Application of Convex Duality to the risk hedging of financial claims

Abstract

We work in an incomplete market with finite states. We obtain all the no arbitrage prices of a financial claim associating them to entropy levels. This is done by means of convex programs with an entropy constraint. We apply Fenchel duality to translate these programs to their duals and we obtain two particular cases. One arises when the dual entropy variable is null and represents the superreplicating case giving as solution the super-replicating portfolio at no risk. The other arises when the dual entropy variable is different from zero and stands for the partial replicating case corresponding to the prices in the interior of the non arbitrage prices interval.