Mean-variance hedging and numéraire

We consider the mean-variance hedging problem when the risky assets price process is a continuous semimartingale. The usual approach deals with self-financed portfolios with respect to the primitive assets family. By adding a numéraire as an asset to trade in, we show how self-financed portfolios may be expressed with respect to this extended assets family, without changing the set of attainable contingent claims. We introduce the hedging numéraire and relate it to the variance-optimal martingale measure. Using this numéraire both as a deflator and to extend the primitive assets family, we are able to transform the original mean-variance hedging problem into an equivalent and simpler one; this transformed quadratic optimization problem is solved by the Galtchouk-Kunita-Watanabe projection theorem under a martingale measure for the hedging numéraire extended assets family. This gives in turn an explicit description of the optimal hedging strategy for the original mean-variance hedging problem.