Variance-Optimal Hedging in Discrete Time

We solve the problem of approximating in ℒ² a given random variable H by stochastic integrals G_T(ϑ) of a given discrete-time process X. We interpret H as a contingent claim to be paid out at time T, X as the price evolution of some risky asset in a financial market, and G(ϑ) as the cumulative gains from trade using the hedging strategy ϑ. As an application, we determine the variance-optimal strategy which minimizes the variance of the net loss H − G_T(ϑ) over all strategies ϑ.