Stochastic Volatility of Volatility and Variance Risk Premia

This paper introduces a new class of stochastic volatility models which allows for stochastic volatility of volatility (SVV): Volatility modulated non-Gaussian Ornstein-Uhlenbeck (VMOU) processes. Various probabilistic properties of (integrated) VMOU processes are presented. Further we study the effect of the SVV on the leverage effect and on the presence of long memory.One of the key results in the paper is that we can quantify the impact of the SVV on the (stochastic) dynamics of the variance risk premium (VRP). Moreover, provided the physical and the risk -- neutral probability measures are related through a structure -- preserving change of measure, we obtain an explicit formula for the VRP