Stochastic dynamics of variance risk premia through stochastic volatility of volatility

This paper introduces a new class of stochastic volatility models which allows for stochastic volatility of volatility. Such models are given by volatility modulated non–Gaussian Ornstein Uhlenbeck processes. We study the probabilistic properties of such models both under the physical and under the risk neutral probability measure, where we focus in particular on the role of the volatility of volatility component in the integrated variance. Further, we show that the volatility of volatility component can be used to account for the leverage effect in a novel way and we discuss how long memory can be introduced into our new model class. The main contribution of the paper is that we derive an explicit formula for the variance risk premium in our new stochastic volatility model. That formula reveals that any stochastic dynamics of the variance risk premium have to come from a stochastic volatility of volatility component — provided the physical and the risk neutral probability measures are related through a structure preserving change of measure. This result indicates that (stochastic) volatility of volatility is a key component in a stochastic volatility model since it essentially drives the stochastic dynamics of the variance risk premium. These results are in accordance with recent empirical findings on the dynamics of variance risk premia.