Preface xi
Acknowledgements xv
Abbreviations xvii
Notation xix
1 Introduction 1
1.1 The world of credit risk 1
1.2 The components of credit risk 2
1.3 Market structure 4
2 Credit Derivatives: Overview and Hedge-Based Pricing 7
2.1 The emergence of a new class of derivatives 7
2.2 Terminology 7
2.3 Underlying assets 10
2.3.1 Loans 10
2.3.2 Bonds 11
2.3.3 Convertible bonds 12
2.3.4 Counterparty risk 12
2.4 Asset swaps 12
2.5 Total return swaps 13
2.6 Credit default swaps 15
2.7 Hedge-based pricing 19
2.7.1 Hedge instruments 20
2.7.2 Short positions in defaultable bonds 20
2.7.3 Asset swap packages 22
2.7.4 Total return swaps 25
2.7.5 Credit default swaps 27
2.8 Exotic credit derivatives 37
2.8.1 Default digital swaps 37
2.8.2 Exotic default payments in credit default swaps 38
2.8.3 Rating-triggered credit default swaps 39
2.8.4 Options on defaultable bonds 40
2.8.5 Credit spread options 41
2.9 Default correlation products and CDOs 43
2.9.1 First-to-default swaps and basket default swaps 43
2.9.2 First loss layers 44
2.9.3 Collateralised debt obligations 46
2.10 Credit-linked notes 49
2.11 Guide to the literature 50
3 Credit Spreads and Bond Price-Based Pricing 51
3.1 Credit spreads and implied default probabilities 52
3.1.1 Risk-neutral probabilities 52
3.1.2 Setup 52
3.1.3 The fundamental relationship 54
3.1.4 The implied survival probability 54
3.1.5 Conditional survival probabilities and implied hazard rates 56
3.1.6 Relation to forward spreads 58
3.2 Recovery modelling 60
3.3 Building blocks for credit derivatives pricing 61
3.4 Pricing with the building blocks 64
3.4.1 Defaultable fixed-coupon bond 64
3.4.2 Defaultable floater 65
3.4.3 Variants of coupon bonds 66
3.4.4 Credit default swaps 66
3.4.5 Forward start CDSs 68
3.4.6 Default digital swaps 68
3.4.7 Asset swap packages 69
3.5 Constructing and calibrating credit spread curves 69
3.5.1 Parametric forms for the spread curves 70
3.5.2 Semi-parametric and non-parametric calibration 72
3.5.3 Approximative and aggregate fits 74
3.5.4 Calibration example 75
3.6 Spread curves: issues in implementation 77
3.6.1 Which default-free interest rates should one use? 77
3.6.2 Recovery uncertainty 79
3.6.3 Bucket hedging 81
3.7 Spread curves: discussion 82
3.8 Guide to the literature 83
4 Mathematical Background 85
4.1 Stopping times 86
4.2 The hazard rate 87
4.3 Point processes 88
4.4 The intensity 88
4.5 Marked point processes and the jump measure 91
4.6 The compensator measure 93
4.6.1 Random measures in discrete time 95
4.7 Examples for compensator measures 97
4.8 Itô's lemma for jump processes 100
4.9 Applications of Itô's lemma 101
4.9.1 Predictable compensators for jump processes 102
4.9.2 Itô product rule and Itô quotient rule 103
4.9.3 The stochastic exponential 104
4.10 Martingale measure, fundamental pricing rule and incompleteness 105
4.11 Change of numeraire and pricing measure 107
4.11.1 The Radon-Nikodym theorem 107
4.11.2 The Girsanov theorem 108
4.12 The change of measure/change of numeraire technique 109
5 Advanced Credit Spread Models 111
5.1 Poisson processes 111
5.1.1 A model for default arrival risk 111
5.1.2 Intuitive construction of a Poisson process 112
5.1.3 Properties of Poisson processes 113
5.1.4 Spreads with Poisson processes 115
5.2 Inhomogeneous Poisson processes 115
5.2.1 Pricing the building blocks 117
5.3 Stochastic credit spreads 118
5.3.1 Cox processes 119
5.3.2 Pricing the building blocks 125
5.3.3 General point processes 126
5.3.4 Compound Poisson processes 128
6 Recovery Modelling 131
6.1 Presentation of the different recovery models 132
6.1.1 Zero recovery 132
6.1.2 Recovery of treasury 133
6.1.3 Multiple defaults and recovery of market value 135
6.1.4 Recovery of par 141
6.1.5 Stochastic recovery and recovery risk 143
6.1.6 Common parametric distribution functions for recoveries 147
6.1.7 Valuation of the delivery option in a CDS 148
6.2 Comparing the recovery models 150
6.2.1 Theoretical comparison of the recovery models 150
6.2.2 Empirical analysis of recovery rates 159
7 Implementation of Intensity-Based Models 165
7.1 Tractable models of the spot intensity 166
7.1.1 The two-factor Gaussian model 167
7.1.2 The multifactor Gaussian model 171
7.1.3 Implied survival probabilities 172
7.1.4 Payoffs at default 174
7.2 The multifactor CIR model 174
7.2.1 Bond prices 175
7.2.2 Affine combinations of independent non-central chi-squared distributed random variables 176
7.2.3 Factor distributions 178
