Malliavin Calculus in Finance: Theory and Practice, Second Edition
Elisa Alòs
David Garcia Lorite
Contents
Foreword xv
Preface xix
Preface to the Second Edition xxiii
About the Authors xxv
Section I A Primer on Option Pricing and Volatility
Modelling
Chapter 1 ■ The Option Pricing Problem 3
1.1 DERIVATIVES 4
1.1.1 Forwards and Futures 4
1.1.2 Options 5
1.2 NON-ARBITRAGE PRICES AND THE BLACK-SCHOLES
FORMULA 6
1.2.1 The Forward Contract 7
1.2.2 The Price of a European Option as a
Risk-Neutral Expectation 8
1.2.3 The Price of a Vanilla Option and the
Black-Scholes Formula 11
1.3 THE BLACK-SCHOLES MODEL 13
1.3.1 From the Black-Scholes Formula to the
Black-Scholes Model 14
1.3.2 Option Replication and Delta Hedging in the
Black-Scholes Model 16
1.4 THE BLACK-SCHOLES IMPLIED VOLATILITY AND
THE NON-CONSTANT VOLATILITY CASE 18
1.4.1 The Implied Volatility Surface 19
1.4.2 The Implied and Spot Volatilities 20
1.5 CHAPTER’S DIGEST 22
Chapter 2 ■ The Volatility Process 25
2.1 THE ESTIMATION OF THE INTEGRATED AND THE
SPOT VOLATILITY 25
2.1.1 Methods Based on the Realised Variance 25
2.1.2 Fourier Estimation of Volatility 26
2.1.3 Properties of the Spot Volatility 29
2.2 LOCAL VOLATILITIES 32
2.2.1 Mimicking Processes 33
2.2.2 Forward Equation and Dupire Formula 34
2.3 STOCHASTIC VOLATILITIES 39
2.3.1 The Heston Model 41
2.3.2 The SABR Model 42
2.4 STOCHASTIC-LOCAL VOLATILITIES 47
2.5 MODELS BASED ON THE FRACTIONAL BROWNIAN
MOTION AND ROUGH VOLATILITIES 49
2.6 VOLATILITY DERIVATIVES 50
2.6.1 Variance Swaps and the VIX 52
2.6.2 Volatility Swaps 53
2.6.3 Weighted Variance Swaps and Gamma Swaps 55
2.7 CHAPTER’S DIGEST 55
Section II Mathematical Tools
Chapter 3 ■ A Primer on Malliavin Calculus 59
3.1 DEFINITIONS AND BASIC PROPERTIES 59
3.1.1 The Malliavin Derivative Operator 60
3.1.2 The Divergence Operator 63
3.2 COMPUTATION OF MALLIAVIN DERIVATIVES 64
3.2.1 The Malliavin Derivative of an Itˆo Process 65
3.2.2 The Malliavin Derivative of a Diffusion Process 66
3.3 MALLIAVIN DERIVATIVES FOR GENERAL SV
MODELS 71
3.3.1 The SABR Volatility 71
3.3.2 The Heston Volatility 72
3.3.3 The 3/2 Heston Volatility 73
3.4 CHAPTER’S DIGEST 74
Chapter 4 ■ Key Tools in Malliavin Calculus 75
4.1 THE CLARK-OCONE-HAUSSMAN FORMULA 75
4.1.1 The Clark-Ocone-Haussman Formula and the
Martingale Representation Theorem 75
4.1.2 Hedging in the Black-Scholes Model 78
4.1.3 A Martingale Representation for Spot and
Integrated Volatilities 81
4.1.4 A Martingale Representation for
Non-Log-Normal Assets 84
4.2 THE INTEGRATION BY PARTS FORMULA 85
4.2.1 The Integration-by-Parts Formula for the
Malliavin Derivative and the Skorohod Integral
Operators 86
4.2.2 Delta, Vega, and Gamma in the Black-Scholes
Model 87
4.2.3 The Delta of an Asian Option in the
Black-Scholes Model 89
4.2.4 The Stochastic Volatility Case 90
4.3 THE ANTICIPATING IT ˆO’S FORMULA 95
4.3.1 The Anticipating Itˆo’s Formula as an Extension
of Itˆo’s Formula 96
4.3.2 The Law of an Asset Price as a Perturbation of
a Mixed Log-Normal Distribution 99
4.3.3 The Moments of Log-Prices in Stochastic
Volatility Models 102
4.3.4 Some Applications to Volatility Derivatives 105
4.4 CHAPTER’S DIGEST 107
Chapter 5 ■ Fractional Brownian Motion and Rough
Volatilities 109
5.1 THE FRACTIONAL BROWNIAN MOTION 109
5.1.1 Correlated Increments 111
5.1.2 Long and Short Memory 111
5.1.3 Stationary Increments and Self-Similarity 113
5.1.4 H¨older Continuity 113
5.1.5 The p-Variation and the Semimartingale
Property 113
5.1.6 Representations of the fBm 114
5.2 THE RIEMANN-LIOUVILLE FRACTIONAL BROWNIAN
MOTION 115
5.3 STOCHASTIC INTEGRATION WITH RESPECT TO THE
FBM 118
5.4 SIMULATION METHODS FOR THE fBM AND THE
