The Volatility Surface: A Practitioner's Guide

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

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

Lévy Processes 56

Examples of Characteristic Functions for Specific Processes 57

Computing Option Prices from the Characteristic Function 58

Proof of (5.6) 58

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

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

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