Abstracts

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. The importance of the VRP is also evident from the proliferation of derivatives products such as the variance swaps, whose market volume has increased steeply over the past few years.
The aim of this paper is to identify a suitable parsimonious model for the stochastic dynamics of equity indices, capable of producing dynamics of the implied VRP in line with the empirical findings that the VRP of equity indices displays stochastic dynamics and jumps, whereas the VRP of individual stocks does not exhibit such stochastic fluctuations.
Existing theoretical and empirical work has so far only advocated univariate models to explain the dynamics of the VRP. However, we argue that dependencies across assets play a key role in explaining the stochastic dynamics of the VRP of stock indices and hence a multivariate stochastic model is needed. This paper presents for the first time explicit analytical formulas for the VRP in a multivariate stochastic volatility framework, which includes multivariate non-Gaussian Ornstein-Uhlenbeck processes and Wishart processes, as well as a new model specification. Moreover, we propose to incorporate self- and mutually exciting multivariate Hawkes processes in the model 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 we can identify the Hawkes intensity process as a possible driver for the stochastic dynamics of the VRP. As a by-product of our work, we also prove useful explicit formulas involving conditional expectations of this popular process.
In addition, we find 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, including the Wishart model, fail.
We finally prove the existence of a structure-preserving risk neutral measure for our model, laying the theoretical foundations for the derivations described above. In particular, we establish the class of equivalent probabilities that preserve the self-affecting structure of the Hawkes process.