  The duality between the robust (or equivalently, model independent) hedging
of path dependent European options and a martingale optimal transport problem
is proved. The financial market is modeled through a risky asset whose price is
only assumed to be a continuous function of time. The hedging problem is to
construct a minimal super-hedging portfolio that consists of dynamically
trading the underlying risky asset and a static position of vanilla options
which can be exercised at the given, fixed maturity. The dual is a
Monge-Kantorovich type martingale transport problem of maximizing the expected
value of the option over all martingale measures that has the given marginal at
maturity. In addition to duality, a family of simple, piecewise constant
super-replication portfolios that asymptotically achieve the minimal
super-replication cost is constructed.
