In this thesis, we study the mean reverting property of the VIX time series, and use the VIX process as the underlying. We employ various mean reverting processes, including the Ornstein-Uhlenbeck (OU) process, the Cox-Ingersoll-Ross (CIR) process and the OU processes driven by Levy processes (Levy OU) to fit historical data of VIX, and calibrate the VIX option prices. The first contribution of this thesis is to use the Levy OU process to model the VIX process, in order to explain the observed high kurtosis. To price the option using the Levy OU process, we develop a FFT method.

The second contribution is to build a joint framework to consistently model the VIX and VIX derivatives together on the entire time series of market data. We choose multi-factor mean-reverting models, in which we model the VIX process as a linear combination of latent factors. To estimate the models, we use Euler approximation to find a discrete approximation for the VIX process. Based on this approximate, we consider various filter methods, namely, the Unscented Kalman Filter (UKF), constrained UKF, mixed Gaussian UKF and Particle Filter (PF) for estimation. The performances of these models are compared and discussed. Radon Nikodym derivatives of the risk-neutral measure are discussed with respect to the physical measure for the jumps. A simple dynamic trading strategy was tested on these models.