Dynamic programming and mean-variance hedging

Abstract

We consider the mean-variance hedging problem when asset prices follow Ito processes in an incomplete market framework. The hedging numeraire and the variance-optimal martingale measure appear to be a key tool for characterizing the optimal hedging strategy (see Gourieroux et al. 1996; Rheinlander and Schweizer 1996). In this paper, we study the hedging numeraire \(\tilde a\) and the variance-optimal martingale measure \(\tilde P\) using dynamic programming methods. We obtain new explicit characterizations of \(\tilde a\) and \(\tilde P\) in terms of the value function of a suitable stochastic control problem. We provide several examples illustrating our results. In particular, for stochastic volatility models, we derive an explicit form of this value function and then of the hedging numeraire and the variance-optimal martingale measure. This provides then explicit computations of optimal hedging strategies for the mean-variance hedging problem in usual stochastic volatility models.