Correct Calculation of Volatility in a Jump-Diffusion Model

Abstract

The jump-diffusion model as an extension of the Black-Scholes pure logarithmic diffusion process was first introduced by Merton and others in the 1970s. The underlying asset follows a regular diffusion, but occasionally experiences a large discrete jump of random size, whose arrival is governed by a Poisson process. To estimate the volatility parameters for a jump-diffusion process, it is important to take into account the impact of both random jump arrival and also the uncertainty over the size of a jump if it should occur. In this article, Navas points out that the influence of jump size uncertainty on stock volatility was left out by a number of the early, and some not-so-early, investigators. The effect on theoretical option values is not huge, but also not negligible, as the results presented here show.