Preprint series: 05-12, Reports on Optimization and Stochastics

MSC:

60H30 Applications of stochastic analysis (to PDE, etc.)

62F03 Hypothesis testing

90A09 Finance, portfolios, investment

Abstract: The problem of hedging in incomplete financial markets is considered, when the risk of a shortfall is measured with a convex risk measure, recently introduced by Föllmer and Schied (2002). The dynamic optimization problem of finding a self-financing strategy that minimizes the convex risk of the shortfall can be split into a static optimization problem and a representation problem.
We show the existence of a solution to the static optimization problem and deduce necessary and sufficient optimality conditions using convex duality methods. The solution of the static optimization problem turns out to be a randomized test with a typical 0-1-structure.
The optimal strategy that solves the dynamic problem consists in superhedging a modified claim. The payoff of this modified claim is the product of the optimal randomized test and the original payoff.

Keywords: hedging, contingent claims, incomplete financial market, shortfall risk, convex risk measures, generalized Neyman-Pearson lemma, convex duality

Upload: 2005-11-23

Update: 2005