Decomposition of the option pricing formula for infinite activity jump-diffusion stochastic volatility models

Abstract

Let the log returns of an asset $X_t=log\left(S_t\right)$ be defined on a risk neutral filtered probability space $(\Omega ,F,{\left(F_t\right)}_{t\in [0,T]},P)$ for some $0<T<\infty$. Assume that $X_t$ is a stochastic volatility jump-diffusion model with infinite activity jumps. In this paper, we obtain an Alós-type decomposition of the plain vanilla option price under a jump-diffusion model with stochastic volatility and infinite activity jumps via two approaches. Firstly, we obtain a closed-form approximate option price formula. The obtained formula is compared with some previous results available in the literature. In the infinite activity but finite variation case jumps of absolute size smaller than a given threshold $\varepsilon$ are approximated by their mean while larger jumps are modeled by a suitable compound Poisson process. A general decomposition is derived as well as a corresponding approximate version. Lastly, numerical approximations of option prices for some examples of Tempered Stable jump processes are obtained. In particular, for the Variance Gamma one, where the approximate price performs well at the money.