The Volatility Surface A Practitioner's Guide

"Written by a practitioner for practitioners, The Volatility Surface examines why options are priced as they are and - starting from a powerful representation of implied volatility in terms of a weighted average of realized volatilities - explores the implications of various popular models for pricing."--BOOK JACKET.

JIM GATHERAL is a Managing Director at Merrill Lynch and also an Adjunct Professor at the Courant Institute of Mathematical Sciences, New York University.Dr. Gatheral obtained a PhD in theoretical physics from Cambridge Universityin 1983. Since then, he has been involved in all of the major derivative product areasas a bookrunner, risk manager, and quantitative analyst in London, Tokyo, and New York. From 1997 to 2005, Dr. Gatheral headed the Equity Quantitative Analytics group at Merrill Lynch. His current research focus is equity market microstructure and algorithmic trading.

List of Figures

xiii

List of Tables

xix

Foreword

xxi

Preface

xxiii

Acknowledgments

xxvii

CHAPTER 1 Stochastic Volatility and Local Volatility

1

(14)

Stochastic Volatility

1

(6)

Derivation of the Valuation Equation

4

(3)

Local Volatility

7

(8)

History

7

(1)

A Brief Review of Dupire's Work

8

(1)

Derivation of the Dupire Equation

9

(2)

Local Volatility in Terms of Implied Volatility

11

(2)

Special Case: No Skew

13

(1)

Local Variance as a Conditional Expectation of Instantaneous Variance

13

(2)

CHAPTER 2 The Heston Model

15

(10)

The Process

15

(1)

The Heston Solution for European Options

16

(4)

A Digression: The Complex Logarithm in the Integration (2.13)

19

(1)

Derivation of the Heston Characteristic Function

20

(1)

Simulation of the Heston Process

21

(4)

Milstein Discretization

22

(1)

Sampling from the Exact Transition Law

23

(1)

Why the Heston Model Is so Popular

24

(1)

CHAPTER 3 The Implied Volatility Surface

25

(18)

Getting Implied Volatility from Local Volatilities

25

(6)

Model Calibration

25

(1)

Understanding Implied Volatility

26

(5)

Local Volatility in the Heston Model

31

(2)

Ansatz

32

(1)

Implied Volatility in the Heston Model

33

(3)

The Term Structure of Black-Scholes Implied Volatility in the Heston Model

34

(1)

The Black-Scholes Implied Volatility Skew in the Heston Model

35

(1)

The SPX Implied Volatility Surface

36

(7)

Another Digression: The SVI Parameterization

37

(3)

A Heston Fit to the Data

40

(2)

Final Remarks on SV Models and Fitting the Volatility Surface

42

(1)

CHAPTER 4 The Heston-Nandi Model

43

(7)

Local Variance in the Heston-Nandi Model

43

(1)

A Numerical Example

44

(5)

The Heston-Nandi Density

45

(1)

Computation of Local Volatilities

45

(1)

Computation of Implied volatilities

46

(3)

Discussion of Results

49

(1)

CHAPTER 5 Adding Jumps

50

(24)

Why Jumps are Needed

50

(2)

Jump Diffusion

52

(4)

Derivation of the Valuation Equation

52

(2)

Uncertain Jump Size

54

(2)

Characteristic Function Methods

56

(9)

Lévy Processes

56

(1)

Examples of Characteristic Functions for Specific Processes

57

(1)

Computing Option Prices from the Characteristic Function

58

(1)

Proof of (5.6)

58

(2)

Computing Implied Volatility

60

(1)

Computing the At-the-Money Volatility Skew

60

(1)

How Jumps Impact the Volatility Skew

61

(4)

Stochastic Volatility Plus Jumps

65

(9)

Stochastic Volatility Plus Jumps in the Underlying Only (SVJ)

65

(1)

Some Empirical Fits to the SPX Volatility Surface

66

(2)

Stochastic Volatility with Simultaneous Jumps in Stock Price and Volatility (SVJJ)

68

(3)

SVJ Fit to the September 15, 2005, SPX Option Data

71

(2)

Why the SVJ Model Wins

73

(1)

CHAPTER 6 Modeling Default Risk

74

(13)

Merton's Model of Default

74

(3)

Intuition

75

(1)

Implications for the Volatility Skew

76

(1)

Capital Structure Arbitrage

77

(2)

Put-Call Parity

77

(1)

The Arbitrage

78

(1)

Local and Implied Volatility in the Jump-to-Ruin Model

79

(3)

The Effect of Default Risk on Option Prices

82

(2)

The CreditGrades Model

84

(3)

Model Setup

84

(1)

Survival Probability

85

(1)

Equity Volatility

86

(1)

Model Calibration

86

(1)

CHAPTER 7 Volatility Surface Asymptotics

87

(14)

Short Expirations

87

(2)

The Medvedev-Scaillet Result

89

(4)

The SABR Model

91

(2)

Including Jumps

93

(2)

Corollaries

94

(1)

Long Expirations: Fouque, Papanicolaou, and Sircar

95

(1)

Small Volatility of Volatility: Lewis

96

(1)

Extreme Strikes: Roger Lee

97

(3)

Example: Black-Scholes

99

(1)

Stochastic Volatility Models

99

(1)

Asymptotics in Summary

100

(1)

CHAPTER 8 Dynamics of the Volatility Surface

101

(6)

Dynamics of the Volatility Skew under Stochastic Volatility

101

(1)

Dynamics of the Volatility Skew under Local Volatility

102

(1)

Stochastic Implied Volatility Models

103

(1)

Digital Options and Digital Cliquets

103

(4)

Valuing Digital Options

104

(1)

Digital Cliquets

104

(3)

CHAPTER 9 Barrier Options

107

(15)

Definitions

107

(1)

Limiting Cases

108

(1)

Limit Orders

108

(1)

European Capped Calls

109

(1)

The Reflection Principle

109

(3)

The Lookback Hedging Argument

112

(1)

One-Touch Options Again

113

(1)

Put-Call Symmetry

113

(1)

QuasiStatic Hedging and Qualitative Valuation

114

(3)

Out-of-the-Money Barrier Options

114

(1)

One-Touch Options

115

(1)

Live-Out Options

116

(1)

Lookback Options

117

(1)

Adjusting for Discrete Monitoring

117

(3)

Discretely Monitored Lookback Options

119

(1)

Parisian Options

120

(1)

Some Applications of Barrier Options

120

(1)

Ladders

120

(1)

Ranges

120

(1)

Conclusion

121

(1)

CHAPTER 10 Exotic Cliquets 122

Locally Capped Globally Floored Cliquet

122

(3)

Valuation under Heston and Local Volatility Assumptions

123

(1)

Performance

124

(1)

Reverse Cliquet

125

(2)

Valuation under Heston and Local Volatility Assumptions

126

(1)

Performance

127

(1)

Napoleon

127

(6)

Valuation under Heston and Local Volatility Assumptions

128

(2)

Performance

130

(1)

Investor Motivation

130

(1)

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The Volatility Surface A Practitioner's Guide

Summary

Author Biography

Table of Contents

Supplemental Materials