Convex duality with transaction costs

Convex duality for two two different super–replication problems in a continuous time financial market with proportional transaction cost is proved. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are allowed. The first one the problems considered is the model–independent hedging that requires the super–replication to hold for every continuous path. In the second one the market model is given through a probability measure P and the inequalities are understood P almost surely. The main result, using the convex duality, proves that the two super–replication problems have the same value provided that P satisfies the conditional full support property. Hence, the transaction costs prevents one from using the structure of a specific model to reduce the super–replication cost.