Hedging in incomplete markets: An approximation procedure for practical application

Derivative financial instruments are frequently used as a tool for influencing the risk ofentrepreneurial uncertain payoff. To this end, an approximation procedure is developed capable ofcalculating the optimal quantity of derivatives to be used. It is assumed that the entrepreneurial cashflow is governed by several stochastic factors and that derivatives are only available as a hedging toolfor one of these factors. In general it is easy to determine optimal hedging payment structures withrespect to this factor, but real-life hedging opportunities will typically not allow to perfectly reproducesuch a fictitious payment structure, thus leading to complex numerical optimisation problems. Insteadof directly approximating the entrepreneurial expected utility maximum we suggest to use the ficti-tious optimal hedging payment structure as a starting point and to minimise the quadratic deviationbetween payment structures realisable by financial derivatives actually available and the resultingentrepreneurial payoff achieving the fictitious optimal hedging payment structure. This approachproves to be rather easy. Indeed, under certain conditions an explicit solution can be reached. Afteranalysing the qualitative properties of our approximation solution we examine its efficiency for twopractical hedging problems. In the first example we get nearly the same solutions by the use of ourapproximation procedure as by a grid programming approach presented by some other authors.Among other things our second example may give an explanation why some special kind of financialderivatives, known as SCOUTs, are not used in international invitations for tenders although they offerhedging opportunities which are otherwise not available.