Coherent Hedging in Incomplete Markets

TITLE: Coherent Hedging in Incomplete Markets.
AUTHOR: Birgit Rudloff
ABSTRACT: In incomplete financial markets not every contingent claim can be perfectly replicated by a self-financing strategy. In this paper, we want to minimize the risk that the value of the hedging portfolio falls below the payoff of the claim at time T. We use a coherent risk measure, introduced by Artzner, Delbaen, Eber, Heath (1999), to measure the risk of the shortfall. 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. We will deduce necessary and sufficient optimality conditions for the static problem using convex duality methods. The solution of the static optimization problem turns out to be a randomized test with a typical 0-1-structure. Our results improve the ones obtained by Nakano (2003). The optimal hedging strategy consists in superhedging a modified claim that is the product of the original payoff and the solution to the static problem.
STATUS: Quantitative Finance 9 (2), 197 - 206, 2009. [pdf]