  In a model free discrete time financial market, we prove the superhedging
duality theorem, where trading is allowed with dynamic and semi-static
strategies. We also show that the initial cost of the cheapest portfolio that
dominates a contingent claim on every possible path $\omega \in \Omega$, might
be strictly greater than the upper bound of the no-arbitrage prices. We
therefore characterize the subset of trajectories on which this duality gap
disappears and prove that it is an analytic set.