RLfBM 121
5.5 THE FRACTIONAL BROWNIAN MOTION IN FINANCE 122
5.6 THE MALLIAVIN DERIVATIVE OF FRACTIONAL
VOLATILITIES 125
5.6.1 Fractional Ornstein-Uhlenbeck Volatilities 126
5.6.2 The Rough Bergomi Model 127
5.6.3 A Fractional Heston Model 127
5.7 CHAPTER’S DIGEST 128
Section III Applications of Malliavin Calculus to the
Study of the Implied Volatility Surface
Chapter 6 ■ The ATM Short-Time Level of the Implied
Volatility 133
6.1 BASIC DEFINITIONS AND NOTATION 134
6.2 THE CLASSICAL HULL AND WHITE FORMULA 135
6.2.1 Two Proofs of the Hull and White Formula 135
6.3 AN EXTENSION OF THE HULL AND WHITE FORMULA
FROM THE ANTICIPATING IT ˆO’S FORMULA 138
6.4 DECOMPOSITION FORMULAS FOR IMPLIED
VOLATILITIES 145
6.5 THE ATM SHORT-TIME LEVEL OF THE IMPLIED
VOLATILITY 146
6.5.1 The Uncorrelated Case 147
6.5.2 The Correlated Case 152
6.5.3 Approximation Formulas for the ATMI 164
6.5.4 Examples 166
6.5.5 Numerical Experiments 176
6.6 CHAPTER’S DIGEST 179
Chapter 7 ■ The ATM Short-Time Skew 182
7.1 THE TERM STRUCTURE OF THE EMPIRICAL IMPLIED
VOLATILITY SURFACE 182
7.2 THE MAIN PROBLEM AND NOTATIONS 185
7.3 THE UNCORRELATED CASE 185
7.4 THE CORRELATED CASE 187
7.5 THE SHORT-TIME LIMIT OF IMPLIED VOLATILITY
SKEW 190
7.6 APPLICATIONS 196
7.6.1 Diffusion Stochastic Volatilities: Finite Limit of
the ATM Skew Slope 196
7.6.2 Local Volatility Models 199
7.6.3 Stochastic-Local Volatility Models 203
7.6.4 Fractional Stochastic Volatility Models 205
7.6.5 Time-Varying Coefficients 208
7.7 IS THE VOLATILITY LONG-MEMORY, SHORT-MEMORY,
OR BOTH? 209
7.8 A COMPARISON WITH JUMP-DIFFUSION MODELS:
THE BATES MODEL 209
7.9 CHAPTER’S DIGEST 213
Chapter 8 ■ The ATM Short-Time Curvature 217
8.1 SOME EMPIRICAL FACTS 217
8.2 THE UNCORRELATED CASE 220
8.2.1 A Representation for the ATM Curvature 220
8.2.2 Limit Results 223
8.2.3 Examples 226
8.3 THE CORRELATED CASE 230
8.3.1 A Representation for the ATM Curvature 230
8.3.2 Limit Results 232
8.3.3 The Convexity of the Short-Time Implied
Volatility 241
8.4 EXAMPLES 242
8.4.1 Regular Local Volatility Models 242
8.4.2 Rough Local Volatilities 244
8.4.3 Diffusion Volatility Models 245
8.4.4 Fractional Volatilities 249
8.5 CHAPTER’S DIGEST 252
Section IV The Implied Volatility of Non-Vanilla Options
Chapter 9 ■ Options with Random Strikes and the
Forward Smile 255
9.1 A DECOMPOSITION FORMULA FOR RANDOM STRIKE
OPTIONS 256
9.2 FORWARD-START OPTIONS AS RANDOM STRIKE
OPTIONS 259
9.3 FORWARD-START OPTIONS AND THE
DECOMPOSITION FORMULA 262
9.4 THE ATM SHORT-TIME LIMIT OF THE IMPLIED
VOLATILITY 264
9.5 AT-THE-MONEY SKEW 271
9.5.1 Local Volatility Models 275
9.5.2 Stochastic Volatility Models 276
9.5.3 Fractional Volatility Models 277
9.5.4 Time-Depending Coefficients 278
9.6 AT-THE-MONEY CURVATURE 279
9.6.1 The Uncorrelated Case 279
9.6.2 The Correlated Case 282
9.7 CHAPTER’S DIGEST 285
Chapter 10 ■ Options on the VIX 290
10.1 THE ATM SHORT-TIME LEVEL AND SKEW OF THE
IMPLIED VOLATILITY 293
10.1.1 The ATMI Short-Time Limit 293
10.1.2 The Short-Time Skew of the ATMI Volatility 296
10.2 VIX OPTIONS 299
10.2.1 The Short-End Level of the ATMI of VIX
Options 300
10.2.2 The ATM Skew of VIX Options 302
10.3 CHAPTER’S DIGEST 311Section V Non-Log Normal Models
Chapter 11 ■ The Bachelier Implied Volatility 315
11.1 BACHELIER-TYPE MODELS 315
11.2 A DECOMPOSITION FORMULA FOR OPTION PRICES 318
11.3 DECOMPOSITION FORMULAS FOR IMPLIED
VOLATILITIES 321
11.4 THE ATM SHORT-TIME LEVEL OF THE IMPLIED
VOLATILITY 322
11.4.1 The Uncorrelated Case 323
11.4.2 The Correlated Case 323
11.5 THE BACHELIER ATM SKEW 334
11.5.1 The Short-Time Limit of Implied Volatility
Skew 336
11.6 CHAPTER’S DIGEST 339
Bibliography 341
Index 363