Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications
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Purdue University Purdue e-Pubs Open Access Dissertations Theses and Dissertations January 2015 Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications. Rolando Dangnanan Navarro Purdue University Follow this and additional works at: https://docs.lib.purdue.edu/open_access_dissertations Recommended Citation Navarro, Rolando Dangnanan, "Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications." (2015). Open Access Dissertations. 1422. https://docs.lib.purdue.edu/open_access_dissertations/1422 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. Graduate School Form 30 Updated 1/15/2015 PURDUE UNIVERSITY GRADUATE SCHOOL Thesis/Dissertation Acceptance This is to certify that the thesis/dissertation prepared By Rolando D. Navarro, Jr. Entitled Malliavin Calculus in the Canonical Levy Process: White Noise Theory and Financial Applications For the degree of Doctor of Philosophy Is approved by the final examining committee: Frederi Viens Chair Jonathon Peterson Michael Levine Jose Figueroa-Lopez To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement, Publication Delay, and Certification Disclaimer (Graduate School Form 32), this thesis/dissertation adheres to the provisions of Purdue University’s “Policy of Integrity in Research” and the use of copyright material. Approved by Major Professor(s): Frederi Viens Approved by: Jun Xie 11/24/2015 Head of the Departmental Graduate Program Date MALLIAVIN CALCULUS IN THE CANONICAL LEVY´ PROCESS: WHITE NOISE THEORY AND FINANCIAL APPLICATIONS A Dissertation Submitted to the Faculty of Purdue University by Rolando D. Navarro, Jr. In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2015 Purdue University West Lafayette, Indiana ii "Stay hungry, stay foolish!" Steve Jobs (1955-2011) iii ACKNOWLEDGMENTS I would like to express my deepest gratitude to the following who made my journey towards the completion of my Ph.D. dissertation possible. The Almighty Father for giving me all the strength and endurance to accomplish this noble endeavor. My parents for their unyielding encouragement for me to pursue graduate studies in Purdue and well as for imparting their invaluable foresight on what it takes to be successful in life and to my relatives in New York City, Lynn Terrell, Edmundo Navarro, and Araceli Galvan Navarro who kept me home away from home. My adviser, Dr. Frederi Viens for the intellectual stimulation and insightful sug- gestions selflessly provided to me during the course of this research as well as for expanding my horizon in research opportunities in mathematical finance. Also, I would like to thank Dr. Michael Levine, Dr. Jose Figueroa-Lopez, and Dr. Jonathon Peterson for their invaluable comments for my dissertation. My academic sibling, Dr. Richard Eden for providing me his meticulously written notes on Malliavin calculus. Without his mentoring and fruitful discussions during my earlier stages in my stay here Purdue, this thesis would not have gone this far. My Boilermaker friends from PQFC and Statistics Department especially to Lin Yang Cheng, Berend Coster, Xiaoguang Wang, Tian Qiu, Jeffrey Nisen, Yao Tang, and Yudong Cao for sharing their individual and collective aspirations so that through hard work and gritty determination, we can achieve our dreams in the exciting world of Quantitative Finance. My brethren in Church of Christ Iglesia ni Cristo of the Locale of Indianapolis especially to the Gumasing family: Bro. Garry, Sis. Carol, and Bro. Paolo who help me remained steadfast to the faith. You are all awesome! 23rd of November 2015, West Lafayette, IN iv TABLE OF CONTENTS Page ABSTRACT ::::::::::::::::::::::::::::::::::: vii 1 Introduction :::::::::::::::::::::::::::::::::: 1 1.1 Motivation :::::::::::::::::::::::::::::::: 1 1.2 Overview of the Dissertation :::::::::::::::::::::: 4 1.3 Main Results :::::::::::::::::::::::::::::: 5 2 Preliminaries ::::::::::::::::::::::::::::::::: 7 2.1 L´evyProcesses ::::::::::::::::::::::::::::: 7 2.2 Moment Inequalities :::::::::::::::::::::::::: 11 2.3 Geometric L´evyProcesses ::::::::::::::::::::::: 12 2.4 Stochastic Differential Equations ::::::::::::::::::: 14 2.5 Canonical L´evySpace ::::::::::::::::::::::::: 15 2.6 Iterated L´evy-It^oIntegral ::::::::::::::::::::::: 16 2.7 Skorohod Integral :::::::::::::::::::::::::::: 18 2.8 Predictable Process ::::::::::::::::::::::::::: 21 3 Canonical L´evyWhite Noise Processes ::::::::::::::::::: 22 3.1 Construction of Canonical L´evyWhite Noise Process :::::::: 22 3.2 Construction of Alternative Chaos Expansion for Canonical L´evyprocesses :::::::::::::::::::::::: 26 3.3 Alternative