Jump Size Distributions of Additive Compound Poisson Processes That Are Closed Under the Esscher Transform

Abstract

We model logarithmic asset price dynamics under the physical probability measure as additive jump-diffusion processes, which exhibit a time-dependent jump intensity and jump size distribution. The corresponding risk-neutral probability measure is defined through an Esscher transform. We are interested in the conditions under which the jump size distributions under the two probability measures fall into the same parametric class. We show that it is necessary and sufficient for the jump size distribution to follow a natural exponential mixture family at all points in time. Immediate applications of these results in financial engineering are discussed.

Keywords: Esscher Transform, Additive Processes, Compound Poisson, Jump Size Distribution, Natural Exponential Family

JEL Classification: G13