ISBN: 3540678050
TITLE: Credit Risk Valuation
AUTHOR: Ammann
TOC:

1. Introduction 1
1.1 Motivation 1
1.1.1 Counterparty Default Risk 2
1.1.2 Derivatives on Defaultable Assets 6
1.1.3 Credit Derivatives 7
1.2 Objectives 8
1.3 Structure 10
2. Contingent Claim Valuation 13
2.1 Valuation in Discrete Time 14
2.1.1 Definitions 14
2.1.2 The Finite Setting 15
2.1.3 Extensions 18
2.2 Valuation in Continuous Time 18
2.2.1 Definitions 19
2.2.2 Arbitrage Pricing 20
2.2.3 Fundamental Asset Pricing Theorem 25
2.3 Applications in Continuous Time 25
2.3.1 Black-Scholes Model 26
2.3.2 Margrabe's Model 30
2.3.3 Heath-Jarrow-Morton Framework 33
2.3.4 Forward Measure 38
2.4 Applications in Discrete Time 41
2.4.1 Geometric Brownian Motion 41
2.4.2 Heath-Jarrow-Morton Forward Rates 43
2.5 Summary 45
3. Credit Risk Models 47
3.1 Pricing Credit-Risky Bonds 47
3.1.1 Traditional Methods 48
3.1.2 Firm Value Models 48
3.1.2.1 Merton's Model 48
3.1.2.2 Extensions and Applications of Merton's Model 51
3.1.2.3 Bankruptcy Costs and Endogenous Default 52
3.1.3 First Passage Time Models 53
3.1.4 Intensity Models 58
3.1.4.1 Jarrow-Turnbull Model 58
3.1.4.2 Jarrow-Lando-Turnbull Model 62
3.1.4.3 Other Intensity Models 65
3.2 Pricing Derivatives with Counterparty Risk 66
3.2.1 Firm Value Models 66
3.2.2 Intensity Models 67

3.2.3 Swaps 68
3.3 Pricing Credit Derivatives 70
3.3.1 Debt Insurance 70
3.3.2 Spread Derivatives 71
3.4 Empirical Evidence 73
3.5 Summary 74
4. A Firm Value Pricing Model for Derivatives with Counter-party Default Risk 77
4.1 The Credit Risk Model 77
4.2 Deterministic Liabilities 79
4.2.1 Prices for Vulnerable Options 80
4.2.2 Special Cases 82
4.2.2.1 Fixed Recovery Rate 83
4.2.2.2 Deterministic Claims 84
4.3 Stochastic Liabilities 85
4.3.1 Prices of Vulnerable Options 87
4.3.2 Special Cases 88
4.3.2.1 Asset Claims 89
4.3.2.2 Debt Claims 89
4.4 Gaussian Interest Rates and Deterministic Liabilities 90
4.4.1 Forward Measure 91
4.4.2 Prices of Vulnerable Stock Options 93
4.4.3 Prices of Vulnerable Bond Options 95
4.4.4 Special Cases 95
4.5 Gaussian Interest Rates and Stochastic Liabilities 96
4.5.1 Prices of Vulnerable Stock Options 97
4.5.2 Prices of Vulnerable Bond Options 99
4.5.3 Special Cases 99
4.6 Vulnerable Forward Contracts 99
4.7 Numerical Examples 100
4.7.1 Deterministic Interest Rates 100
4.7.2 Stochastic Interest Rates 103
4.7.3 Forward Contracts 110
4.8 Summary 113
4.9 Proofs of Propositions 115
4.9.1 Proof of Proposition 4.2.1 115
4.9.2 Proof of Proposition 4.3.1 120
4.9.3 Proof of Proposition 4.4.1 125
4.9.4 Proof of Proposition 4.5.1 132
5. A Hybrid Pricing Model for Contingent Claims with Credit Risk 141
5.1 The General Credit Risk Framework 141
5.1.1 Independence and Constant Parameters 143
5.1.2 Price Reduction and Bond Prices 145
5.1.3 Model Specifications 146
5.1.3.1 Arrival Rate of Default 146
5.1.3.2 Recovery Rate 147
5.1.3.3 Bankruptcy Costs 148
5.2 Implementations 149
5.2.1 Lattice with Deterministic Interest Rates 149
5.2.2 The Bankruptcy Process 153
5.2.3 An Extended Lattice Model 155
5.2.3.1 Stochastic Interest Rates 157
5.2.3.2 Recombining Lattice versus Binary Tree 158
5.3 Prices of Vulnerable Options 159
5.4 Recovering Observed Term Structures 160
5.4.1 Recovering the Risk-Free Term Structure 160
5.4.2 Recovering the Defaultable Term Structure 161
5.5 Default-Free Options on Risky Bonds 162
5.5.1 Put-Call Parity 163
5.6 Numerical Examples 164
5.6.1 Deterministic Interest Rates 164
5.6.2 Stochastic Interest Rates 168
5.7 Computational Cost 171
5.8 Summary 173
6. Pricing Credit Derivatives 175
6.1 Credit Derivative Instruments 176
6.1.1 Credit Derivatives of the First Type 176
6.1.2 Credit Derivatives of the Second Type 178
6.1.3 Other Credit Derivatives 178
6.2 Valuation of Credit Derivatives 178
6.2.1 Payoff Functions 180
6.2.1.1 Credit Forward Contracts 180
6.2.1.2 Credit Spread Options 182
6.3 The Compound Pricing Approach 183
6.3.1 Firm Value Model 183
6.3.2 Stochastic Interest Rates 187
6.3.3 Intensity and Hybrid Credit Risk Models 188
6.4 Numerical Examples 189
6.4.1 Deterministic Interest Rates 189
6.4.2 Stochastic Interest Rates 193
6.5 Pricing Spread Derivatives with a Reduced-Form Model 194
6.6 Credit Derivatives as Exchange Options 198
6.6.1 Process Specifications 198
6.6.2 Price of an Exchange Option 200
6.7 Credit Derivatives with Counterparty Default Risk 205
6.7.1 Price of an Exchange Option with Counterparty Default Risk 205
6.8 Summary 215
7. Conclusion 217
7.1 Summary 218
7.2 Practical Implications 220
7.3 Future Research 220
A. Useful Tools from Martingale Theory 223
A.1 Probabilistic Foundations 223
A.2 Process Classes 225
A.3 Martingales 225
A.4 Brownian Motion 227
A.5 Stochastic Integration 229
A.6 Change of Measure 233
References 237
List of Figures 247
List of Tables 249
Index 251
END
