Modelling Single-Name and Multi-Name Credit Derivatives

by O'Kane, Dominic

• ISBN13: 9780470519288
• ISBN10: 0470519282
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-08-04
• Publisher: WILEY

Summary

The credit derivatives market has grown significantly and rapidly over the past 8 years. However, it is only in the last 4 years that we have seen the advent of the CDS index portfolios such as CDX and iTraxx and their rapid domination of the market. These have revolutionised the credit derivatives market, increasing liquidity and significantly broadening the user base. On top of this, we also have the arrival of CDO tranches linked to these portfolios which has resulted in large and liquid market in default correlation, something that was almost unimaginable 4 years ago. This has created significant challenges for credit modellers as it has essentially rendered earlier correlation models unusable. The search for new approaches has produced a set of new and competing models, all vying to become the new standard. Other products have arrived which have also presented new challenges to credit modellers. These include constant maturity default swaps, options on CDS indices, and CDO squareds, to name just a few. This book covers all of the theory of credit in detail and also covers all of the new developments listed above. Most of the current treatments of the credit derivatives markets are authored by academics (Schonbucher, Duffie&Singleton, Hull) and so are strong on the mathematics. They are usually weak on products descriptions, discussions of the risks, realistic examples, development of model intuition and discussions of the practicalities of implementing the model for the pricing and risk management of credit derivatives in a real industrial setting like an investment bank. Other treatments by practitioners (Chaplin, Tavakoli) tend to be out of date and tend not to get highly technical in the mathematical and implementational detail. O'Kane's aim is to fill the gap by with something which is technical, practical and accessible in the sense that the mathematical treatment is more of a physics/engineering style. It should therefore be useful to the practising quant, risk-manager or student seeking information on how these models are selected and used in reality.

Author Biography

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.

