Minimal entropy-hellinger martingale measure in incomplete markets

Abstract

This paper defines an optimization criterion for the set of all martingale measures for an incomplete market model when the discounted price process is bounded and quasi-left continuous. This criterion is based on the entropy-Hellinger process for a nonnegative Doléans-Dade exponential local martingale. We develop properties of this process and establish its relationship to the relative entropy "distance." We prove that the martingale measure, minimizing this entropy-Hellinger process, is unique. Furthermore, it exists and is explicitly determined under some mild conditions of integrability and no arbitrage. Different characterizations for this extremal risk-neutral measure as well as immediate application to the exponential hedging are given. If the discounted price process is continuous, the minimal entropy-Hellinger martingale measure simply is the minimal martingale measure of Föllmer and Schweizer. Finally, the relationship between the minimal entropy-Hellinger martingale measure (MHM) and the minimal entropy martingale measure (MEM) is provided. We also give an example showing that in contrast to the MHM measure, the MEM measure is not robust with respect to stopping. © 2005 Blackwell Publishing Inc.