Arbitrage Pricing in Convex, Cash-Additive Markets

We consider superhedging and no-arbitrage pricing in markets with a convex and cash-additive structure and derive an explicit functional form for the super-replication price. Using convex duality methods, we show that the superhedging price maximizes the difference between the expected payoff and a confidence function that accounts for the reliability of the probability used in pricing. We demonstrate that the existence of a strictly positive probability within the domain of the confidence function, which maximizes the super-replication price for a specific payoff and acts as a lower bound for all other payoffs, is necessary and sufficient to prevent arbitrage opportunities. Furthermore, we explore entropy pricing as a notable example of a super-replication pricing functional and provide conditions on the market structure under which the super-replication price takes the form of entropy pricing. We show that the confidence function in entropy pricing can be expressed using the Kullback-Leibler divergence.