The variance risk premium (VRP) refers to the premium demanded for holding assets
whose variance is exposed to stochastic shocks.
This paper identifies a new modelling framework for equity indices and presents for the
first time explicit analytical formulas for their VRP in a multivariate stochastic volatility
setting, which includes multivariate non-Gaussian Ornstein-Uhlenbeck processes and Wishart
processes. Moreover, we propose to incorporate contagion within the equity index via a
multivariate Hawkes process and find that the resulting dynamics of the VRP represent a
convincing alternative to the models studied in the literature up to date. We show that our
new model can explain the key stylised facts of both equity indices and individual assets and
their corresponding VRP, while some popular (multivariate) stochastic volatility models may
fail.