Local risk-minimization under transaction costs

We propose a new approach to the pricing and hedging of contingent claims u nder transaction costs in a general incomplete market in discrete time. Und er the assumptions of a bounded mean-variance tradeoff, substantial risk an d a nondegeneracy condition on the conditional variances of asset returns, we prove the existence of a locally risk-minimizing strategy inclusive of t ransaction costs for every square-integrable contingent claim. Then we show that local risk-minimization is robust under the inclusion of transaction costs: The preceding strategy which is locally risk-minimizing inclusive of transaction costs in a model with bid-ask spreads on the underlying asset is also locally risk-minimizing without transaction costs in a fictitious m odel which is frictionless and where the fictitious asset price lies betwee n the bid and ask price processes of the original model. In particular, our results apply to any nondegenerate model with a finite state space if the transaction cost parameter is sufficiently small.