Equity-to-Credit Problem Philippe Henrotte ITO 33 and HEC Paris Equity-to-Credit Arbitrage Gestion Alternative, Evry, April 2004

Or how to optimally hedge your credit risk exposure with equity, equity options and credit default swaps

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

I – Traditional approach

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)

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

In the risk-neutral world We solve the PDE opposite X is the jump in value of the hedge portfolio S def is the recovery value of the underlying share V def is the recovery value of the derivative. Example : Convertible Bond Game is over upon default

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

Example: Credit Default Swap Credit protection buyer pays a premium u until maturity or default event We model this as asset U

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

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

Example: Equity Options PDE for a Call under default risk PDE for a Put under default risk

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

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?

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

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

Conclusion: the status of default/no default is the second state variable

II – Incomplete Markets

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

Inhomogeneous Default State No Default State p(S,t) (S,t) Homogeneous Default State p 1 Default 3 1 p 3 Default 1 2 p 2 Default 1 3

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)

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

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

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

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

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

Tyco Volatility Smile

Tyco CDS Calibration

Calibrated Regime Switching Model

Tyco Convertible Vanilla convertible bond Maturing in 5 years Conversion ratio 4.38, corresponding to a conversion price of $22.8

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

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

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