Table Of ContentMalliavin Calculus in
Finance
Chapman & Hall/CRC Financial Mathematics Series
Aims and scope:
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Centre for Financial Research
Department of Pure Mathematics and Statistics
University of Cambridge
Dilip B. Madan
Robert H. Smith School of Business
University of Maryland
Rama Cont
Department of Mathematics
Imperial College
Handbook of Financial Risk Management
Thierry Roncalli
Optional Processes
Stochastic Calculus and Applications
Mohamed Abdelghani, Alexander Melnikov
Machine Learning for Factor Investing
Guillaume Coqueret and Tony Guida
Malliavin Calculus in Finance
Theory and Practice
Elisa Alos, David Garcia Lorite
Risk Measures and Insurance Solvency Benchmarks
Fixed-Probability Levels in Renewal Risk Models
Vsevolod K. Malinovskii
For more information about this series please visit: https://www.crcpress.
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CHFINANCMTH
Malliavin Calculus in
Finance
Theory and Practice
Elisa Alòs
David García Lorite
Foreword by Dariusz Gatarek
First edition published 2021
by CRC Press
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and by CRC Press
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Library of Congress Cataloging‑in‑Publication Data
Names: Alòs, Elisa, author. | Lorite, David Garcia, author.
Title: Malliavin calculus in finance : theory and practice / Elisa Alòs,
David Garcia Lorite.
Description: First edition. | Boca Raton : Chapman & Hall/CRC Press, 2021.
| Series: Chapman and Hall/CRC financial mathematics series | Includes
bibliographical references and index.
Identifiers: LCCN 2021000977 (print) | LCCN 2021000978 (ebook) | ISBN
9780367893446 (hardback) | ISBN 9781003018681 (ebook)
Subjects: LCSH: Finance--Mathematical models. | Malliavin calculus. |
Stochastic analysis.
Classification: LCC HG106 .A54 2021 (print) | LCC HG106 (ebook) | DDC
332.01/51922--dc23
LC record available at https://lccn.loc.gov/2021000977
LC ebook record available at https://lccn.loc.gov/2021000978
ISBN: 978-0-367-89344-6 (hbk)
ISBN: 978-0-367-86325-8 (pbk)
ISBN: 978-1-003-01868-1 (ebk)
Typeset in Nimbus
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Contents
Foreword xv
Preface xix
Section I A primer on option pricing and volatility
modelling
Chapter 1(cid:4) The option pricing problem 3
1.1 DERIVATIVES 4
1.1.1 Forwards and futures 4
1.1.2 Options 5
1.2 NON-ARBITRAGEPRICESANDTHEBLACK-SCHOLES
FORMULA 7
1.2.1 The forward contract 7
1.2.2 ThepriceofaEuropeanoptionasarisk-neutral
expectation 9
1.2.3 The price of a vanilla option and the Black-
Scholes formula 11
1.3 THEBLACK-SCHOLESMODEL 14
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
THENON-CONSTANTVOLATILITYCASE 18
1.4.1 The implied volatility surface 18
1.4.2 The implied and spot volatilities 19
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viii (cid:4) Contents
1.5 CHAPTER’SDIGEST 21
Chapter 2(cid:4) The volatility process 25
2.1 THE ESTIMATION OF THE INTEGRATED AND THE
SPOTVOLATILITY 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 LOCALVOLATILITIES 32
2.2.1 Mimicking processes 33
2.2.2 Forward equation and Dupire formula 34
2.3 STOCHASTICVOLATILITIES 38
2.3.1 The Heston model 40
2.3.2 The SABR model 41
2.4 STOCHASTIC-LOCALVOLATILITIES 46
2.5 MODELS BASED ON THE FRACTIONAL BROWNIAN
MOTIONANDROUGHVOLATILITIES 47
2.6 VOLATILITYDERIVATIVES 49
2.6.1 Variance swaps and the VIX 50
2.6.2 Volatility swaps 51
2.6.3 Weighted variance swaps and gamma swaps 53
2.7 CHAPTER’SDIGEST 53
Section II Mathematical tools
Chapter 3(cid:4) A primer on Malliavin Calculus 57
3.1 DEFINITIONSANDBASICPROPERTIES 57
3.1.1 The Malliavin derivative operator 58
3.1.1.1 Basic properties 59
3.1.2 The divergence operator 61
3.2 COMPUTATIONOFMALLIAVINDERIVATIVES 62
3.2.1 The Malliavin derivative of an Itoˆ process 63
3.2.2 The Malliavin derivative of a diffusion process 63
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Contents (cid:4) ix
3.2.2.1 TheMalliavinderivativeofadiffusion
process as a solution of a linear SDE 64
3.2.2.2 RepresentationformulasfortheMalliavin
derivative of a diffusion process 65
3.3 MALLIAVINDERIVATIVESFORGENERALSV MOD-
ELS 68
3.3.1 The SABR volatility 68
3.3.2 The Heston volatility 69
3.3.3 The 3/2 Heston volatility 71
3.4 CHAPTER’SDIGEST 72
Chapter 4(cid:4) Key tools in Malliavin Calculus 73
4.1 THECLARK-OCONE-HAUSSMANFORMULA 73
4.1.1 The Clark-Ocone-Haussman formula and the
martingale representation theorem 73
4.1.2 Hedging in the Black-Scholes model 76
4.1.3 A martingale representation for spot and
integrated volatilities 79
4.1.3.1 The SABR volatility 79
4.1.3.2 The Heston volatility 80
4.1.4 A martingale representation for non-log-normal
assets 82
4.2 THEINTEGRATIONBYPARTSFORMULA 84
4.2.1 Theintegration-by-partsformulafortheMalliavin
derivative and the Skorohod integral operators 84
4.2.2 Delta, Vega, and Gamma in the Black-Scholes
model 85
4.2.2.1 The delta 85
4.2.2.2 The vega 86
4.2.2.3 The gamma 87
4.2.3 The Delta of an Asian option in the Black-
Scholes model 88
4.2.4 The Stochastic volatility case 89
4.2.4.1 Thedeltainstochasticvolatilitymodels 90
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