On two-parametric Esscher transform for geometric CGMY Lévy processes

Abstract

We present a new transformation measure depending on two parameters, called two-parametric Esscher transform, for CGMY processes. This transform defines a class of equivalent martingale measures for CGMY processes, preserving the CGMY character. We follow a decomposition approach by writing any CGMY process as sum of two independent CGMY ones. Next, following our approach, we focus on the relative entropy with respect to the initial probability and we provide optimal parameters, by minimising the relative entropy, which defines an equivalent martingale measure called ‘model preserving minimal entropy martingale measure’. This measure has the advantage of preserving the CGMY character of any CGMY process. As application, we finally show that Asian option’s price function, in a CGMY market model, is a solution of a time-dependent partial-integro differential equation.