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).

A simple implementation in Matlab of the algorithm in the case of variance gamma model
is available here (see first comments in file main.m).

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