Modelling Single-name and Multi-name Credit Derivatives

Dominic O'Kane is an affiliated Professor of Finance at the French business school EDHEC which is based in Nice, France. Until May 2006, Dominic O'Kane was a managing director and ran the European Fixed Income Quantitative Research group at Lehman Brothers, the US investment bank. Dominic spent seven of his nine years at Lehman Brothers working as a quant for the credit derivatives trading desk.

• Acknowledgements xiiiAbout the Author xvIntroduction viiNotation xix1 The Credit Derivatives Market 11.1 Introduction 11.2 Market Growth 21.3 Products 41.4 Market Participants 61.5 Summary 72 Building the Libor Discount Curve 92.1 Introduction 92.2 The Libor Index 92.3 Money Market Deposits 102.4 Forward Rate Agreements 122.5 Interest Rate Futures 132.6 Interest Rate Swaps 162.7 Bootstrapping the Libor Curve 212.8 Summary 262.9 Technical Appendix 26Part I Single-name Credit Derivatives 293 Single-name Credit Modelling 313.1 Introduction 313.2 Observing Default 323.3 Risk-neutral Pricing Framework 353.4 Structural Models of Default 383.5 Reduced Form Models 423.6 The Hazard Rate Model 443.7 Modelling Default as a Cox Process 463.8 A Gaussian Short Rate and Hazard Rate Model 493.9 Independence and Deterministic Hazard Rates 513.10 The Credit Triangle 543.11 The Credit Risk Premium 553.12 Summary 573.13 Technical Appendix 574 Bonds and Asset Swaps 594.1 Introduction 594.2 Fixed Rate Bonds 604.3 Floating Rate Notes 684.4 The Asset Swap 724.5 The Market Asset Swap 784.6 Summary 805 The Credit Default Swap 815.1 Introduction 815.2 The Mechanics of the CDS Contract 825.3 Mechanics of the Premium Leg 845.4 Mechanics of the Protection Leg 855.5 Bonds and the CDS Spread 905.6 The CDS–Cash basis 925.7 Loan CDS 945.8 Summary 956 A Valuation Model for Credit Default Swaps 976.1 Introduction 976.2 Unwinding a CDS Contract 976.3 Requirements of a CDS Pricing Model 996.4 Modelling a CDS Contract 1006.5 Valuing the Premium Leg 1016.6 Valuing the Protection Leg 1056.7 Upfront Credit Default Swaps 1086.8 Digital Default Swaps 1106.9 Valuing Loan CDS 1116.10 Summary 1127 Calibrating the CDS Survival Curve 1137.1 Introduction 1137.2 Desirable Curve Properties 1137.3 The Bootstrap 1147.4 Interpolation Quantities 1157.5 Bootstrapping Algorithm 1177.6 Behaviour of the Interpolation Scheme 1187.7 Detecting Arbitrage in the Curve 1217.8 Example CDS Valuation 1237.9 Summary 1258 CDS Risk Management 1278.1 Introduction 1278.2 Market Risks of a CDS Position 1278.3 Analytical CDS Sensitivities 1288.4 Full Hedging of a CDS Contract 1388.5 Hedging the CDS Spread Curve Risk 1398.6 Hedging the Libor Curve Risk 1458.7 Portfolio Level Hedging 1478.8 Counterparty Risk 1488.9 Summary 1499 Forwards, Swaptions and CMDS 1519.1 Introduction 1519.2 Forward Starting CDS 1519.3 The Default Swaption 1569.4 Constant Maturity Default Swaps 1699.5 Summary 180Part II Multi-name Credit Derivatives 18110 CDS Portfolio Indices 18310.1 Introduction 18310.2 Mechanics of the Standard Indices 18410.3 CDS Portfolio Index Valuation 18810.4 The Index Curve 19010.5 Calculating the Intrinsic Spread of an Index 19210.6 The Portfolio Swap Adjustment 19510.7 Asset-backed and Loan CDS Indices 20010.8 Summary 20111 Options on CDS Portfolio Indices 20311.1 Introduction 20311.2 Mechanics 20311.3 Valuation of an Index Option 20711.4 An Arbitrage-free Pricing Model 20911.5 Examples of Pricing 21311.6 Risk Management 21511.7 Black’s Model Revisited 21511.8 Summary 21712 An Introduction to Correlation Products 21912.1 Introduction 21912.2 Default Baskets 21912.3 Leveraging the Spread Premia 22712.4 Collateralised Debt Obligations 23012.5 The Single-tranche Synthetic CDO 23212.6 CDOs and