  We provide a model-free pricing-hedging duality in continuous time. For a
frictionless market consisting of $d$ risky assets with continuous price
trajectories, we show that the purely analytic problem of finding the minimal
superhedging price of a path dependent European option has the same value as
the purely probabilistic problem of finding the supremum of the expectations of
the option over all martingale measures. The superhedging problem is formulated
with simple trading strategies, the claim is the limit inferior of continuous
functions, which allows for upper and lower semi-continuous claims, and
superhedging is required in the pathwise sense on a $\sigma$-compact sample
space of price trajectories. If the sample space is stable under stopping, the
probabilistic problem reduces to finding the supremum over all martingale
measures with compact support. As an application of the general results we
deduce dualities for Vovk's outer measure and semi-static superhedging with
finitely many securities.
