Hedging Diffusion Processes by Local Risk-Minimisation with Applications to Index Tracking

The solution to the problem of hedging contingent claims by local risk-minimisation has been considered in detail in Follmer and Sondermann (1986), Follmer and Schweizer (1991) and Schweizer (1991). However, given a stochastic process Xt and tau1 <> tau2, the strategy that is locally risk-minimising for Xtau1 is in general not locally risk-minimising for Xtau2. In the case of diffusion processes, this paper considers the problem of determining a strategy that is simultaneously locally risk-minimising for Xtau for all tau. That is, a strategy that is locally risk-minimising for the entire process Xt. The necessary and sufficient conditions under which this is possible are obtained, and applied to the problem of index tracking. In particular, a close connection between the local risk-minimising and the tracking error variance minimising strategies for index tracking is established, and leads to a simple criterion for the selection of optimal set of assets from which to form a tracker portfolio, as well as a value-at-risk type measure for the set of assets used.