Correlation 23612.7 The Tranche Survival Curve 23712.8 The Standard Index Tranches 24012.9 Summary 24013 The Gaussian Latent Variable Model 24113.1 Introduction 24113.2 The Model 24113.3 The Multi-name Latent Variable Model 24313.4 Conditional Independence 24613.5 Simulating Multi-name Default 24813.6 Default Induced Spread Dynamics 25313.7 Calibrating the Correlation 25713.8 Summary 25814 Modelling Default Times using Copulas 26114.1 Introduction 26114.2 Definition and Properties of a Copula 26114.3 Measuring Dependence 26414.4 Rank Correlation 26514.5 Tail Dependence 26914.6 Some Important Copulae 27014.7 Pricing Credit Derivatives from Default Times 27814.8 Standard Error of the Breakeven Spread 28014.9 Summary 28114.10 Technical Appendix 28215 Pricing Default Baskets 28315.1 Introduction 28315.2 Modelling First-to-default Baskets 28315.3 Second-to-default and Higher Default Baskets 29115.4 Pricing Baskets using Monte Carlo 29415.5 Pricing Baskets using a Multi-Factor Model 29615.6 Pricing Baskets in the Student-t Copula 29815.7 Risk Management of Default Baskets 29915.8 Summary 30116 Pricing Tranches in the Gaussian Copula Model 30316.1 Introduction 30316.2 The LHP Model 30316.3 Drivers of the Tranche Spread 30816.4 Accuracy of the LHP Approximation 31216.5 The LHP Model with Tail Dependence 31316.6 Summary 31416.7 Technical Appendix 31417 Risk Management of Synthetic Tranches 31717.1 Introduction 31717.2 Systemic Risks 31817.3 The LH+ Model 32417.4 Idiosyncratic Risks 32817.5 Hedging Tranches 33417.6 Summary 33917.7 Technical Appendix 33918 Building the Full Loss Distribution 34318.1 Introduction 34318.2 Calculating the Tranche Survival Curve 34318.3 Building the Conditional Loss Distribution 34518.4 Integrating over the Market Factor 35318.5 Approximating the Conditional Portfolio Loss Distribution 35418.6 A Comparison of Methods 36018.7 Perturbing the Loss Distribution 36218.8 Summary 36419 Implied Correlation 36519.1 Introduction 36519.2 Implied Correlation 36519.3 Compound Correlation 36719.4 Disadvantages of Compound Correlation 37019.5 No-arbitrage Conditions 37119.6 Summary 37420 Base Correlation 37520.1 Introduction 37520.2 Base Correlation 37520.3 Building the Base Correlation Curve 37720.4 Base Correlation Interpolation 38220.5 Interpolating Base Correlation using the ETL 38920.6 A Base Correlation Surface 39320.7 Risk Management of Index Tranches 39420.8 Hedging the Base Correlation Skew 39520.9 Base Correlation for Bespoke Tranches 39820.10 Risk Management of Bespoke Tranches 40520.11 Summary 40621 Copula Skew Models 40921.1 Introduction 40921.2 The Challenge of Fitting the Skew 40921.3 Calibration 41121.4 Random Recovery 41221.5 The Student-t Copula 41321.6 The Double-t Copula 41521.7 The Composite Basket Model 41821.8 The Marshall–Olkin Copula 42021.9 The Mixing Copula 42121.10 The Random Factor Loading Model 42321.11 The Implied Copula 42721.12 Copula Comparison 42921.13 Pricing Bespokes 43121.14 Summary 43122 Advanced Multi-name Credit Derivatives 43322.1 Introduction 43322.2 Credit CPPI 43322.3 Constant Proportion Debt Obligations 43622.4 The CDO-squared 44122.5 Tranchelets 44822.6 Forward Starting Tranches 44922.7 Options on Tranches 44922.8 Leveraged Super Senior 45022.9 Summary 45123 Dynamic Bottom-up Correlation Models 45323.1 Introduction 45323.2 A Survey of Dynamic Models 45523.3 The Intensity Gamma Model 45823.4 The Affine Jump Diffusion Model 46623.5 Summary 47023.6 Technical Appendix 47024 Dynamic Top-down Correlation Models 47124.1 Introduction 47124.2 The Markov Chain Approach 47224.3 Markov Chain: Initial Generator 47424.4 Markov Chain: Stochastic Generator 47924.5 Summary 483Appendix A Useful Formulae 485Bibliography 487Index 491