The hunt variance gamma process with applications to option pricing

TLDR: In this article, the Hunt variance gamma process (VGPM) is used to model the risk-neutral distribution of future stock prices, and a continuous-time Markov chain approximation is found to fit the S&P 500 futures option surface.

Abstract: In this dissertation we develop a spatially inhomogeneous Markov process as a model for financial asset prices. This model is called the Hunt variance gamma process. We define it via its infinitesimal generator and prove that this generator induces a unique measure on the space ofcádì ag functions. We next describe a procedure to do computations with this model by finding a continuous-time Markov chain approximation. This approximation is used to calibrate the model to fit the S&P 500 futures option surface. Next we investigate specific characteristics of the process, showing how it differs from both Lévy and Sato processes. We conclude by using the calibrated model to answer questions about properties of the risk-neutral distribution of future stock prices. We observe a more accurate fit to the risk-neutral term structure of volatility, skewness, and kurtosis, and the presence of mean-reversion in conditional probabilities involving large jumps. Dedication There is only one person to whom I can dedicate this thesis. And though she will probably never read more than a page of it, I couldn't have done it without her. Mary, thanks for everything. I love you. ii Acknowledgments There are so many people who have contributed either directly or indirectly to this dissertation. It is impossible for me to acknowledge everyone who has helped me in some manner during my time at the University of Maryland. Nevertheless, I would be negligent if I didn't at least make an effort to mention those who have contributed so much. To start, I have to mention my adviser, Dr. Dilip Madan. His enthusiasm and expertise piqued my interest in mathematical finance initially. As I learned more, I realized just how knowledgeable he was. But what impressed me most was his willingness to share that knowledge with his students in seminars and private conversations. Without his ideas, suggestions, and recommendations, I never would have reached this point. I would also like to acknowledge the members of my committee for their work in reviewing and improving my work. and Dr. Alt, thank you for taking time to help me. During the last few years, Dr. Balan has been especially kind in assisting me with a number of administrative details, for which I thank him. Along these lines, I am grateful to the entire Norbert Wiener Center for allowing me to attend conferences, classes, and seminars throughout my time in Maryland. It is fitting that …