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

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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

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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 ?

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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

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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

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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 ?

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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

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

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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.

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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…

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