  We consider the martingale optimal transport duality for c\`adl\`ag processes
with given initial and terminal laws. Strong duality and existence of dual
optimizers (robust semi-static superhedging strategies) are proved for a class
of payoffs that includes American, Asian, Bermudan, and European options with
intermediate maturity. We exhibit an optimal superhedging strategy for which
the static part solves an auxiliary problem and the dynamic part is given
explicitly in terms of the static part.