7.3 Credit derivatives in the CIR model 179
7.3.1 Default digital payoffs 180
7.3.2 Calculations to the Gaussian model 180
7.3.3 Calculations to the CIR model 184
7.4 Tree models 187
7.4.1 The tree implementation: inputs 187
7.4.2 Default branching 188
7.4.3 The implementation steps 190
7.4.4 Building trees: the Hull-White algorithm 190
7.4.5 Fitting the tree: default-free interest rates 193
7.4.6 Combining the trees 194
7.4.7 Fitting the combined tree 197
7.4.8 Applying the tree 198
7.4.9 Extensions and conclusion 199
7.5 PDE-Based implementation 200
7.6 Modelling term structures of credit spreads 204
7.6.1 Intensity models in a Heath, Jarrow, Morton framework 206
7.7 Monte Carlo simulation 211
7.7.1 Pathwise simulation of diffusion processes 214
7.7.2 Simulation of recovery rates 219
7.8 Guide to the literature 220
8 Credit Rating Models 223
8.1 Introduction 223
8.1.1 Empirical observations 224
8.1.2 An example 225
8.2 The rating process and transition probabilities 226
8.2.1 Discrete-time Markov chains 229
8.2.2 Continuous-time Markov chains 229
8.2.3 Connection to Poisson processes 231
8.3 Estimation of transition intensities 233
8.3.1 The cohort method 233
8.3.2 The embedding problem: finding a generator matrix 234
8.4 Direct estimation of transition intensities 238
8.5 Pricing with deterministic generator matrix 239
8.5.1 Pricing zero-coupon bonds 239
8.5.2 Pricing derivatives on the credit rating 240
8.5.3 General payoffs 241
8.5.4 Rating trees 242
8.5.5 Downgrade triggers 243
8.5.6 Hedging rating transitions 245
8.6 The calibration of rating transition models 246
8.6.1 Deterministic intensity approaches 246
8.6.2 Incorporating rating momentum 249
8.6.3 Stochastic rating transition intensities 250
8.7 A general HJM framework 251
8.8 Conclusion 253
9 Firm Value and Share Price-Based Models 255
9.1 The approach 255
9.1.1 Modelling philosophy 255
9.1.2 An example 256
9.1.3 State variables and modelling 259
9.1.4 The time of default 261
9.2 Pricing equations 263
9.2.1 The firm's value model 263
9.2.2 The pricing equation 264
9.2.3 Some other securities 265
9.2.4 Hedging 268
9.3 Solutions to the pricing equation 269
9.3.1 The T-forward measure 269
9.3.2 Time change 270
9.3.3 The hitting probability 270
9.3.4 Putting it together 271
9.3.5 The Longstaff-Schwartz results 271
9.3.6 Strategic default 273
9.4 A practical implementation: KMV 275
9.4.1 The default point 275
9.4.2 The time horizon 275
9.4.3 The initial value of the firm's assets and its volatility 275
9.4.4 The distance to default 276
9.5 Unobservable firm's values and CreditGrades 277
9.5.1 A simple special case: delayed observation 280
9.5.2 The idea of Lardy and Finkelstein: CreditGrades and E2C 281
9.6 Advantages and disadvantages 284
9.6.1 Empirical evidence 284
9.6.2 Discussion 286
9.7 Guide to the literature 286
10 Models for Default Correlation 289
10.1 Default correlation basics 290
10.1.1 Empirical evidence 290
10.1.2 Terminology 291
10.1.3 Linear default correlation, conditional default probabilities, joint default probabilities 292
10.1.4 The size of the impact of default correlation 293
10.1.5 Price bounds for FtD swaps 293
10.1.6 The need for theoretical models of default correlations 297
10.2 Independent defaults 298
10.2.1 The binomial distribution function 298
10.2.2 Properties of the binomial distribution function 299
10.2.3 The other extreme: perfectly dependent defaults 300
10.3 The binomial expansion method 301
10.4 Factor models 305
10.4.1 One-factor dependence of defaults 305
10.4.2 A simplified firm's value model 305
10.4.3 The distribution of the defaults 307
10.4.4 The large portfolio approximation 309
10.4.5 Generalisations 312
10.4.6 Portfolios of two asset classes 313
10.4.7 Some remarks on implementation 314
10.5 Correlated defaults in intensity models 315
10.5.1 The intensity of the default counting process 315
10.5.2 Correlated intensities 316
10.5.3 Stress events in intensity models 318
10.5.4 Default contagion/infectious defaults 321
10.6 Correlated defaults in firm's value models 321
10.7 Copula functions and dependency concepts 326
10.7.1 Copula functions 327
10.7.2 Examples of copulae 330
10.7.3 Archimedean copulae 333
10.8 Default modelling with copula functions 337
10.8.1 Static copula models for default correlation 337
10.8.2 Large portfolio loss distributions for Archimedean copulae 340
10.8.3 A semi-dynamic copula model 343
10.8.4 Dynamic copula-dependent defaults 349
Bibliography 361
Index 369
