬁnally pays for the Véga ! CDO Option Implementation (1) Market Model DORN J. Preliminaries Ta =4,Tb =5 Motivation Recall : CDO Structure CDO Spread 4 x 10 Determinants 2.5 Previous Research The Model 2 Overview Option Pay-Off Spread Dynamics 1.5 Conclusion call 1 Implementation 0.5 1 0 0 0.8 0.1 0.2 0.6 0.3 0.4 0.4 0.5 0.6 0.7 0.2 0.8 0.9 1 0 correl tranches Fig.: Call Value based on 4y - 5y Spreads. CDO Option Implementation (2) Market Model DORN J. Preliminaries Ta =8,Tb =9 Motivation Recall : CDO Structure CDO Spread Determinants 4 Previous x 10 Research 2.5 The Model 2 Overview Option Pay-Off Spread Dynamics 1.5 Conclusion call 1 Implementation 0.5 1 0 0.8 0 0.1 0.2 0.6 0.3 0.4 0.4 0.5 0.6 0.7 0.2 0.8 0.9 1 0 correl tranches Fig.: Call Value based on 8y - 9y Spreads. CDO Option Implementation (3) Market Model DORN J. Preliminaries Ta =1,Tb =2 Motivation Recall : CDO Structure 5 x 10 CDO Spread 2 Determinants 1.8 Previous Research 1.6 The Model 1.4 Overview 1.2 Option Pay-Off 1 Spread Dynamics call 0.8 Conclusion 0.6 0.4 Implementation 0.2 0 1 0.8 0.6 1 0.9 0.8 0.4 0.7 0.6 0.5 0.2 0.4 0.3 0.2 0 0.1 0 correl tranches Fig.: Cumulative Call Value based on 1y - 2y Spreads. CDO Option Implementation (4) Market Model DORN J. Preliminaries Tj =2,Ti =15,Tranche(%) = [0.5,0.6],correl =0.09 Motivation Recall : CDO 4 x 10 Structure 5 CDO Spread Determinants Previous 4 Research The Model 3 Overview Option Pay-Off 2 Spread Dynamics call Conclusion 1 Implementation 0 -1 0 500 16 1000 14 12 10 1500 8 6 4 2000 2 time horizon iterations Fig.: Converging one-period spreads. CDO Option Implementation (5) Market Model DORN J. Preliminaries differences(closed form - monte carlo)T a = 1,Tb = 6 Motivation Recall : CDO Structure CDO Spread Determinants 1500 Previous Research 1000 The Model 500 Overview Option Pay-Off 0 Spread Dynamics -500 Conclusion -1000 Implementation -1500 1 0.9 0.8 0.7 0.6 0.9 0.5 0.8 0.4 0.7 0.6 0.3 0.5 0.2 0.4 0.3 0.1 0.2 correl 0.1 0 0 tranches Fig.: Diﬀerential Closed-form formula vs. Dynamics. CDO Option Outlook Market Model DORN J. Preliminaries Motivation Recall : CDO Structure CDO Spread Determinants Approach might serve to model bespoke CDOs. Previous Research The spread on a CDO tranche can be replicated by a The Model Overview Call Spread on the CDO’s cumulative Loss Given Option Pay-Off Spread Dynamics Default (LGD) with strikes being the respective Conclusion Attachment/Detachment Points. Implementation Hence by modeling the LGD dynamics there should be a way to price bespoke CDO tranches. CDO Option Market Model DORN J. Preliminaries Motivation Recall : CDO Structure CDO Spread Determinants Previous Research THANK YOU FOR YOUR ATTENTION ! The Model Overview Option Pay-Off Spread Dynamics Conclusion Implementation

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 2 |

posted: | 10/5/2011 |

language: | English |

pages: | 31 |

OTHER DOCS BY zhangyun

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.