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Equity-to-Credit Problem

Learn how to hedge your credit risk exposure using equity, equity options, and credit default swaps in this informative article.

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Equity-to-Credit Problem

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  1. Equity-to-Credit Problem Philippe Henrotte ITO 33 and HEC Paris Equity-to-Credit Arbitrage Gestion Alternative, Evry, April 2004 http://www.ito33.com

  2. Or how to optimally hedge your credit risk exposure with equity, equity options and credit default swaps http://www.ito33.com

  3. Agenda • Traditional approach: diffusion + jump to default • The notion of hazard rate • Inhomogeneous model (local vol & hazard rate) • Calibration and hedging problems • More robust approach: jump-diffusion + stochastic volatility • Incomplete markets • Homogeneous model • Optimal hedging http://www.ito33.com

  4. I – Traditional approach http://www.ito33.com

  5. The equity price is the sole state variable • Structural models of the firm:Default is triggered by a bankruptcy threshold (certain or uncertain: Merton, KMV, CreditGrades) • Reduced-form model:Default is triggered by a Poisson process of given intensity, a.k.a. “hazard rate” • Synthesis:making the hazard rate a function of the underlying equity value (and time) http://www.ito33.com

  6. Default is a jump with intensity p(S, t) • Given no default before t: • With probability (1 – pdt): no default • With probability pdt: default • Taking expectations (in the risk-neutral probability) • Risk-free growth of the hedged portfolio http://www.ito33.com

  7. In the risk-neutral world • We solve the PDE opposite • X is the jump in value of the hedge portfolio • Sdefis the recovery value of the underlying share • Vdef is the recovery value of the derivative. Example : Convertible Bond • Game is over upon default http://www.ito33.com

  8. Example: Convertible Bond • We recover a fraction of face value N • We may have the right to convert at the recovery value of the underlying share http://www.ito33.com

  9. Example: Credit Default Swap • Credit protection buyer pays a premium u until maturity or default event • We model this as asset U http://www.ito33.com

  10. Example: Credit Default Swap • Credit protection seller pays a contingent amount at the time of default • We model this as asset  • V is the insured security http://www.ito33.com

  11. Example: Credit Default Swap • R recovery rate • CDS guarantees we recover par at maturity • Simple closed forms when hazard rate is time dependent only: • u is such that U(0,T) = (0,T) at inception http://www.ito33.com

  12. Example: Equity Options • PDE for a Call under default risk • PDE for a Put under default risk http://www.ito33.com

  13. Example: Equity Options • The jump to default generates an implied volatility skew • Problem of the joint calibration to implied volatility data and credit spread data • Calibrate (S, t) and p(S, t)? • In practice, we use parametric forms and p  as S  0 http://www.ito33.com

  14. Hedging (traditional approach) • The hazard rate is expressed in the risk-neutral world (calibrated from market data) • Collapse of the bond floor (negative gamma) • The delta-hedge presupposes that credit risk has been hedged with a CDS (or a put, …) • Volatility hedge? http://www.ito33.com

  15. What if there were a life after default?(Convertible bond case) • Share does not jump to zero • Issuer reschedules the debt • Holder retains conversion rights • It may not be optimal to convert a the time of default http://www.ito33.com

  16. Switch to “default regime” • The default regime and the no-default regime are coupled through the Poisson transition • Two coupled PDEs, with different process parameters and different initial and boundary conditions http://www.ito33.com

  17. Conclusion: the status of default/no default is the second state variable http://www.ito33.com

  18. II – Incomplete Markets http://www.ito33.com

  19. Incomplete markets • The state of no-default decomposes into sub-regimes of different diffusion components and different hazard rates • This replaces (S, t) and p(S, t) with stochastic  and stochastic p • It turns the model into a homogenous model • Markov transition matrix between regimes • Stock jumps between regimes yield the needed correlations with vol and default http://www.ito33.com

