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Credit Risk in Derivative Pricing

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.

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Credit Risk in Derivative Pricing

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

  2. 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

  3. 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 ?

  4. 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

  5. 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

  6. 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 ?

  7. 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

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

  9. 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.

  10. 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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