On the Minimal Entropy Martingale Measure for L\'evy Processes

Abstract

In the present paper, a new and simple approach is provided for proving rigorously that for general L\'evy financial markets the minimal entropy martingale measure and the Esscher martingale measure coincide. The method consists in approximating the probability measure P by a sequence of L\'evy preserving probability measures P_n with exponential moments of all order. As a by-product, it turns out that the problem of finding the minimal entropy martingale measure for the L\'evy market is equivalent to the corresponding problem but for a certain one-step financial market. The existence of the Esscher martingale measure (and hence the minimal entropy martingale measure) will be characterized by using moment generating functions of the L\'evy process. Keywords: L\'evy financial markets; minimal entropy martingale measure; Esscher martingale measure; no-arbitrage conditions; moment generating functions. MSC (2010) Classification: Primary 60G51, 60G44, 91G99; Secondary 91B25, 91B16