Application of Malliavin Calculus and Wiener Chaos to Option Pricing Theory by Eric Ben-Hamou The London School of Economics and Political Science Thesis submitted in partial fulfilment of the requirements of the degree of Doctor of Philosophy at the University of London UMI Number: U615447 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U615447 Published by ProQuest LLC 2014. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 weseS F 7855 % * A bstract This dissertation provides a contribution to the option pricing literature by means of some recent developments in probability theory, namely the Malliavin Calculus and the Wiener chaos theory. It concentrates on the issue of faster convergence of Monte Carlo and Quasi-Monte Carlo simulations for the Greeks, on the topic of the Asian option as well as on the approximation for convexity adjustment for fixed income derivatives. The first part presents a new method to speed up the convergence of Monte- Carlo and Quasi-Monte Carlo simulations of the Greeks by means of Malliavin weighted schemes. We extend the pioneering works of Fournie et al. (1999), (2000) by deriving necessary and sufficient conditions for a function to serve as a weight function and by providing the weight function with minimum variance. To do so, we introduce its generator defined as its Skorohod integrand. On a numerical example, we find evidence of spectacular efficiency of this method for corridor options, especially for the gamma calculation. The second part brings new insights on the Asian option. We first show how to price discrete Asian options consistent with different types of underlying densities, especially non-normal returns, by means of the Fast Fourier Transform algorithm. We then extends Malliavin weighted schemes to continuous time Asian options. In the last part, we first prove that the Black Scholes convexity adjustment (Brotherton-Ratcliffe and Iben (1993)) can be consistently derived in a mar­ tingale framework. As an application, we examine the convexity bias between CMS and forward swap rates. However, for more complicated term structures assumptions, this approach does not hold any more. We offer a solution to this, thanks to an approximation formula, in the case of multi-factor lognormal zero coupon models, using Wiener chaos theory. 7 7 Z $ 3 L f Acknowledgements I enjoyed the help and support of numerous friends and colleagues while working on my dissertation. My greatest debt, however, goes to Doctor Pierre Mella-Barral, my supervisor at the London School of Economics and to Profes­ sor Nicole El Karoui, who initiated me to financial mathematics and nurtured my passion for this fantastic research field. Doctor Pierre Mella-Barral, whose fantastic breadth of knowledge and intuition, I have been privileged enough to have access to, guided me in countless meetings. He helped me to structure and clarify my thought and suggested valuable insightful comments. Professor Nicole El Karoui provided invaluable support in reading nearly all versions of all chapters of this Thesis. I am very much indebted to her for being thorough, where I was sloppy, and ingenious, where I was lost. I also thank her for numer­ ous occasions on which her encouragement helped me have the stamina to get to the end of the task. I am very thankful to Professor David Webb, Head of the Financial Markets Group for his friendship and for his trust. I would like to thank Margaret Bray and Bob Nobay for their support and useful remarks. I am particularly indebted to my girlfriend, Beatrice. She assisted me at each stage of the dissertation process, starting with early discussions of ideas through reading every draft of every chapter. She deserves more love and gratitude than could fit possibly on this page. I would like to express my gratitude to Alexandre Duguet for fruitful conversations and Grigorios Mamalis, whose kindly spent a couple of dozen of hours reading the chapters of this Thesis and giving me helpful and relevant feedback. Of course, all remaining errors are mine. I appreciated the remarks of the participants of the Ph.D. seminar of the Financial Markets Group, the ones of the Ph.D. seminar of the Centre de Mathematiques Appliquees at Ecole Polytechnique as well as the ones of the Ph.D. seminar in Economics. I am also indebted to several institutions for their supports: the Finan­ cial Markets Group and the Centre de Mathematiques Appliquees de l’Ecole Polytechnique. I thank all of the F.M.G. members including the administra­ tive staff, Sarah Vink, Tarah Lehane and Pippa Maddicott, for creating and maintaining such a wonderfully friendly place, which puts pressure off rather than on students and provides such an excellent environment for research and discussions. I am very appreciative of my fellow students Simon Emrich and Alexander Murmann. At the Ecole Polytechnique, I was among the privileged ones to become a member of the research group in Financial Mathematics at the Centre de Mathematiques Appliquees at Ecole Polytechnique. I would like to mention Rama Comt, Jeanne Bailleul for their nice support. Last but not least, I am especially grateful to all my family and especially to my parents, Gilbert and Sylvie Ben-Hamou, for their support and for the confidence they have put in me. I am very much indebted to my sister Agnes Ben-Hamou who kindly spent some hours correcting my english. I am thankful to my brother Joel Ben-Hamou for providing me a computer and an internet access every time I came back to France. And finally I would like to thank other relatives, friends, and teachers who have positively influenced my life. London, June 07, 2000 Eric Ben-Hamou Contents Introduction 13 Brief Review of Literature 17 Notation 21 First Part: Malliavin weighted schemes 26 1 Malliavin Weighted Scheme 26 1.1 Introduction ....................................... 27 1.2 Mathematical framework and preliminary results ........................ 29 1.2.1 Intuition ................................................................................ 29 1.2.2 Notations and hypotheses .................................................. 31 1.2.3 Generalizing Greeks ........................................................... 33 1.2.4 Results on the first variation process . ......................... 35 1.3 A new method for computing the G r e e k ..................................... 36 1.3.1 State of a r t .......................................................................... 36 1.3.2 Determination of the Malliavin w eights ............................ 37 1.3.3 Extension to models with stochastic interest rates .... 39 1.3.4 The minimal variance weighting problem ......................... 39 1.4 Examples of Malliavin weights ....................................................... 40 1.4.1 Foumie et al. solutions ........................................................ 41 1.4.2 Other exam ples .................................................................... 42 5 CONTENTS 6 1.4.3 Choice of the generator ......................................................... 44 1.5 Conclusion ........................................................................................ 44 2 Black Model 46 2.1 Introduction ..................................................................................... 47 2.2 Why a new method for the estimation of the G reeks? ............... 48 2.2.1 Description of the Black pricing m o d e l ............................. 48 2.2.2 Failure of finite difference for discontinuous payoff .... 49 2.3 Determination of the Malliavin Weights ....................................... 51 2.3.1 D elta....................................................................................... 52 2.3.2 G a m m a ................................................................................. 53 2.3.3 R h o ....................................................................................... 54 2.3.4 V ega....................................................................................... 55 2.4 Numerical Result on the Efficiency of Malliavin weights ........... 57 2.4.1 Comparative analysis: Finite Difference versus Malliavin weighted scheme ................................................................ 58 2.4.2 Typology of options requiring Malliavin weighted scheme 61 2.4.3 Local Malliavin formulae ...................................... 64 2.5 Conclusion ........................................................................................ 66 Second Part: The Asian option revisited 72 3 Discrete Asian Options 72 3.1 Introduction ..................................................................................... 73 3.2 Description of the m e th o d ............................................................ 75 3.2.1 FVamework ........................................................................... 75 3.2.2 Why Fast Fourier Transform ? ............................................ 76 3.2.3 A lgorithm .............................................................................. 77 3.3 Efficiency