Pricing variance swaps under subordinated Jacobi stochastic volatility models

Abstract

In this paper, we first study the pricing of variance swaps under the Jacobi stochastic volatility model, where the squared volatility of the log-price follows a Jacobi process with values in some compact interval. The Jacobi model is versatile and highly tractable and belongs to the class of polynomial diffusions. The attractive property for this class is that the moments of the finite-dimensional distributions of the log-prices can be calculated explicitly. We employ this property and are able to obtain the closed-form solutions to variance swap prices. To introduce jumps in the asset price processes, we further extend the Jacobi model to subordinated Jacobi models where the subordination can come from either Lévy or additive subordinators. Based on an important observation that the spatial and temporal variables are separated in the moment formula of the polynomial processes, we express the moment formula under the subordinated models explicitly, which enables us to calculate the prices of variance swaps in an efficient way. The empirical analysis demonstrates that the Jacobi model can potentially outperform the Heston model and the inclusion of subordination can greatly improve the goodness of fit of the Jacobi model.