MEAN-VARIANCE HEDGING AND NUMERAIRE

We consider the mean-variance hedging problem when the risky assets pr ice process is a continuous semimartingale. The usual approach deals w ith self-financed portfolios with respect to the primitive assets fami ly. By adding a numeraire as an asset to trade in, we show how self-fi nanced portfolios may be expressed with respect to this extended asset s family, without changing the set of attainable contingent claims. We introduce the hedging numeraire and relate it to the variance-optimal martingale measure. Using this numeraire both as a deflator and to ex tend the primitive assets family, we are able to transform the origina l mean-variance hedging problem into an equivalent and simpler one; th is transformed quadratic optimization problem is solved by the Galtcho uk-Kunita-Watanabe projection theorem under a martingale measure for t he hedging numeraire extended assets family. This gives in turn an exp licit description of the optimal hedging strategy for the original mea n-variance hedging problem.