Modelling the Variance Risk Premium of Equity Indices: The Role of Dependence and Contagion

Abstract

Understanding variance risk is of key importance in mathematical finance since it affects risk management, asset allocation and derivative pricing. Variance risk is priced in financial markets by the so-called variance risk premium (VRP), which 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 popular (multivariate) stochastic volatility models fail.We finally prove the existence of a structure-preserving risk neutral measure for our model. In particular, we establish the class of equivalent probabilities that preserve the self-affecting structure of the Hawkes process.