Volatility derivatives in market models with jumps

Abstract

It is well documented that a model for the underlying asset price process that seeks to capture the behaviour of the market prices of vanilla options needs to exhibit both diffusion and jump features. In this paper we assume that the asset price process $S$ is Markov with cadlag paths and propose a scheme for computing the law of the realized variance of the log returns accrued while the asset was trading in a prespecified corridor. We thus obtain an algorithm for pricing and hedging volatility derivatives and derivatives on the corridor-realized variance in such a market. The class of models under consideration is large, as it encompasses jump-diffusion and Levy processes. We prove the weak convergence of the scheme and describe in detail the implementation of the algorithm in the characteristic cases where $S$ is a CEV process (continuous trajectories), a variance gamma process (jumps with independent increments) or an infinite activity jump-diffusion (discontinuous trajectories with dependent increments).