Chaos Expansion for Canonical L´evyprocesses ::::: 35 3.4 Stochastic Test and Distribution Function :::::::::::::: 41 3.4.1 The spaces G and G∗ :::::::::::::::::::::: 42 3.4.2 Kontratiev and Hida spaces :::::::::::::::::: 43 3.5 White Noise Processes from Canonical L´evyProcesses :::::::: 44 3.6 Wick Product and Hermite Transform :::::::::::::::: 49 v Page 3.7 Stochastic Derivative :::::::::::::::::::::::::: 52 3.8 Generalized Expectation and Generalized Conditional Expectation : 58 3.9 Skorohod Integration on G∗ :::::::::::::::::::::: 63 3.10 Clark-Ocone Theorem in L2(P ) :::::::::::::::::::: 71 3.11 Multivariate Extension ::::::::::::::::::::::::: 79 3.11.1 Notations :::::::::::::::::::::::::::: 80 3.11.2 Chaos Expansion :::::::::::::::::::::::: 81 3.11.3 Stochastic Test and Distribution Functions :::::::::: 82 3.11.4 Wick Product :::::::::::::::::::::::::: 84 3.11.5 Stochastic Derivatives ::::::::::::::::::::: 84 3.11.6 Generalized Conditional Expectation ::::::::::::: 85 3.11.7 Skorohod Integration on G∗ :::::::::::::::::: 86 3.11.8 Clark Ocone Theorem in L2(P ) :::::::::::::::: 88 4 Clark-Ocone Theorem Under The Change of Measure and Mean-Variance Hedging :::::::::::::::::::::::::::::::::::: 89 4.1 Girsanov Theorem for L´evyProcesses ::::::::::::::::: 89 4.2 Clark-Ocone Theorem in L2(P ) \ L2(Q) ::::::::::::::: 92 4.3 Mean Variance Hedging :::::::::::::::::::::::: 104 4.3.1 Financial Modeling Under a L´evyMarket ::::::::::: 104 4.3.2 Quadratic Hedging ::::::::::::::::::::::: 109 4.3.3 Geometric L´evyProcesses ::::::::::::::::::: 115 4.3.4 Minimal Martingale Measure :::::::::::::::::: 121 4.3.5 The Bates Model :::::::::::::::::::::::: 126 5 Donsker Delta and Its Applications to Finance ::::::::::::::: 137 5.1 Donsker Delta :::::::::::::::::::::::::::::: 137 5.2 Evaluation of E[Dt;zg(Y (T ))jFt] :::::::::::::::::::: 138 5.2.1 Case I: E[Dt;0g(Y (T ))jFt] ::::::::::::::::::: 139 5.2.2 Case II: E[Dt;zg(Y (T ))jFt]; z 6= 0 ::::::::::::::: 141 vi Page 5.3 Examples :::::::::::::::::::::::::::::::: 143 5.3.1 Merton Model :::::::::::::::::::::::::: 143 5.3.2 Continuous Case :::::::::::::::::::::::: 146 6 Evaluating Greeks In Exotic Options :::::::::::::::::::: 149 6.1 Preliminaries :::::::::::::::::::::::::::::: 149 6.2 Markovian Property of the Payoff ::::::::::::::::::: 153 6.3 Malliavin Derivatives of the Supremum and Infimum :::::::: 154 6.4 Some Important Identities ::::::::::::::::::::::: 165 6.5 Delta ::::::::::::::::::::::::::::::::::: 166 6.6 Gamma ::::::::::::::::::::::::::::::::: 170 6.7 Construction of Dominating Processes :::::::::::::::: 173 6.7.1 Continuous-Time Monitoring ::::::::::::::::: 174 6.7.2 Discrete-Time Monitoring ::::::::::::::::::: 176 6.8 Example: Merton Model :::::::::::::::::::::::: 178 6.8.1 Continuous Monitoring ::::::::::::::::::::: 178 6.8.2 Discrete Monitoring ::::::::::::::::::::::: 179 REFERENCES :::::::::::::::::::::::::::::::::: 181 A Wiener and Poisson Chaos Expansions ::::::::::::::::::: 186 A.1 Hermite Polynomial and Hermite Function :::::::::::::: 186 A.2 Wiener Chaos Expansions ::::::::::::::::::::::: 187 A.3 Poisson Chaos Expansions ::::::::::::::::::::::: 188 VITA ::::::::::::::::::::::::::::::::::::::: 191 vii ABSTRACT Navarro, Rolando, Jr. D. PhD, Purdue University, December 2015. Malliavin Calcu- lus in the Canonical L´evyProcess: White Noise Theory and Financial Applications. Major Professor: Frederi G. Viens. We constructed a white noise theory for the Canonical L´evyprocess by Sol´e, Utzet, and Vives. The construction is based on the alternative construction of the chaos expansion of square integrable random variable. Then, we showed a Clark- Ocone theorem in L2(P ) and under the change of measure. The result from the Clark-Ocone theorem was used for the mean-variance hedging problem and applied it to stochastic volatility models such as the Barndorff-Nielsen and Shepard model model and the Bates model. A Donsker Delta approach is employed on a Binary option to solve the mean-variance hedging problem. Finally, we are able to derive the Delta and Gamma for a barrier and lookback options for an exp-L´evyprocess using the methodology of Bernis, Gobet, and Kohatsu-Higa by employing a dominating process. 1 1. INTRODUCTION 1.1 Motivation Financial modeling of risky assets is assumed to follow the classical Black-Scholes- Merton model where the log-returns risky asset follows a normal distribution. How- ever, stylized facts suggests that the Black-Scholes-Merton model is inadequate. There is a growing interest that suggests that financial modeling under a L´evyprocess is bet- ter suited in capturing market behavior. This includes skewness and long-tailed dis- tribution of the asset returns, presence of jumps,