A Counterexample Concerning the Variance-Optimal Martingale Measure

The present note addresses an open question concerning a sufficient characterization of the variance-optimal martingale measure. Denote by S the discounted price process of an asset and suppose that Q* is an equivalent martingale measure whose density is a multiple of 1 minus; \varphi S_T for some S-integrable process \varphi. We show that Q* does not necessarily coincide with the variance-optimal martingale measure, not even if \varphi S is a uniformly integrable Q*-martingale