Jump-Diffusion Models with Correlated Brownian Motion and Compound Poisson Processes

Abstract

In this paper we provide a closed form option pricing model with underlying uncertainty modeled as an exponential Lévy process. The stochastic structure of our model relaxes the restrictive assumption of zero covariance between the Brownian motion and Poisson process jump size found in all traditional jump-diffusion models. The resulting equity diffusion and jump risk premia of the equilibrium consumption capital asset pricing relationship are non-linear functions of time. The option pricing model is developed using Rubinstein’s method of pricing by substitution in equilibrium and it is shown that the sign and magnitude of this covariance plays a crucial role in determining the slope of the implied volatility term structure.