Table of Contents

Acknowledgements	p. xiii
About the Author	p. xv
Introduction	p. xvii
Notation	p. xix
The Credit Derivatives Market	p. 1
Introduction	p. 1
Market Growth	p. 2
Products	p. 4
Market Participants	p. 6
Summary	p. 7
Building the Libor Discount Curve	p. 9
Introduction	p. 9
The Libor Index	p. 9
Money Market Deposits	p. 10
Forward Rate Agreements	p. 12
Interest Rate Futures	p. 13
Interest Rate Swaps	p. 16
Bootstrapping the Libor Curve	p. 21
Summary	p. 26
Technical Appendix	p. 26
Single-Name Credit Derivatives	p. 29
Single-name Credit Modelling	p. 31
Introduction	p. 31
Observing Default	p. 32
Risk-neutral Pricing Framework	p. 35
Structural Models of Default	p. 38
Reduced Form Models	p. 42
The Hazard Rate Model	p. 44
Modelling Default as a Cox Process	p. 46
A Gaussian Short Rate and Hazard Rate Model	p. 49
Independence and Deterministic Hazard Rates	p. 51
The Credit Triangle	p. 54
The Credit Risk Premium	p. 55
Summary	p. 57
Technical Appendix	p. 57
Bonds and Asset Swaps	p. 59
Introduction	p. 59
Fixed Rate Bonds	p. 60
Floating Rate Notes	p. 68
The Asset Swap	p. 72
The Market Asset Swap	p. 78
Summary	p. 80
The Credit Default Swap	p. 81
Introduction	p. 81
The Mechanics of the CDS Contract	p. 82
Mechanics of the Premium Leg	p. 84
Mechanics of the Protection Leg	p. 85
Bonds and the CDS Spread	p. 90
The CDS-Cash basis	p. 92
Loan CDS	p. 94
Summary	p. 95
A Valuation Model for Credit Default Swaps	p. 97
Introduction	p. 97
Unwinding a CDS Contract	p. 97
Requirements of a CDS Pricing Model	p. 99
Modelling a CDS Contract	p. 100
Valuing the Premium Leg	p. 101
Valuing the Protection Leg	p. 105
Upfront Credit Default Swaps	p. 108
Digital Default Swaps	p. 110
Valuing Loan CDS	p. 111
Summary	p. 112
Calibrating the CDS Survival Curve	p. 113
Introduction	p. 113
Desirable Curve Properties	p. 113
The Bootstrap	p. 114
Interpolation Quantities	p. 115
Bootstrapping Algorithm	p. 117
Behaviour of the Interpolation Scheme	p. 118
Detecting Arbitrage in the Curve	p. 121
Example CDS Valuation	p. 123
Summary	p. 125
CDS Risk Management	p. 127
Introduction	p. 127
Market Risks of a CDS Position	p. 127
Analytical CDS Sensitivities	p. 128
Full Hedging of a CDS Contract	p. 138
Hedging the CDS Spread Curve Risk	p. 139
Hedging the Libor Curve Risk	p. 145
Portfolio Level Hedging	p. 147
Counterparty Risk	p. 148
Summary	p. 149
Forwards, Swaptions and CMDS	p. 151
Introduction	p. 151
Forward Starting CDS	p. 151
The Default Swaption	p. 156
Constant Maturity Default Swaps	p. 169
Summary	p. 180
Multi-Name Credit Derivatives	p. 181
CDS Portfolio Indices	p. 183
Introduction	p. 183
Mechanics of the Standard Indices	p. 184
CDS Portfolio Index Valuation	p. 188
The Index Curve	p. 190
Calculating the Intrinsic Spread of an Index	p. 192
The Portfolio Swap Adjustment	p. 195
Asset-backed and Loan CDS Indices	p. 200
Summary	p. 201
Options on CDS Portfolio Indices	p. 203
Introduction	p. 203
Mechanics	p. 203
Valuation of an Index Option	p. 207
An Arbitrage-free Pricing Model	p. 209
Examples of Pricing	p. 213
Risk Management	p. 215
Black's Model Revisited	p. 215
Summary	p. 217
An Introduction to Correlation Products	p. 219
Introduction	p. 219
Default Baskets	p. 219
Leveraging the Spread Premia	p. 227
Collateralised Debt Obligations	p. 230
The Single-tranche Synthetic CDO	p. 232
CDOs and Correlation	p. 236
The Tranche Survival Curve	p. 237
The Standard Index Tranches	p. 240
Summary	p. 240
The Gaussian Latent Variable Model	p. 241
Introduction	p. 241
The Model	p. 241
The Multi-name Latent Variable Model	p. 243
Conditional Independence	p. 246
Simulating Multi-name Default	p. 248
Default Induced Spread Dynamics	p. 253
Calibrating the Correlation	p. 257
Summary	p. 258
Modelling Default Times using Copulas	p. 261
Introduction	p. 261
Definition and Properties of a Copula	p. 261
Measuring Dependence	p. 264
Rank Correlation	p. 265
Tail Dependence	p. 269
Some Important Copulae	p. 270
Pricing Credit Derivatives from Default Times	p. 278
Standard Error of the Breakeven Spread	p. 280
Summary	p. 281
Technical Appendix	p. 282
Pricing Default Baskets	p. 283
Introduction	p. 283
Modelling First-to-default Baskets	p. 283
Second-to-default and Higher Default Baskets	p. 291
Pricing Baskets using Monte Carlo	p. 294
Pricing Baskets using a Multi-Factor Model	p. 296
Pricing Baskets in the Student-t Copula	p. 298
Risk Management of Default Baskets	p. 299
Summary	p. 301
Pricing Tranches in the Gaussian Copula Model	p. 303
Introduction	p. 303
The LHP Model	p. 303
Drivers of the Tranche Spread	p. 308
Accuracy of the LHP Approximation	p. 312
The LHP Model with Tail Dependence	p. 313
Summary	p. 314
Technical Appendix	p. 314
Risk Management of Synthetic Tranches	p. 317
Introduction	p. 317
Systemic Risks	p. 318
The LH+ Model	p. 324
Idiosyncratic Risks	p. 328
Hedging Tranches	p. 334
Summary	p. 339
Technical Appendix	p. 339
Building the Full Loss Distribution	p. 343
Introduction	p. 343
Calculating the Tranche Survival Curve	p. 343
Building the Conditional Loss Distribution	p. 345
Integrating over the Market Factor	p. 353
Approximating the Conditional Portfolio Loss Distribution	p. 354
A Comparison of Methods	p. 360
Perturbing the Loss Distribution	p. 362
Summary	p. 364
Implied Correlation	p. 365
Introduction	p. 365
Implied Correlation	p. 365
Compound Correlation	p. 367
Disadvantages of Compound Correlation	p. 370
No-arbitrage Conditions	p. 371
Summary	p. 374
Base Correlation	p. 375
Introduction	p. 375
Base Correlation	p. 375
Building the Base Correlation Curve	p. 377
Base Correlation Interpolation	p. 382
Interpolating Base Correlation using the ETL	p. 389
A Base Correlation Surface	p. 393
Risk Management of Index Tranches	p. 394
Hedging the Base Correlation Skew	p. 395
Base Correlation for Bespoke Tranches	p. 398
Risk Management of Bespoke Tranches	p. 405
Summary	p. 406
Copula Skew Models	p. 409
Introduction	p. 409
The Challenge of Fitting the Skew	p. 409
Calibration	p. 411
Random Recovery	p. 412
The Student-t Copula	p. 413
The Double-t Copula	p. 415
The Composite Basket Model	p. 418
The Marshall-Olkin Copula	p. 420
The Mixing Copula	p. 421
The Random Factor Loading Model	p. 423
The Implied Copula	p. 427
Copula Comparison	p. 429
Pricing Bespokes	p. 431
Summary	p. 431
Advanced Multi-name Credit Derivatives	p. 433
Introduction	p. 433
Credit CPPI	p. 433
Constant Proportion Debt Obligations	p. 436
The CDO-squared	p. 441
Tranchelets	p. 448
Forward Starting Tranches	p. 449
Options on Tranches	p. 449
Leveraged Super Senior	p. 450
Summary	p. 451
Dynamic Bottom-up Correlation Models	p. 453
Introduction	p. 453
A Survey of Dynamic Models	p. 455
The Intensity Gamma Model	p. 458
The Affine Jump Diffusion Model	p. 466
Summary	p. 470
Technical Appendix	p. 470
Dynamic Top-down Correlation Models	p. 471
Introduction	p. 471
The Markov Chain Approach	p. 472
Markov Chain: Initial Generator	p. 474
Markov Chain: Stochastic Generator	p. 479
Summary	p. 483
Useful Formulae	p. 485
Bibliography	p. 487
Index	p. 491
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