Modelling Single-name and Multi-name Credit Derivatives

Modelling Single-name and Multi-name Credit Derivatives presents an up-to-date, comprehensive, accessible and practical guide to the pricing and risk-management of credit derivatives. It is both a detailed introduction to credit derivative modelling and a reference for those who are already practitioners. This book is up-to-date as it covers many of the important developments which have occurred in the credit derivatives market in the past 4-5 years. These include the arrival of the CDS portfolio indices and all of the products based on these indices. In terms of models, this book covers the challenge of modelling single-tranche CDOs in the presence of the correlation skew, as well as the pricing and risk of more recent products such as constant maturity CDS, portfolio swaptions, CDO squareds, credit CPPI and credit CPDOs.

Acknowledgements xiii About the Author xv Introduction vii Notation xix 1 The Credit Derivatives Market 1 1.1 Introduction 1 1.2 Market Growth 2 1.3 Products 4 1.4 Market Participants 6 1.5 Summary 7 2 Building the Libor Discount Curve 9 2.1 Introduction 9 2.2 The Libor Index 9 2.3 Money Market Deposits 10 2.4 Forward Rate Agreements 12 2.5 Interest Rate Futures 13 2.6 Interest Rate Swaps 16 2.7 Bootstrapping the Libor Curve 21 2.8 Summary 26 2.9 Technical Appendix 26 Part I Single-name Credit Derivatives 29 3 Single-name Credit Modelling 31 3.1 Introduction 31 3.2 Observing Default 32 3.3 Risk-neutral Pricing Framework 35 3.4 Structural Models of Default 38 3.5 Reduced Form Models 42 3.6 The Hazard Rate Model 44 3.7 Modelling Default as a Cox Process 46 3.8 A Gaussian Short Rate and Hazard Rate Model 49 3.9 Independence and Deterministic Hazard Rates 51 3.10 The Credit Triangle 54 3.11 The Credit Risk Premium 55 3.12 Summary 57 3.13 Technical Appendix 57 4 Bonds and Asset Swaps 59 4.1 Introduction 59 4.2 Fixed Rate Bonds 60 4.3 Floating Rate Notes 68 4.4 The Asset Swap 72 4.5 The Market Asset Swap 78 4.6 Summary 80 5 The Credit Default Swap 81 5.1 Introduction 81 5.2 The Mechanics of the CDS Contract 82 5.3 Mechanics of the Premium Leg 84 5.4 Mechanics of the Protection Leg 85 5.5 Bonds and the CDS Spread 90 5.6 The CDS–Cash basis 92 5.7 Loan CDS 94 5.8 Summary 95 6 A Valuation Model for Credit Default Swaps 97 6.1 Introduction 97 6.2 Unwinding a CDS Contract 97 6.3 Requirements of a CDS Pricing Model 99 6.4 Modelling a CDS Contract 100 6.5 Valuing the Premium Leg 101 6.6 Valuing the Protection Leg 105 6.7 Upfront Credit Default Swaps 108 6.8 Digital Default Swaps 110 6.9 Valuing Loan CDS 111 6.10 Summary 112 7 Calibrating the CDS Survival Curve 113 7.1 Introduction 113 7.2 Desirable Curve Properties 113 7.3 The Bootstrap 114 7.4 Interpolation Quantities 115 7.5 Bootstrapping Algorithm 117 7.6 Behaviour of the Interpolation Scheme 118 7.7 Detecting Arbitrage in the Curve 121 7.8 Example CDS Valuation 123 7.9 Summary 125 8 CDS Risk Management 127 8.1 Introduction 127 8.2 Market Risks of a CDS Position 127 8.3 Analytical CDS Sensitivities 128 8.4 Full Hedging of a CDS Contract 138 8.5 Hedging the CDS Spread Curve Risk 139 8.6 Hedging the Libor Curve Risk 145 8.7 Portfolio Level Hedging 147 8.8 Counterparty Risk 148 8.9 Summary 149 9 Forwards, Swaptions and CMDS 151 9.1 Introduction 151 9.2 Forward Starting CDS 151 9.3 The Default Swaption 156 9.4 Constant Maturity Default Swaps 169 9.5 Summary 180 Part II Multi-name Credit Derivatives 181 10 CDS Portfolio Indices 183 10.1 Introduction 183 10.2 Mechanics of the Standard Indices 184 10.3 CDS Portfolio Index Valuation 188 10.4 The Index Curve 190 10.5 Calculating the Intrinsic Spread of an Index 192 10.6 The Portfolio Swap Adjustment 195 10.7 Asset-backed and Loan CDS Indices 200 10.8 Summary 201 11 Options on CDS Portfolio Indices 203 11.1 Introduction 203 11.2 Mechanics 203 11.3 Valuation of an Index Option 207 11.4 An Arbitrage-free Pricing Model 209 11.5 Examples of Pricing 213 11.6 Risk Management 215 11.7 Black’s Model Revisited 215 11.8 Summary 217 12 An Introduction