This paper develops robust pricing and hedging dualities for contingent claims in discrete-time financial markets. The authors consider a general framework where uncertainty is modeled by a set of possible asset price trajectories, without a fixed probability measure. They define super- and subhedging functionals using an acceptance set A and a set of trading gains G, allowing for both strict hedging (A = non-negative outcomes) and relaxed hedging where shortfall risk is reduced to an acceptable level. The main result, Theorem 2.1, provides dual representations of these functionals in terms of expectations under probability measures, under a growth condition (2.1) and a regularity condition (2.2). This yields a robust fundamental theorem of asset pricing, linking no-arbitrage conditions to the existence of generalized martingale measures.

The paper applies the general theory to several important settings. For strict superhedging, it considers semi-static strategies with convex transaction costs and trading constraints, deriving explicit dualities involving martingale or supermartingale measures. For relaxed hedging, it uses robust versions of the average value at risk and entropic risk measure to define acceptance sets, leading to robust good deal bounds that are tighter than strict bounds. The approach unifies and extends existing results on robust hedging, transaction costs, and risk-based hedging, providing a flexible framework for pricing under model uncertainty.