Parameter Estimation in Stochastic Volatility Models Via Approximate Bayesian Computing A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Achal Awasthi, B.S. Graduate Program in Department of Statistics The Ohio State University 2018 Master's Examination Committee: Radu Herbei,Ph.D., Advisor Laura S. Kubatko, Ph.D. c Copyright by Achal Awasthi 2018 Abstract In this thesis, we propose a generalized Heston model as a tool to estimate volatil- ity. We have used Approximate Bayesian Computing to estimate the parameters of the generalized Heston model. This model was used to examine the daily closing prices of the Shanghai Stock Exchange and the NIKKEI 225 indices. We found that this model was a good fit for shorter time periods around financial crisis. For longer time periods, this model failed to capture the volatility in detail. ii This is dedicated to my grandmothers, Radhika and Prabha, who have had a significant impact in my life. iii Acknowledgments I would like to thank my thesis supervisor, Dr. Radu Herbei, for his help and his availability all along the development of this project. I am also grateful to Dr. Laura Kubatko for accepting to be part of the defense committee. My gratitude goes to my parents, without their support and education I would not have had the chance to study worldwide. I would also like to express my gratitude towards my uncles, Kuldeep and Tapan, and Mr. Richard Rose for helping me transition smoothly to life in a different country. In addition, my deepest appreciation goes to my friends at the department of Statistics who have been there for me since my first day of class at the Ohio State University. Finally, I am extremely thankful to my housemates for bearing with me during the past one year. iv Vita 2016 . .B.S. Physics 2016-present . .Graduate Teaching Associate, The Ohio State University. Publications Fields of Study Major Field: Department of Statistics v Contents Page Abstract....................................... ii Dedication...................................... iii Acknowledgments.................................. iv Vita.........................................v List of Tables.................................... viii List of Figures................................... xii 1. Introduction..................................1 1.1 Motivation...............................1 1.2 Emerging Markets during Financial Crisis..............2 1.3 Structure of Thesis...........................5 2. Background..................................6 2.1 Introduction..............................6 2.2 Brownian Motion............................8 2.3 Geometric Brownian Motion (GBM)................. 10 2.3.1 Parameter Estimation for the GBM process using Maximum Likelihood Estimation..................... 14 2.4 The Ornstein-Uhlenbeck Process................... 17 2.4.1 Simulation of the OU Process................. 18 2.4.2 Parameter Estimation for OU Process using Maximum Like- lihood.............................. 20 vi 2.4.3 Parameter Estimation for OU Process using Ordinary Least Squares............................. 23 2.5 Cox-Ingersoll-Ross Process....................... 26 2.5.1 Simulation of CIR process................... 27 2.5.2 Parameter Estimation for CIR Process using Maximum Like- lihood.............................. 32 2.6 Generalized Cox-Ingersoll-Ross model................ 35 2.6.1 Parameter Estimation for generalized CIR Process using Max- imum Likelihood........................ 37 2.6.2 Distribution of R t2 W (s) ds .................. 40 t1 3. Approximate Bayesian Computing for Stochastic Volatility Models... 43 3.1 Heston Model.............................. 43 3.1.1 Simulation of sample paths of the Heston Model...... 45 3.1.2 Euler-Maruyama (EM) Approximation............ 46 3.1.3 Euler-Maruyama scheme with Lord et al'.s modification.. 47 3.1.4 Milstein scheme......................... 47 3.1.5 Broadie and Kaya's Exact Algorithm............. 48 3.2 A generalized Heston Model...................... 51 3.2.1 Simulation of sample paths of the generalized Heston model 53 3.3 Approximate Bayesian Computing (ABC).............. 60 3.3.1 ABC for Heston Model..................... 61 3.3.2 ABC for generalized Heston Model.............. 83 4. Application: Modeling Volatility in Financial Markets........... 100 4.1 Introduction.............................. 100 4.1.1 Stock Index........................... 100 4.2 Exploratory Data Analysis....................... 104 4.3 Parameter estimation of the Generalized Heston model using ABC 107 4.3.1 Parameter estimation using ABC for SSE.......... 107 4.3.2 Parameter estimation using ABC for NIKKEI 225..... 134 5. Contributions and Future Work....................... 142 5.1 Results Overview............................ 