to Correlation Products 219 12.1 Introduction 219 12.2 Default Baskets 219 12.3 Leveraging the Spread Premia 227 12.4 Collateralised Debt Obligations 230 12.5 The Single-tranche Synthetic CDO 232 12.6 CDOs and Correlation 236 12.7 The Tranche Survival Curve 237 12.8 The Standard Index Tranches 240 12.9 Summary 240 13 The Gaussian Latent Variable Model 241 13.1 Introduction 241 13.2 The Model 241 13.3 The Multi-name Latent Variable Model 243 13.4 Conditional Independence 246 13.5 Simulating Multi-name Default 248 13.6 Default Induced Spread Dynamics 253 13.7 Calibrating the Correlation 257 13.8 Summary 258 14 Modelling Default Times using Copulas 261 14.1 Introduction 261 14.2 Definition and Properties of a Copula 261 14.3 Measuring Dependence 264 14.4 Rank Correlation 265 14.5 Tail Dependence 269 14.6 Some Important Copulae 270 14.7 Pricing Credit Derivatives from Default Times 278 14.8 Standard Error of the Breakeven Spread 280 14.9 Summary 281 14.10 Technical Appendix 282 15 Pricing Default Baskets 283 15.1 Introduction 283 15.2 Modelling First-to-default Baskets 283 15.3 Second-to-default and Higher Default Baskets 291 15.4 Pricing Baskets using Monte Carlo 294 15.5 Pricing Baskets using a Multi-Factor Model 296 15.6 Pricing Baskets in the Student-t Copula 298 15.7 Risk Management of Default Baskets 299 15.8 Summary 301 16 Pricing Tranches in the Gaussian Copula Model 303 16.1 Introduction 303 16.2 The LHP Model 303 16.3 Drivers of the Tranche Spread 308 16.4 Accuracy of the LHP Approximation 312 16.5 The LHP Model with Tail Dependence 313 16.6 Summary 314 16.7 Technical Appendix 314 17 Risk Management of Synthetic Tranches 317 17.1 Introduction 317 17.2 Systemic Risks 318 17.3 The LH+ Model 324 17.4 Idiosyncratic Risks 328 17.5 Hedging Tranches 334 17.6 Summary 339 17.7 Technical Appendix 339 18 Building the Full Loss Distribution 343 18.1 Introduction 343 18.2 Calculating the Tranche Survival Curve 343 18.3 Building the Conditional Loss Distribution 345 18.4 Integrating over the Market Factor 353 18.5 Approximating the Conditional Portfolio Loss Distribution 354 18.6 A Comparison of Methods 360 18.7 Perturbing the Loss Distribution 362 18.8 Summary 364 19 Implied Correlation 365 19.1 Introduction 365 19.2 Implied Correlation 365 19.3 Compound Correlation 367 19.4 Disadvantages of Compound Correlation 370 19.5 No-arbitrage Conditions 371 19.6 Summary 374 20 Base Correlation 375 20.1 Introduction 375 20.2 Base Correlation 375 20.3 Building the Base Correlation Curve 377 20.4 Base Correlation Interpolation 382 20.5 Interpolating Base Correlation using the ETL 389 20.6 A Base Correlation Surface 393 20.7 Risk Management of Index Tranches 394 20.8 Hedging the Base Correlation Skew 395 20.9 Base Correlation for Bespoke Tranches 398 20.10 Risk Management of Bespoke Tranches 405 20.11 Summary 406 21 Copula Skew Models 409 21.1 Introduction 409 21.2 The Challenge of Fitting the Skew 409 21.3 Calibration 411 21.4 Random Recovery 412 21.5 The Student-t Copula 413 21.6 The Double-t Copula 415 21.7 The Composite Basket Model 418 21.8 The Marshall–Olkin Copula 420 21.9 The Mixing Copula 421 21.10 The Random Factor Loading Model 423 21.11 The Implied Copula 427 21.12 Copula Comparison 429 21.13 Pricing Bespokes 431 21.14 Summary 431 22 Advanced Multi-name Credit Derivatives 433 22.1 Introduction 433 22.2 Credit CPPI 433 22.3 Constant Proportion Debt Obligations 436 22.4 The CDO-squared 441 22.5 Tranchelets 448 22.6 Forward Starting Tranches 449 22.7 Options on Tranches 449 22.8 Leveraged Super Senior 450 22.9 Summary 451 23 Dynamic Bottom-up Correlation Models 453 23.1 Introduction 453 23.2 A Survey of Dynamic Models 455 23.3 The Intensity Gamma Model 458 23.4 The Affine Jump Diffusion Model 466 23.5 Summary 470 23.6 Technical Appendix 470 24 Dynamic Top-down Correlation Models 471 24.1 Introduction 471 24.2 The Markov Chain Approach 472 24.3 Markov Chain: Initial Generator 474 24.4 Markov Chain: Stochastic Generator 479 24.5 Summary 483 Appendix A Useful Formulae 485 Bibliography 487 Index 491