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CREDIT CONTINGENT OPTIONS Frédéric Abergel Ecole Centrale de paris Introduction Credit Contingent option: a payout is delivered conditionally to a credit event not occuring before the maturity, nor any coupon date, of the option Credit Contingent Option Some examples: Option with a counterparty which may default (go bankrupt) Convertible Bond Extinguishable swap that terminates when the counterparty fails to fulfill one of its obligations A note paying equity-based coupons (say) as long as a given bond does not default MODELLING BY REPLICATION Single asset, single credit to begin with Replicating strategy based on delta- hedging the asset and dynamically trading CDS Three sources of risk: asset volatility, spread volatility, default risk CREDIT MODELLING Book by Duffie and Singleton “Credit Risk” Numerous papers by Schönbucher on Credit Derivatives A few references (Rutkowsky, Jeanblanc- Pique) on modelling by replication MODELLING BY REPLICATION Modelling assumptions A jump-diffusion for the asset A diffusion for the intensity process governing both the default process and the jump of the asset The asset may or may not jump, depending on whether there is a default correlation or not The same Cox-Poisson process drives the credit event and the potential jump of the asset The asset and spread driving factors are correlated dSt t St dWt t St dt J 1 St d t 1 d t at dWt bt dt 2 d W1 , W2 t dt MODELLING BY REPLICATION Hedge portfolio approach: consider a “credit contingent” contingent claim use the asset and (in theory) two risky instruments (CDS, bonds with different maturities) to eliminate the risk factors Determine the risk premia via calibration Get the pricing equation and the hedges HEDGE PORTFOLIO SET UP 1 unit of contract Vt - t units of one risky bond - t units of another risky bond - t units of asset HEDGE PORTFOLIO SET UP V c V dPt dSt d t V V d t S V 1 2 2 2V 1 2 2V 2V S a a S dt t 2 S 2 2 2 S t dBt1c t dBt2 c t Bt1 Bt1 d t t Bt2 Bt2 d t t dStc t J 1 St d t RISK FREE PORTFOLIO V t S V V t J 1 S t Bt1 Bt1 t Bt2 Bt2 V B1 B 2 t t V 1 2 2 2V 1 2 2V 2V S a a S t 2 S 2 2 2 S dPt dt t B1 a B1 t B2 a B2 2 2 2 2 2 2 t t 2 2 RISK FREE PORTFOLIO V V V S J 1 S t Bt1 Bt1 t Bt2 Bt2 V B1 B 2 t t V 1 2 2 2V 1 2 2V 2V B1 a 2 2 B1 S a a S t t 2 S 2 2 2 S t 2 2 B2 a 2 2 B2 t t 2 2 RISK FREE PORTFOLIO Three equations in two unknowns: there exists a zero linear combination of the rows of the matrix with non zero weights (basic algebra) Hence there exist two functions k,q of the independent variables (the market prices of risk) such that the following set of equations holds: B1 a 2 2 B1 B1 k Bt Bt q 1 1 t 2 2 B2 a 2 2 B2 B2 k Bt Bt q 2 2 t 2 2 V 1 2 2 2V 1 2 2V 2V S a a S t 2 S 2 2 2 S V V k V V J 1 S q S PRICING EQUATION V 1 2 2 V 1 2 V 2 2 2 V S a a S t 2 S 2 2 2 S V V k V V J 1 S q S MARKET PRICE OF RISK(S) k is the “market price of jump risk, q is the “market price of spread risk” Once they are fixed, the bond price obtains T formally as E exp k u , u du t under the risk-neutral t diffusion for the hazard rate d t adWt qdt 2 MARKET PRICE OF RISK(S) Question: how to choose and estimate the risk premia ? Answer(s): motivated by Model tractability Market information Example: affine functions of the hazard rate Leads to simple formula for the risky bond CIR process so as to ensure nonnegativity of the hazard rate MODELLING BY REPLICATION This approach is interesting because: Gives a price associated with a hedge Helps one determine which parameters can be calibrated, which one should be estimated from historical parameters. Separates between the spread risk and the default risk (relevant since there is dynamic trading in CDS) SOME EXAMPLES Credit Contingent Stock: the contract is to deliver 1 unit of a given stock subject to no credit event. Payoff ST 1 T Going back to the pricing equation, one can easily see that variables can be separated CREDIT CONTINGENT STOCK V Q , t S Particular solution where Q is a “risky bond” with correlation Q 1 2 2Q Q a a and jump corrections, t 2 2 solution to the 1D Q q JQ 0 equation CREDIT CONTINGENT STOCK In the deterministic case, one recovers the simple pricing formula: stock * survival probability The contract is not identically delta-1: as maturity approaches and no credit event occurs, it will tend to 1. The continuous rebalancing in stock is financed by being structurally seller of CDS. SOME EXAMPLES Extinguishable (XCCY) swap: a type of deal where we pay (receive) coupons in one currency and the counteparty receives (pays) coupons in another, with full notional exchange at the beginning and at the end. Equivalent to a series of risky FX forwards as in the previous example XCCY SWAP There are potentially two credit curves for B F a 2 2 B F B F a k2 the counterparty, one t 2 2 in each currency. rF Jk3 B F 0 Shifting from one to the other involves a B D a 2 2 B D B D k2 compo-bond type of t 2 2 correction: rD Jk3 B D 0 SOME EXAMPLES Convertible and Exchangeable Bond: a bond that can be converted at any time during a predefined conversion period into shares of the company (convertible) or of another company (exchangeable) Typically, a 2 factor, american-style option CONCLUDING REMARKS First steps towards joint modelling of credit- and asset-based derivatives Single asset, single credit well advanced Future developments: Make our generic derivatives credit contingent by incorporating more sophisticated dynamics and calibrators for the asset Simulate in a consistent and dynamically tractable correlation of default times

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posted: | 7/22/2011 |

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