DERIVATIVES PRICING VIA p-OPTIMAL MARTINGALE MEASURES: SOME EXTREME CASES

In an incomplete financial market in which the dynamics of the asset prices is driven by a d-dimensional continuous semimartingale X, we consider the problem of pricing European contingent claims embedded in a power utility framework. This problem reduces to identifying the p-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 the p-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, the p-optimal martingale measure can be simply expressed in terms of a Doléans-Dade exponential involving X.