We consider the Robust Pricing Problem for a class of options encompassing American, Asian, Bermudan and European Options in a martingale optimal transport setting for càdlàg processes. We prove strong duality of the pricing and hedging problem and the existence of an optimal pathwise hedge. Our approach provides insight into the structure of primal and dual optimizers and provides a remarkable parity of the price bounds in this setting. For finitely supported marginal laws we are able to reduce the problem to a semi-infinite linear program and in the case of piecewise linear payoffs (risk reversal, butterfly spread, etc.) a finite linear program, making it very amenable to numerical optimization methods.
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