## Credit Risk Valuation: Methods, Models, and ApplicationsCredit risk is an important consideration in most financial transactions. As for any other risk, the risk taker requires compensation for the undiversifiable part of the risk taken. In bond markets, for example, riskier issues have to promise a higher yield to attract investors. But how much higher a yield? Using methods from contingent claims analysis, credit risk valuation models attempt to put a price on credit risk. This monograph gives an overview of the current methods for the valu ation of credit risk and considers several applications of credit risk models in the context of derivative pricing. In particular, credit risk models are in corporated into the pricing of derivative contracts that are subject to credit risk. Credit risk can affect prices of derivatives in a variety of ways. First, financial derivatives can be subject to counterparty default risk. Second, a derivative can be written on a security which is subject to credit risk, such as a corporate bond. Third, the credit risk itself can be the underlying vari able of a derivative instrument. In this case, the instrument is called a credit derivative. Fourth, credit derivatives may themselves be exposed to counter party risk. This text addresses all of those valuation problems but focuses on counterparty risk. The book is divided into six chapters and an appendix. Chapter 1 gives a brief introduction into credit risk and motivates the use of credit risk models in contingent claims pricing. |

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### Contents

Introduction | 1 |

12 Objectives | 8 |

13 Structure | 10 |

Contingent Claim Valuation | 13 |

21 Valuation in Discrete Time | 14 |

22 Valuation in Continuous Time | 18 |

23 Applications in Continuous Time | 25 |

24 Applications in Discrete Time | 41 |

54 Recovering Observed Term Structures | 158 |

55 DefaultFree Options on Risky Bonds | 160 |

56 Numerical Examples | 162 |

57 Computational Cost | 169 |

58 Summary | 171 |

Pricing Credit Derivatives | 173 |

61 Credit Derivative Instruments | 174 |

62 Valuation of Credit Derivatives | 176 |

25 Summary | 45 |

Credit Risk Models | 47 |

32 Pricing Derivatives with Counterparty Risk | 66 |

33 Pricing Credit Derivatives | 70 |

34 Empirical Evidence | 73 |

35 Summary | 74 |

A Firm Value Pricing Model for Derivatives with Counterparty Default Risk | 77 |

42 Deterministic Liabilities | 79 |

43 Stochastic Liabilities | 85 |

44 Gaussian Interest Rates and Deterministic Liabilities | 90 |

45 Gaussian Interest Rates and Stochastic Liabilities | 96 |

46 Vulnerable Forward Contracts | 99 |

47 Numerical Examples | 100 |

48 Summary | 113 |

49 Proofs of Propositions | 115 |

A Hybrid Pricing Model for Contingent Claims with Credit Risk | 139 |

52 Implementations | 147 |

53 Prices of Vulnerable Options | 157 |

63 The Compound Pricing Approach | 181 |

64 Numerical Examples | 187 |

65 Pricing Spread Derivatives with a ReducedForm Model | 192 |

66 Credit Derivatives as Exchange Options | 196 |

67 Credit Derivatives with Counterparty Default Risk | 203 |

68 Summary | 213 |

Conclusion | 215 |

71 Summary | 216 |

72 Practical Implications | 218 |

Useful Tools from Martingale Theory | 221 |

A2 Process Classes | 223 |

A4 Brownian Motion | 225 |

A5 Stochastic Integration | 227 |

A6 Change of Measure | 231 |

References | 235 |

245 | |

Index | 247 |

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### Common terms and phrases

American options approach arbitrage assumed bankruptcy process bivariate Black-Scholes bond options bond price call option Chapter compound option contingent claim correlation counterparty default risk counterparty risk credit derivatives credit risk model credit spread options credit-risky bonds debt default intensity default-free options defined denotes deterministic interest rates evaluated example exchange option exogenous expression firm value model firm's assets forward contract forward price forward rates geometric Brownian motion Girsanov's theorem given implies indicator functions intensity models Jarrow Jarrow and Turnbull lattice maturity measure Q Merton money market account numeraire obtain option prices parameters passage time models price process Price reductions pricing formulae put option recovery rate risk-free risk-neutral measure riskless risky bond Section short rate specification spread derivatives stochastic interest rates strike price swap Table term structure trading strategy underlying valuation variable Vasicek volatility function vulnerable options yield spread zero zero-coupon bond