Markov Chain Monte Carlo Estimation of Stochastic Volatility Models with Finite and Infinite Activity Lévy Jumps: Evidence for Efficient Models and Algorithms

A financial model plays a key role in the valuation and risk management of financial derivatives, and it serves as an important tool for investors to measure the risk exposure of their portfolios and make predictions and decisions. However, the popular a ne stochastic volatility models without jumps, such as the Heston model, have been questioned in the finance literature in terms of their appropriateness for modelling stock prices and pricing derivatives. Many alternative model specifications have been proposed in recent decades, including the specification of non-a ne variance dynamics and the inclusion of Lévy jumps. However, the complexity introduced by further model specifications leads to poor probabilistic properties, and hence most popular estimation methods are not applicable. The Bayesian estimation method is among the few that work. In this thesis, I discuss the role of new model specifications and investigate the performance of Bayesian estimation methods. First, I use an extensive empirical data set to study how the use of infinite-activity Lévy jumps in stock returns and variance improves model performance. The stock returns and variance are driven by di usions and di erent Lévy jumps, including the finite-activity compound Poisson jump and infinite-activity Variance Gamma and Normal Inverse Gaussian (NIG) jumps. Moreover, the non-a ne linear variance process is compared to the a ne square-root stochastic process. With the conventional Markov Chain Monte Carlo (MCMC) algorithms, including the Gibbs sampler and Metropolis-Hastings (MH) methods, and the Damien-Wakefield-Walker method to cope with complicated posteriors, eighteen di erent model specifications are estimated using the joint information of the S&P 500 index and the VIX index for 1996–2009. There is clear evidence that in terms of the goodness of fit and option pricing performance, a relatively parsimonious model with infinite-activity NIG jumps in returns and non-a ne variance dynamics is particularly competitive. In the second part of the thesis, I examine the performance of advanced MCMC algorithms. The e ciency of the MH algorithm has been questioned because of its slow mixing speed, especially in the presence of high dimensions and a strong dependence between model parameters and state variables. Generally, a class of algorithms seeks to improve the MH by constructing more e ective proposals, and another combines the MCMC with the Sequential Monte Carlo algorithms. To investigate, I first conduct simulation studies to compare the estimation performance of seven advanced Bayesian estimation methods against the MH. Specifically, I use the a ne Heston model, the a ne Bates model, and an a ne model with NIG return jumps, and examine whether the di erent jump structures a ect the estimation results. Second, I estimate the non-a ne model with NIG return jumps using the joint information of the S&P 500 index and the VIX for 2002–2005 with selected algorithms that perform well in the simulation studies. The results of the simulation and empirical studies are mixed about the performance of the algorithms. The i ii Abstract Fast Universal Self-tuned Sampler algorithms are particularly competitive in generating virtually independent samples and achieving the fastest mixing with a fixed number of MCMC runs, and their performance is stable regardless of the model specifications. However, they are computationally expensive. The computational costs of the Particle Markov Chain Monte Carlo (PMCMC) methods are much cheaper and also e cient in mixing, and they perform best when estimating the models without jumps/with NIG jumps in the simulation studies, as well as in the fit to the VIX in the empirical studies. However, the PMCMC methods are more vulnerable to model specifications than the other algorithms; in particular, the rare large compound Poisson jumps in the Bates model significantly reduce the acceptance rate and worsen the estimation performance of the PMCMC methods. Preface The research presented in this thesis was carried out at the Department of Industrial Management at Tampere University of Technology between 2013 and 2015, within the Marie Curie Initial Training Network HPCFinance. During the writing of this thesis, I was financially supported by the European Union Seventh Framework Programme under grant agreement No. 289032. I would like to express my sincere gratitude to my supervisor, Professor Juho Kanniainen, for giving me such an invaluable opportunity to work on the HPCFinance project, and more importantly, for his insightful guidance and consistent support that helped me manage many di cult problems. In my research, I have always been encouraged by his enthusiasm and honest criticism and advice. I am also especially grateful for the inspiring cooperation with, and support from, Professor Luca Martino, who proposed and developed the powerful FUSS algorithms and guided me in applying the algorithms to financial problems. In addition, I would like to express my gratitude to the pre-examiners, Professor Michael Johannes and Professor Andreas Kaeck, for their helpful comments and suggestions. Furthermore, I wish to thank my colleagues at the Department of Industrial Management and in the HPCFinance project. In particular, I thank Milla Siikanen, Sindhuja Ranganathan, Jaakko Valli, and Binghuan Lin in the finance group for the helpful discussions and activities outside of work. Special thanks go to my o cemates, Jun Hu and Ye Yue, for providing a fun and stimulating work environment. I am also thankful to the project managers, Dr. Santiago Velasquez and Dr. Hanna-Riikka Sundberg, and the department secretary, Ms. Heli Kiviranta, for their assistance and administrative support. I want to thank Jukka Annala and his wife Tuija for inviting me to their beautiful home and summer cottage. I will never forget the journey into the deep forest and the experiences of taking a sauna and jumping into the cold lake, picking mushrooms, and tasting karjalanpiirakka. A special mention goes to the lovely couple Huiling Wang and Tinghuai Wang. I am glad to have friends like you, who have made my life here unforgettable. Last but not least, I owe my deepest thanks to my parents and my dearest grandmother for their endless encouragement and love. Thank you for sharing your lives with me. You remain my best friends, no matter how far away you are. iii