L 2 -discrete hedging in a continuous-time model

In the setting of the Black-Scholes option pricing market model, the seller of a European option must trade continuously in time. This is, of course, unrealistic from the practical viewpoint. He must then follow a discrete trading strategy. However, it does not seem natural to hedge at deterministic times regardless of moves of the spot price. In this paper, it is supposed that the hedger trades at a fixed number N of rebalancing (stopping) times. The problem (P^N) of selecting the optimal hedging times and ratios which allow one to minimize the variance of replication error is considered. For given N rebalancing, the discrete optimal hedging strategy is identified for this criterion. The problem (P^N) is then transformed into a multidimensional optimal stopping problem with boundary constraints. The restrictive problem (P^N _BS) of selecting the optimal rebalancing for the same criterion is also considered when the ratios are given by Black-Scholes. Using the vector-valued optimal stopping theory, the existence is shown of an optimal sequence of rebalancing for each one of the problems (P^N) and (P^N _BS). It also shown BS that they are asymptotically equivalent when the number of rebalances becomes large and an optimality criterion is stated for the problem (P^N). The same study is made when more realistic restrictions are imposed on the hedging times. In the special case of two rebalances, the problem (P² _BS) is solved and the problems (P² _BS) and (P²) are transformed into two optimal stopping problems. This transformation is useful for numerical purposes.