What is the cheapest way to superhedge a path-dependent derivative security? If liquid European calls and the underlying risky stock can be used for hedging, then the lowest superhedging price corresponds to the highest expected cost of the exotic claim. Each expected cost is associated to a probabilistic model which makes the risky stock a martingale and such that the model-based prices of European calls perfectly match the prices observed in the market. We formulate our problem as a constrained optimal transport problem and show how to explicitly construct extremal models together with optimal hedging strategies.