Table of Contents
Preface vii
Acknowledgments xi
1 Introduction 1
1.1 Equity, fairness and arbitrage 1
1.2 Arbitrage Pricing Theory 3
1.3 The Black-Scholes model 6
1.4 The problems of the real world 10
1.4.1 Discrete hedging 10
1.4.2 Transaction costs 13
1.4.3 The breakdown of log-normality 14
1.5 The importance of model choice 17
1.5.1 Wrong hedge, wrong price 17
1.5.2 Right hedge, wrong price 20
1.6 The approach of this book 25
1.7 Exercises 27
2 Preliminaries 31
2.1 Mathematical formulation 32
2.2 Volatility surface arbitrage 38
2.3 Implied volatility surface dynamics 40
2.4 Implied forward volatility 44
2.5 Local volatility as an effective model 46
2.6 Exercises 49
3 Dividends 51
3.1 History of dividend modelling 52
3.1.1 Discrete proportional dividends 54
3.1.2 Discrete cash dividends 56
3.1.3 Mixed dividend modelling 61
3.2 How do dividends actually evolve? Historical analysis 63
3.3 The mixed dividend model 68
3.4 Barrier options and lookbacks 74
3.4.1 Basic barrier and lookback valuation 75
3.4.2 Barrier options priced with different dividend models 79
3.4.3 What have we learned? 85
3.5 Conclusions 86
3.6 Exercises 87
4 Volatility 89
4.1 Historical analysis of realised volatility 90
4.2 Historical analysis of implied volatility 94
4.2.1 Vega hedging in a volatile environment 97
4.3 Stochastic volatility modelling 104
4.4 Stochastic Local Volatility 111
4.4.1 Initial observations 115
4.5 Trading realised volatility: Variance swaps 120
4.5.1 Definition and trading rationale 120
4.5.2 Pricing 122
4.6 Trading implied volatility: Cliquets 130
4.7 Barriers revisited 143
4.8 But does it make sense? 149
4.9 Conclusions 150
4.10 Exercises 151
5 Default 153
5.1 How to model default 155
5.1.1 Structural models of default 156
5.1.2 Reduced form models of default 159
5.2 The Andersen-Buffum model 164
5.2.1 Calibration 170
5.3 Equity Default Swaps 174
5.4 Convertible Bonds 180
5.4.1 A brief history of conversion 182
5.4.2 The impact of state independent hazard rates 188
5.4.3 The impact of state dependent hazard rates 193
5.5 Conclusions 197
5.6 Exercises 198
6 Jumps 201
6.1 Historical and market inference 201
6.2 How to hedge a continuum of jumps 206
6.3 The Andersen-Andreasen model 210
6.3.1 The Merton model 211
6.3.2 Local volatility in the presence of jumps 218
6.4 The effects of jumps on equity exotics 222
6.4.1 Variance swaps 222
6.4.2 Investment strategies 226
6.4.3 Forward skew 229
6.4.4 Smile dynamics 232
6.5 Conclusions 233
6.6 Exercises 233
7 Rates 237
7.1 Historical analysis 237
7.2 Market analysis 242
7.3 Interest rate models 245
7.3.1 Preliminaries 245
7.3.2 Market models 248
7.3.3 The Heath-Jarrow-Morton model 255
7.3.4 From HJM to HW 258
7.4 Equity-rate hybrid modelling 260
7.4.1 Initial observations 263
7.5 Implicit hybrids 269
7.5.1 Autocallables 271
7.5.2 Callables: Immediate redemption 274
7.5.3 Callables: Delayed redemption 279
7.6 Explicit hybrids 282
7.7 A note on vol skew 285
7.8 Conclusions 287
7.9 Exercises 287
8 Correlation 291
8.1 Historical analysis 292
8.2 Hedging correlation 298
8.3 A simple model for dynamic correlation 300
8.3.1 Implied distributions 305
8.3.2 Implied correlation 307
8.4 Rainbow products 310
8.5 Correlation products 315
8.5.1 Covariance swaps 317
8.5.2 Correlation swaps 322
8.6 Conclusions 325
8.7 Exercises 325
9 Control 329
9.1 Elimination 330
9.2 Reserving 331
9.3 Model-free approaches 333
9.3.1 Calibration to observable dynamics 333
9.3.2 Static replication 337
9.4 Conclusions 338
Appendix A Solutions to Exercises 341
A.l Chapter 1 341
A.2 Chapter 2 348
A.3 Chapter 3 354
A.4 Chapter 4 357
A.5 Chapter 5 367
A.6 Chapter 6 373
A.7 Chapter 7 380
A.8 Chapter 8 392
Bibliography 405
Index 415
About the Author 421
