An empirical comparison of transformed diffusion models for VIX and VIX futures

Transformed diffusions (TDs) are nonlinear functions of continuous-time affine diffusion processes. Since they are flexible models with tractable analytic properties, financial modelling with TDs has become increasing popular in recent years. We first provide a formal classification of TD models into drift-driven, diffusion-driven, and distribution-driven according to their empirical emphases and specification strategies. Motivated by the stylized distributional features of VIX such as skewness and excess kurtosis, we then propose a pair of new distribution-driven TDs for modelling VIX dynamics and pricing VIX futures by directly incorporating such information into the specification of the transformation. We conduct a comprehensive empirical investigation into the relative performance of the three classes of models against several empirically relevant criteria. Our focus is on the in-sample goodness-of-fit measure and the out-of-sample forecast accuracy for modelling VIX and pricing VIX futures, as well as the stock return predictability of the implied Variance Risk Premium. Our findings demonstrate that the newly proposed distribution-driven models have clear advantages over well-established alternatives in most of our exercises.