Coherent Hedging in Incomplete Markets

In incomplete financial markets not every given contingent claim can be replicated by a self-financing strategy. The risk of the resulting shortfall can be measured by coherent risk measures, introduced by Artzner et al. [1]. The dynamic optimization problem of finding a self-financing strategy that minimizes the coherent risk of the shortfall can be split into a static optimization problem and a representation problem. In this paper, we will deduce necessary and sufficient optimality conditions for the static problem using convex duality methods. The solution of the static optimization problem tourns out to be a randomized test with a typical 0-1-structure. Our results improve the ones obtained by Nakano [7].