Mean-Variance Hedging for General Claims

0. Introduction. In this paper, we solve the continuous-time hedging problem with a mean-variance objective for general contingent claims. A special case of this problem was treated by Duffie and Richardson (1991) and provided the motivation for this work. There are two assets whose prices are both modelled by exponential Brownian motions with time-dependent random coefficients. The rates of return between assets are correlated. At a fixed time, the hedger faces a random loss which may depend in full generality on the entire evolution of both asset prices. For the purpose of hedging against this risk, however, only one asset is available. This implies that markets are incomplete and contingent claims cannot be replicated by trading. The goal of the hedger is to minimize his total expected quadratic costs, or equivalently to maximize his expected utility from terminal wealth for a quadratic utility function. A precise statement is given in Section 1 and the solution is presented in Section 3. We remark that the same arguments would also work for any number N of driving assets with n hedging assets, where 1 < n < N. Our approach to this problem follows the method of Duffie and Richardson (1991): We show that the inner product associated with the normal equations for orthogonal projection is defined by an ordinary differential equation in time with an explicit solution. This is done by choosing a suitable tracking process for the contingent claim under consideration. The essential difference from the above paper lies in two points: We are able to solve the hedging problem for a general contingent claim and we do not have to conjecture the solution from discrete-time reasoning. In fact, our approach shows that the natural choice for the optimal tracking process is provided by the intrinsic value process associated to the given contingent claim. This process is defined in terms of the minimal equivalent martingale measure for that asset price which is used for hedging. Both of these concepts are explained in more detail in Section 2. We conclude the paper in Section 4 with a class of examples where explicit