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PORTFOLIO OPTIMIZATION on JUMP DIFFUSION by WANRUDEE SKULPAKDEE a Dissertation Submitted in Partial Fulfillment of the Requi PORTFOLIO OPTIMIZATION ON JUMP DIFFUSION By WANRUDEE SKULPAKDEE A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Mathematics MAY 2016 © Copyright by WANRUDEE SKULPAKDEE, 2016 All Rights Reserved © Copyright by WANRUDEE SKULPAKDEE, 2016 All Rights Reserved To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of WANRUDEE SKULPAKDEE find it satisfactory and recommend that it be accepted. ___________________________________ Hong Ming Yin, Ph.D., Chair ___________________________________ Haijun Li Ph.D. ___________________________________ Charles Moore, Ph.D. ii ACKNOWLEDGMENT Though only my name appears on the cover of this dissertation, a great many people have contributed to its production. I owe my gratitude to all those people who have made this dissertation possible and because of whom my graduate experience has been one that I will cherish forever. My deepest gratitude is to my advisor, Dr. Hong Ming Yin. I have been amazingly fortunate to have an advisor who gave me a lot of useful suggestion. His patience and support helped me overcome many crisis situations and finish this dissertation. Most importantly, none of this would have been possible without the love and patience of my husband, Mongkol Hunkrajok, and my family. Finally, I appreciate the support from my colleagues and the Department of Mathematics and Statistics, Washington State University. iii PORTFOLIO OPTIMIZATION ON JUMP DIFFUSION Abstract by Wanrudee Skulpakdee, Ph.D. Washington State University May 2016 Chair: Hong Ming Yin Portfolio optimization problem is an important research topic in financial applications. In this research, we focus on a consumption process and a terminal wealth problem. A portfolio including a riskless asset, a zero coupon bond and a stock is presented. All assets are modeled by continuous time dynamic. Bond is modeled by a classical Vasicek’s model. A stock is modeled using Merton Jump diffusion (MJD) model. This work is new because an economic inflation rate and consumption price index (CPI) are taken into consideration to evaluate a real value of assets. A Hamilton-Jacobi-Bellman (HJB) equation that satisfies an optimal solution is derived. Then, the solution is proved to exist and to be unique under certain conditions. Finally, the solution is found by using numerical method. iv TABLE OF CONTENTS Page ACKNOWLEDGEMENTS............................................................................................................iii ABSTRACT................................................................................................................................... iv LIST OF TABLES.........................................................................................................................vii LIST OF FIGURES......................................................................................................................viii CHAPTER 1. INTRODUCTION........................................................................................................... 1 1.1 Overview............................................................................................................1 1.2 Portfolio Optimization on Discrete time Model……………………………… 3 1.3 The Efficient Frontier………………………………………………………… 6 2. STOCHASTIC MODELS FORMULATION................................................................. 9 2.1 Stochastic Processes and Filtrations…………………….……………………. 9 2.2 Continuous Time Models………………………………...…………………..11 2.3 Utilities Functions……………………………………………………………14 2.4 Portfolio Optimization…………………………….………………………… 16 2.5 Bond Models…………………………………………………………......…. 21 3. PORTFOLIO OPTIMIZATION AND HJB EQUATION............................................28 3.1 Stock Models……………………………………………..…………………. 28 3.2 Ito Calculus…………………………………………….……………………. 33 3.3 Real Value of Assets…………………………………………………………35 3.4 Consumption and Terminal Wealth Problem……………………………….. 38 3.5 Wealth Process……………………………………………………………….39 v 3.6 Hamilton-Jacobi-Bellman Equation for Consumption and Terminal Wealth Problem……………….……………………………………………………. 40 3.7 A Model Reduction…………………………………..………………………43 4. EXISTENCE AND UNIQUENESS OF THE OPTIMAL SOLUTION…………….. 47 4.1 Existence of the Optimal Solution for Cauchy Problem……………..………47 4.2 Existence and Uniqueness of the Portfolio………………………..…………51 5. A COMPUTATIONAL SOLUTION FOR THE OPTIMIZATION PROBLEM….…56 5.1 Parameters Calibration………………………………………………….……56 5.2 A Numerical Solution for a HJB equation………………………………...…66 BIBLIOGRAPHY..........................................................................................................................73 APPENDIX A. DATA AND MATERIALS..........................................................................................76 B. MATLAB CODE FOR NUMERICAL SOLUTION………………………............... 