142 5.2 Future Work.............................. 144 5.2.1 Moments of generalized Heston model............ 145 Bibliography.................................... 147 vii List of Tables Table Page 3.1 Table showing the number of simulations vs number of accepted pa- rameters for different = 100....................... 62 3.2 Table showing the number of simulations vs number of accepted pa- rameters for different = 200....................... 62 3.3 Table showing the number of simulations vs number of accepted pa- rameters for different = 500....................... 63 3.4 Table showing the number of simulations vs number of accepted pa- rameters for different = 800....................... 63 3.5 Table showing the number of simulations vs number of accepted pa- rameters for different = 1000...................... 63 viii 3.6 Table showing the number of simulations vs number of accepted pa- rameters for different = 1500...................... 64 3.7 Table showing the number of simulations vs number of accepted pa- rameters for = 100............................ 84 3.8 Table showing the number of simulations vs number of accepted pa- rameters for = 200............................ 84 3.9 Table showing the number of simulations vs number of accepted pa- rameters for = 500............................ 85 3.10 Table showing the number of simulations vs number of accepted pa- rameters for = 800............................ 85 3.11 Table showing the number of simulations vs number of accepted pa- rameters for = 1; 000........................... 85 3.12 Table showing the number of simulations vs number of accepted pa- rameters for = 1; 500........................... 86 4.1 Table showing the number of simulations vs number of accepted pa- rameters for different = 10; 000..................... 108 ix 4.2 Table showing the number of simulations vs number of accepted pa- rameters for different levels....................... 111 4.3 Table showing the estimated parameters for different levels (100 sim- ulations).................................. 114 4.4 Table showing the number of simulations vs number of accepted pa- rameters for different levels....................... 116 4.5 Table showing the estimated parameters for different levels (100 sim- ulations).................................. 120 4.6 Table showing the number of simulations vs number of accepted pa- rameters for different levels....................... 122 4.7 Table showing the estimated parameters for different levels (100 sim- ulations).................................. 126 4.8 Table showing the number of simulations vs number of accepted pa- rameters for different levels....................... 128 4.9 Table showing the estimated parameters for different levels...... 132 x 4.10 Table showing the number of simulations vs number of accepted pa- rameters for different levels....................... 135 4.11 Table showing the estimated parameters for different levels...... 139 xi List of Figures Figure Page 2.1 Simulated paths of the GBM process with parameters as described in algorithm1................................ 13 2.2 Histogram of log of GBM at the 50th time-step. The orange curve repre- sents the superimposed normal density curve with parameters obtained from simulated data at the 50th time-step................ 14 2.3 Histogram of estimated values of µ of the GBM as simulated above. The dashed red line represents the true value of the parameter..... 16 2.4 Histogram of estimated values of σ of the GBM as simulated above. The dashed red line represents the true value of the parameter..... 16 2.5 Simulated paths of the OU process with parameters as described above 19 xii 2.6 Histogram of estimated values of β of the OU process as simulated above. The dashed red line represents the true value of the parameter. 21 2.7 Histogram of estimated values of θ of the OU process as simulated above. The dashed red line represents the true value of the parameter. 22 2.8 Histogram of estimated values of σ of the OU process as simulated above. The dashed red line represents the true value of the parameter. 22 2.9 Histogram of estimated values of β of the OU process using least squares approximation. The dashed red line represents the true value of the parameter.............................. 24 2.10 Histogram of estimated values of θ of the OU process using least squares approximation. The dashed red line represents the true value of the parameter.................................. 25 2.11 Histogram of estimated values of σ of the OU process using least squares approximation. The dashed red line represents the true value of the parameter.............................. 25 2.12 Simulated paths of the CIR process with parameters as