Pricing Volatility Derivatives Under Lévy Processes
In this thesis, we study the pricing of the volatility derivatives, including VIX options, VIX futures, VXX options and S&P 500 variance futures, under Lévy processes with stochastic volatility. In particular, we investigate the role of different types of jump structures, such as finite-activity jump, infinite-activity jump and double jump structures, as well as the role of variance processes with time-varying mean in the valuation of volatility derivatives. In our models, we assume that the long-term mean of the variance process follows an Ornstein–Uhlenbeck process and specify the infinity-activity jump component of the main process in four cases: the variance gamma process (VG), the normal inverse Gaussian process (NIG), the tempered stable process (TS) and the generalized tempered stable process (GTS). Then, we apply the combined estimation approach of an unscented Kalman filter (UKF) and maximum log-likelihood estimation (MLE) to our models and make an extensive comparison analysis on the performance among the different models.
Our empirical studies reveal three important results. First, the models with infiniteactivity jumps are superior to the models with finite-activity jumps, particularly in pricing VIX options and VXX options. Thus, the infinite-activity jumps cannot be ignored in pricing volatility derivatives. Second, both the infinite-activity jump and diffusion components play important roles in modelling the dynamics of the underlying asset returns for the volatility derivatives. Third, the mean of the variance process for the S&P 500 index returns varies stochastically toward to its long-term mean.