The minimal entropy martingale measure and the valuation problem in incomplete markets

Abstract

Let x be a family of stochastic processes on a given filtered probability space (Ω,ℱ,(ℱl)l∈script T, P) with script T⊆ℝ+. Under the assumption that the set script Me of equivalent martingale measures for x is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to P, in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy.