Derivatives pricing viap-optimal martingale measures: some extreme cases

Abstract

In an incomplete financial market in which the dynamics of the asset prices is driven by ad-dimensional continuous semimartingaleX, we consider the problem of pricing European contingent claims embedded in a power utility framework. This problem reduces to identifying thep-optimal martingale measure, which can be given in terms of the solution to a semimartingale backward equation. We use this characterization to examine two extreme cases. In particular, we find a necessary and sufficient condition, written in terms of the mean-variance trade-off, for thep-optimal martingale measure to coincide with the minimal martingale measure. Moreover, if and only if an exponential function of the mean-variance trade-off is a martingale strongly orthogonal to the asset price process, thep-optimal martingale measure can be simply expressed in terms of a Doléans-Dade exponential involvingX.