Title: On two-parametric Esscher transform for geometric CGMY Lévy processes

Authors: Wissem Boughamoura; Faouzi Trabelsi

Addresses: Department of Mathematics, Laboratory of Mathematical and Numerical Modelling in Engineering Science, National Engineering School of Tunis, Tunis El Manar University, B.P. 37, 1002 Tunis-Belvédère, Tunisia; Higher Institute of Applied Mathematics and Computer Science of Kairouan, Kairouan University, Avenue Assad Iben Fourat – 3100 Kairouan, Tunisia ' Department of Mathematics, Laboratory of Mathematical and Numerical Modelling in Engineering Science, National Engineering School of Tunis, Tunis El Manar University, B.P. 37, 1002 Tunis-Belvédère, Tunisia; Higher Institute of Computer Sciences and Mathematics of Monastir, Monastir University, Avenue de la Korniche, B.P. 223, 5000 Monastir, Tunisia

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.

Keywords: CGMY processes; equivalent martingale measures; EMMs; relative entropy; model preserving minimal entropy martingale measure; MPMEMM; Asian option; partial integro-differential equation; PIDE; transformation measures; two-parametric Esscher transform.

DOI: 10.1504/IJOR.2014.059504

International Journal of Operational Research, 2014 Vol.19 No.3, pp.280 - 301

Received: 11 Feb 2012
Accepted: 21 Aug 2012
Published online: 17 Jun 2014 *

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