List of Figures xiii
List of Tables xix
Foreword xxi
Preface xxiii
Acknowledgments xxvii
CHAPTER 1
Stochastic Volatility and Local Volatility 1
Stochastic Volatility 1
Derivation of the Valuation Equation 4
Local Volatility 7
History 7
A Brief Review of Dupire’s Work 8
Derivation of the Dupire Equation 9
Local Volatility in Terms of Implied Volatility 11
Special Case: No Skew 13
Local Variance as a Conditional Expectation
of Instantaneous Variance 13
CHAPTER 2
The Heston Model 15
The Process 15
The Heston Solution for European Options 16
A Digression: The Complex Logarithm
in the Integration (2.13) 19
Derivation of the Heston Characteristic Function 20
Simulation of the Heston Process 21
Milstein Discretization 22
Sampling from the Exact Transition Law 23
Why the Heston Model Is so Popular 24
vii
viii CONTENTS
CHAPTER 3
The Implied Volatility Surface 25
Getting Implied Volatility from Local Volatilities 25
Model Calibration 25
Understanding Implied Volatility 26
Local Volatility in the Heston Model 31
Ansatz 32
Implied Volatility in the Heston Model 33
The Term Structure of Black-Scholes Implied Volatility
in the Heston Model 34
The Black-Scholes Implied Volatility Skew
in the Heston Model 35
The SPX Implied Volatility Surface 36
Another Digression: The SVI Parameterization 37
A Heston Fit to the Data 40
Final Remarks on SV Models and Fitting
the Volatility Surface 42
CHAPTER 4
The Heston-Nandi Model 43
Local Variance in the Heston-Nandi Model 43
A Numerical Example 44
The Heston-Nandi Density 45
Computation of Local Volatilities 45
Computation of Implied Volatilities 46
Discussion of Results 49
CHAPTER 5
Adding Jumps 50
Why Jumps are Needed 50
Jump Diffusion 52
Derivation of the Valuation Equation 52
Uncertain Jump Size 54
Characteristic Function Methods 56
L´evy Processes 56
Examples of Characteristic Functions
for Specific Processes 57
Computing Option Prices from the
Characteristic Function 58
Proof of (5.6) 58
Contents ix
Computing Implied Volatility 60
Computing the At-the-Money Volatility Skew 60
How Jumps Impact the Volatility Skew 61
Stochastic Volatility Plus Jumps 65
Stochastic Volatility Plus Jumps in the Underlying
Only (SVJ) 65
Some Empirical Fits to the SPX Volatility Surface 66
Stochastic Volatility with Simultaneous Jumps
in Stock Price and Volatility (SVJJ) 68
SVJ Fit to the September 15, 2005, SPX Option Data 71
Why the SVJ Model Wins 73
CHAPTER 6
Modeling Default Risk 74
Merton’s Model of Default 74
Intuition 75
Implications for the Volatility Skew 76
Capital Structure Arbitrage 77
Put-Call Parity 77
The Arbitrage 78
Local and Implied Volatility in the Jump-to-Ruin Model 79
The Effect of Default Risk on Option Prices 82
The CreditGrades Model 84
Model Setup 84
Survival Probability 85
Equity Volatility 86
Model Calibration 86
CHAPTER 7
Volatility Surface Asymptotics 87
Short Expirations 87
The Medvedev-Scaillet Result 89
The SABR Model 91
Including Jumps 93
Corollaries 94
Long Expirations: Fouque, Papanicolaou, and Sircar 95
Small Volatility of Volatility: Lewis 96
Extreme Strikes: Roger Lee 97
Example: Black-Scholes 99
Stochastic Volatility Models 99
Asymptotics in Summary 100
x CONTENTS
CHAPTER 8
Dynamics of the Volatility Surface 101
Dynamics of the Volatility Skew under Stochastic Volatility 101
Dynamics of the Volatility Skew under Local Volatility 102
Stochastic Implied Volatility Models 103
Digital Options and Digital Cliquets 103
Valuing Digital Options 104
Digital Cliquets 104
CHAPTER 9
Barrier Options 107
Definitions 107
Limiting Cases 108
Limit Orders 108
European Capped Calls 109
The Reflection Principle 109
The Lookback Hedging Argument 112
One-Touch Options Again 113
Put-Call Symmetry 113
QuasiStatic Hedging and Qualitative Valuation 114
Out-of-the-Money Barrier Options 114
One-Touch Options 115
Live-Out Options 116
Lookback Options 117
Adjusting for Discrete Monitoring 117
Discretely Monitored Lookback Options 119
Parisian Options 120
Some Applications of Barrier Options 120
Ladders 120
Ranges 120
Conclusion 121
CHAPTER 10
Exotic Cliquets 122
Locally Capped Globally Floored Cliquet 122
Valuation under Heston and Local
Volatility Assumptions 123
Performance 124
Reverse Cliquet 125
Contents xi
Valuation under Heston and Local
Volatility Assumptions 126
Performance 127
Napoleon 127
Valuation under Heston and Local
Volatility Assumptions 128
Performance 130
Investor Motivation 130
More on Napoleons 131
CHAPTER 11
Volatility Derivatives 133
Spanning Generalized European Payoffs 133
Example: European Options 134
Example: Amortizing Options 135
The Log Contract 135
Variance and Volatility Swaps 136
Variance Swaps 137
Variance Swaps in the Heston Model 138
Dependence on Skew and Curvature 138
The Effect of Jumps 140
Volatility Swaps 143
Convexity Adjustment in the Heston Model 144
Valuing Volatility Derivatives 146
Fair Value of the Power Payoff 146
The Laplace Transform of Quadratic Variation under
Zero Correlation 147
The Fair Value of Volatility under Zero Correlation 149
A Simple Lognormal Model 151
Options on Volatility: More on Model Independence 154
Listed Quadratic-Variation Based Securities 156
The VIX Index 156
VXB Futures 158
Knock-on Benefits 160
Summary 161
Postscript 162
Bibliography 163
Index 169