85 vi LIST OF TABLES 1. The Reduction Model Parameters..................................................................................71 2. Correlation Parameters...................................................................................................71 vii LIST OF FIGURES 1. Figure1; Minimal variance portfolios (Markowitz bullet) on mean-variance plane....................6 2. Figure 2; The efficient frontier on Markowitz bullet………………………………...................7 3. Figure 3; A linear mixture between a risk-free and a risky asset.................................................8 4. Figure 4; A graph of the U.S. monthly interest rate…………………………………….……..58 5. Figure 5; The forecasted interest rate………………………………………………….………58 6. Figure 6; The predicted zero coupon bond yield…………………………………….……….. 60 7. Figure 7; A nominal price and a real value price for 7-year maturity bond………………….. 61 8. Figure 8; A histogram of ln S ……………………………………………………..……….. 64 9. Figure 9; Nominal and real value of SP500 stock index…………………………...………… 66 10. Figure 10; A value function for HJB equation……………………………………………….72 11. Figure 11; An optimal consumption for the portfolio problem………………….……….…. 72 viii CHAPTER ONE INTRODUCTION 1.1 Overview Portfolio optimization is an important research topic in financial applications. One question always asked is how to manage financial assets so that the portfolio performs well in comparison to general markets. We will answer this question of an investment problem under uncertainty to optimize investors’ return with reasonable risk. Techniques used separate this field of study into two major categories, discrete-time models and continuous-time models. Methods for discrete-time models are previously well developed. The mean-variance method is the pioneer work introduced by Harry Markowitz in 1952. By optimizing a portfolio mean of return while considering a portfolio variance as a level of uncertainty. This makes a one-period decision of assets allocation. His work was awarded the Nobel Prize in economic sciences in 1990. However, his model itself is overly simplified in practice since parameters in the model often depends on time. Merton’s continuous-time approach was considered the continuous time model in 1969. R.C. Merton adopted stochastic control techniques to find an optimal utility of the portfolio combined with risky and riskless assets for both finite and infinite lifetime consumption problems. This method is required to solve Hamilton-Jacobi-Bellman equations (HJB equation) which are typically highly nonlinear partial differential equations (PDE). They rarely obtain explicit form solutions unless models are simplified. Nevertheless, his work earned him the Nobel prize in economic sciences in 1997. The stochastic differential equation (SDE) model for the risky assets in Merton’s approach has been studied and refining to be close to reality including transaction cost, dividend, or special events. Other preliminary works should be regarded to put here included a use of stochastic distribution by E.F. Fama,1960-1980, the use of 1 stochastic integral by Harrison and Pliska,1981, the martingale approach by Karatzas,1986, and Cox and Huang, 1989. In this dissertation, we consider the portfolio optimization problem with Merton’s continuous time model. Unlike Merton’s and other related research, our model includes the risk assets with jump diffusions which are more realistic in the financial markets due to special events and corporations’ announcements. Moreover, our model also incorporates the changes of interest rate and consumption rate. This dissertation is organized as follows. In the first chapter, we will review Markowitz’s discrete-time model. In the second chapter, classical stochastic models for risky assets and portfolio optimization using stochastic control are presented. In the third chapter, Jump diffusion model including economic inflation in consideration is introduced. A portfolio optimization problem including a riskless asset, a bond and a risky stock is illustrated. HJB equation satisfies a solution of the optimal problem is derived. In Chapter Four, we prove that a solution of the HJB equation exists and unique. Finally, in Chapter Five, a reduced model is served as an example of a numerical solution. We would like to point out that the model does not include option values of risk assets. It is our plan that future research will include this topic in the model. See recent research papers in [10] and [11]. In this chapter, we mainly introduce Markowitz’s discrete-time models portfolio optimization as a fundamental
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