Credit Risk in Derivative Pricing Frédéric Abergel Chair of Quantitative Finance École Centrale de Paris

Implicit Credit Risk  Every financial product is subject to credit risk  Example : a contract with A on the stock S of the company B  Payoff  Counterparty risk  Default risk

Implicit Credit Risk  Several issues :  Is credit risk priced in the derivatives ?  Can it be hedged out using market instruments?  Should it be made more explicit ?

Implicit Credit Risk  A basic example : consistent pricing of CDS, Bonds and Vanilla options  An intensity-based credit model  are preferrably stochastic.  Convertible J = 0, exchangeable J = 1  Model calibration On options, CDS, convertible bonds Joint calibration not always possible  Credit risk is generally not priced in the long vanillas

Hedging out the credit risk  A simple joint modelling for the stock of company A and the default of company B  spread/stock correlation  (J-1): size of the jump of the stock of A if B defaults

Hedging out the credit risk  Continuous trading in stock and CDS’s allows a theoretical risk neutralization  Identification of the market risk premiums risk-neutral hazard rate risk-neutral drifts  Practical concerns Liquidity of the CDS market Shape of the curve : roll or hold ? Estimate of J when J is not 0 nor 1 ? Price aggressiveness ?

Making the risk more explicit  Trivial example : Credit Contingent Stock  the contract is to deliver one stock of company A subject to company B not defaulting  Payoff  Variables can be separated

Making the risk more explicit  Particular solution  where Q is a “risky bond” with correlation and jump corrections

Making the risk more explicit  In the “fully decorrelated case” case, one recovers a simple pricing formula: stock * survival probability  In general, the hedge is not identically “long one stock”:  as maturity approaches and no credit event occurs, it will tend to 1  the rebalancing in stock is financed by being structurally seller of CDS.

Making the risk more explicit  Cancellable options Payoff conditional to a credit event not occuring –Equity financing –Cancellable swaps  Contingent CDS Pay : fixed quarterly coupon until default Receive : Option value at default time  Many generalizations…