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We consider the problem of superhedging under volatility uncertainty for aninvestor allowed to dynamically trade the underlying asset, and staticallytrade European call options for all possible strikes with some given maturity.This problem is classically approached by means of the Skorohod EmbeddingProblem (SEP). Instead, we provide a dual formulation which converts thesuperhedging problem into a continuous martingale optimal transportationproblem. We then show that this formulation allows us to recover previouslyknown results about lookback options. In particular, our methodology induces anew proof of the optimality of Az\'{e}ma-Yor solution of the SEP for a certainclass of lookback options. Unlike the SEP technique, our approach applies to alarge class of exotics and is suitable for numerical approximation techniques.

A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options