Hedging in Incomplete Markets: An Approximation Procedure for Practical Application

Derivative financial instruments are frequently used as a tool for influencing the risk of entrepreneurial uncertain payoff. To this end, an approximation procedure is developed capable of calculating the optimal quantity of derivatives to be used. It is assumed that the entrepreneurial cash flow is governed by several stochastic factors and that derivatives are only available as a hedging tool for one of these factors. In general, it is easy to determine optimal hedging payment structures with respect to this factor, but real‐life hedging opportunities will typically not allow to perfectly reproduce such a fictitious payment structure, thus leading to complex numerical optimization problems. Instead of directly approximating the entrepreneurial expected utility maximum, we suggest using the fictitious optimal hedging payment structure as a starting point and to minimize the quadratic deviation between payment structures realizable by financial derivatives actually available and the resulting entrepreneurial payoff achieving the fictitious optimal hedging payment structure. This approach proves to be rather easy. Indeed, under certain conditions an explicit solution can be reached. After analyzing the qualitative properties of our approximation solution, we examine its efficiency for two practical hedging problems. In the first example, we get nearly the same solutions with our approximation procedure as with a grid programming approach presented by some other authors. Among other things, our second example may explain why some special kinds of financial derivatives, known as shared currency option under tenders, are not used in international invitations for tenders even though they offer hedging opportunities that are otherwise not available. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21: 599–631, 2001