A superconvergent partial differential equation approach to price variance swaps under regime switching models

Abstract

We present a superconvergent finite difference algorithm to price discretely sampled variance swaps. We consider the Black–Scholes model, the Merton’s jump–diffusion model, stochastic volatility models that use constant-elasticity of variance for the instantaneous variance and corresponding regime switching models. PDE approach provides a universal and efficient framework for pricing under these models. To obtain extremely accurate results, we solve PDEs whose associated terminal conditions can be represented as second-order polynomials based on the two popular definitions of realised variance and for which the spatial derivatives greater than second-order are all zero. We then apply second-order finite difference discretisations in space with an exponential time integration. We also derive analytical solutions under the Merton’s model and some regime switching models to validate our superconvergent results.