Hedging index options with few assets

We consider hedging strategies against contingent claims depending on a large number of assets (typically options on an index). We introduce strategies involving a limited number of assets and give explicit formulae to characterize optimal strategies. Numerical methods to compute these formulae are also discussed. The purpose of this paper is to apply some ideas that appeared in the recent developments of mathematical nance to the study of options depending on several assets, particularly options on indices. The approach used by many practitioners to evaluate these options is based on the Black-Scholes model, applied to the index. The main drawback of this approach is that, when the index is deened by the arithmetic mean of the prices of some assets , it is not consistent to assume that the index and the prices of the assets behave like geometric Brownian motions. Therefore, we want to demonstrate that it is possible to use models that are consistent for both the index and its components and that the pricing and hedging formulae involved in these models can be computed at a reasonable cost. In the rst section, we set our basic assumptions. The model we start with is the standard complete model of Bensoussan, Karatzas and others (see 2] 3]). In this model, every contingent claim has a \fair price", which is obtained by constructing a perfect hedging strategy. However, when the claim depends on many assets, the hedging strategy may involve a large number of assets and therefore be hard to use in practise. For that reason, we introduce Research supported in part by a contract from Banque Internationale de Placement.