Contents 9
Preface 11
ACKNOWLEDGMENTS 13
Chapter 1: Mathematical Preliminaries 15
INTRODUCTION 15
COMPLEX NUMBERS 15
FINDING ROOTS OF FUNCTIONS 21
OLS AND WLS 26
NELDER-MEAD ALGORITHM 34
MAXIMUM LIKELIHOOD ESTIMATION 41
CUBIC SPLINE INTERPOLATION 41
SUMMARY 45
EXERCISES 46
SOLUTIONS TO EXERCISES 47
Chapter 2: Numerical Integration 53
INTRODUCTION 53
NEWTON-COATES FORMULAS 54
IMPLEMENTING NEWTON-COTES FORMULAS IN VBA 63
GAUSSIAN QUADRATURES 71
SUMMARY 76
EXERCISES 77
SOLUTIONS TO EXERCISES 78
APPENDIX 79
Chapter 3: Tree-Based Methods 84
INTRODUCTION 84
CRR BINOMIAL TREE 85
LEISEN-REIMER BINOMIAL TREE 89
EDGEWORTH BINOMIAL TREE 91
FLEXIBLE BINOMIAL TREE 95
TRINOMIAL TREE 97
ADAPTIVE MESH METHOD 101
COMPARING TREES 105
IMPLIED VOLATILITY TREES 106
ALLOWING FOR DIVIDENDS AND THE COST-OF-CARRY 118
SUMMARY 122
EXERCISES 122
SOLUTIONS TO EXERCISES 123
Chapter 4: The Black-Scholes, Practitioner Black-Scholes, and Gram-Charlier Models 126
INTRODUCTION 126
THE BLACK-SCHOLES MODEL 126
IMPLIED VOLATILITY AND THE DVF 130
THE PRACTITIONER BLACK-SCHOLES MODEL 134
THE GRAM-CHARLIER MODEL 138
SUMMARY 143
EXERCISES 144
SOLUTIONS TO EXERCISES 145
Chapter 5: The Heston (1993) Stochastic Volatility Model 150
INTRODUCTION 150
THE HESTON (1993) MODEL 151
INCREASING INTEGRATION ACCURACY 160
THE FUNDAMENTAL TRANSFORM 163
SENSITIVITY ANALYSIS 165
SUMMARY 170
EXERCISES 171
SOLUTIONS TO EXERCISES 171
APPENDIX 174
Chapter 6: The Heston and Nandi (2000) GARCH Model 177
INTRODUCTION 177
PERSISTENT VOLATILITY IN ASSET RETURNS 177
GARCH VARIANCE MODELING 180
THE HESTON AND NANDI (2000) MODEL 188
SUMMARY 196
EXERCISES 196
SOLUTIONS TO EXERCISES 198
Chapter 7: The Greeks 201
INTRODUCTION 201
BLACK-SCHOLES GREEKS 202
GREEKS FROM THE TREES 206
GREEKS FROM THE GRAM-CHARLIER MODEL 211
GREEKS FROM THE HESTON (1993) MODEL 217
GREEKS FROM THE HESTON AND NANDI (2000) MODEL 224
GREEKS BY FINITE DIFFERENCES 231
SUMMARY 235
EXERCISES 236
SOLUTIONS TO EXERCISES 236
APPENDIX 240
Chapter 8: Exotic Options 244
INTRODUCTION 244
SINGLE-BARRIER OPTIONS 244
DIGITAL OPTIONS 270
ASIAN OPTIONS 275
FLOATING-STRIKE LOOKBACK OPTIONS 279
SUMMARY 282
EXERCISES 283
SOLUTIONS TO EXERCISES 284
Chapter 9: Parameter Estimation 289
INTRODUCTION 289
UNCONDITIONAL MOMENTS 289
MAXIMUM LIKELIHOOD FOR GARCH MODELS 292
ESTIMATION BY LOSS FUNCTIONS 297
OTHER ESTIMATION METHODS 307
SUMMARY 307
EXERCISES 307
SOLUTIONS TO EXERCISES 309
Chapter 10: Implied Volatility 318
INTRODUCTION 318
OBTAINING IMPLIED VOLATILITY 318
EXPLAINING SMILES AND SMIRKS 326
SUMMARY 330
EXERCISES 331
SOLUTIONS TO EXERCISES 331
Chapter 11: Model-Free Implied Volatility 336
INTRODUCTION 336
THEORETICAL FOUNDATION 336
IMPLEMENTATION 337
INTERPOLATION-EXTRAPOLATION METHOD 343
MODEL-FREE IMPLIED FORWARD VOLATILITY 350
THE VIX INDEX 353
SUMMARY 360
EXERCISES 361
SOLUTIONS TO EXERCISES 361
Chapter 12: Model-Free Higher Moments 364
INTRODUCTION 364
THEORETICAL FOUNDATION 364
IMPLEMENTATION 368
VERIFYING IMPLIED MOMENTS 377
GRAM-CHARLIER IMPLIED MOMENTS 379
SUMMARY 382
EXERCISES 384
SOLUTIONS TO EXERCISES 384
Chapter 13: Volatility Returns 388
INTRODUCTION 388
STRADDLE RETURNS 389
DELTA-HEDGED GAINS 400
VOLATILITY EXPOSURE 402
VARIANCE SWAPS 406
SUMMARY 411
EXERCISES 412
SOLUTIONS TO EXERCISES 413
Appendix A: A VBA Primer 418
References 423
About the CD-ROM 427
INTRODUCTION 427
SYSTEM REQUIREMENTS 427
USING THE CD WITH WINDOWS 427
WHAT\u2019S ON THE CD 428
TROUBLESHOOTING 429
CUSTOMER CARE 429
About the Authors 431
Index 433