  20. Inhomogeneous Homogeneous 1 12 21 No Default State (S,t) 31 2 13 23 32 p(S,t) 3 p2Default Default State p1Default p3Default Default State http://www.ito33.com

  21. Incomplete markets • In a Black-Scholes world without hedging, you can use the BS formula with any implied volatility value • Perfect replication in the BS world imposes uniqueness: the implied volatility had better be the volatility of the underlying • Under a general process (jump-diffusion, stochastic volatility, default process, etc.), perfect replication is not possible… • …and many non arbitrage pricing systems are possible (risk neutral probabilities) http://www.ito33.com

  22. Pricing and calibration • If we wish to price one contingent claim relative to another, we can work in the risk-neutral probability. This is called “calibration”: • Reverse engineer the prices of the Arrow-Debreu securities from the market prices of a given set of contingent claims • Use the AD prices, or risk-neutral probability measure, to price a new contingent claim • Whenever we wish to price a contingent claim “against the underlying” (by expressing the optimal hedging strategy), we have to work in the real probability http://www.ito33.com

  23. Pricing through optimal hedge • The “fair value” of a contingent claim is the initial cost of its optimal dynamic replication strategy (for some optimality measure) • This requires the knowledge of the historic or real probability measure… • …while calibration only recovers a risk neutral probability • We need to know the drift or the Sharpe ratio of the underlying • The drift of the underlying drops out of the Black-Scholes pricing formula, not of the Black-Scholes world http://www.ito33.com

  24. Calibration is just a pricing shortcut(It has nothing to say about hedging) • Examples: • Calibration of the risk-neutral default intensity function p(S, t) from the market prices of vanilla CDSs, or risky bonds • Calibration of the risk-neutral jump-diffusion stochastic volatility process from the market prices of vanilla options • To express the hedge, we have to transform back the risk-neutral probability into the real probability http://www.ito33.com

  25. Hedging credit risk • Using the underlying only • The notion of HERO • Correlation between regimes and stock price • Reducing the HERO • Using the CDS to hedge credit risk and an option to hedge volatility risk (typically, hedging the CB) • Using an out-of-the-money Put to hedge default risk (typically, hedging the CDS) • Completing the market http://www.ito33.com

  26. Tyco • Tyco, 3 February 2003 • Stock price $16 • Sharpe ratio 0.3 • Joint calibration of options and CDS • Option prices fitted with a maximum error of 4 cents • CDS up to 10 years http://www.ito33.com

  27. Tyco Volatility Smile http://www.ito33.com

  28. Tyco CDS Calibration http://www.ito33.com

  29. Calibrated Regime Switching Model http://www.ito33.com

  30. Tyco Convertible • Vanilla convertible bond • Maturing in 5 years • Conversion ratio 4.38, corresponding to a conversion price of $22.8 http://www.ito33.com

  31. Optimal Dynamic Hedge • With the underlying alone HERO is $9.8 • If one uses the CDS with a maturity of 5 years on top of the underlying, the HERO falls to $5 • If we add the Call with the same maturity and strike price $22.5, the HERO falls down to a few cents and an almost exact replication is achieved http://www.ito33.com

  32. Optimal Dynamic Hedge • As a result, the convertible bond has been dynamically decomposed into an equity call option and a pure credit instrument • This is the essence of the Equity to Credit paradigm http://www.ito33.com

  33. References • E. Ayache, P. Forsyth, and K. Vetzal: Valuation of Convertible Bonds with Credit Risk. The Journal of Derivatives, Fall 2003 • E. Ayache, P. Forsyth, and K. Vetzal: Next Generation Models for Convertible Bonds with Credit Risk. Wilmott, December 2002 • E. Ayache, P. Henrotte, S. Nassar, and X. Wang: Can Anyone Solve the Smile Problem?. Wilmott magazine, January 2004 • P. Henrotte: Pricing and Hedging in the Equity to Credit Paradigm. FOW, January 2004 http://www.ito33.com

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