  We pursue robust approach to pricing and hedging in mathematical finance. We
consider a continuous time setting in which some underlying assets and options,
with continuous paths, are available for dynamic trading and a further set of
European options, possibly with varying maturities, is available for static
trading. Motivated by the notion of prediction set in Mykland (2003), we
include in our setup modelling beliefs by allowing to specify a set of paths to
be considered, e.g. super-replication of a contingent claim is required only
for paths falling in the given set. Our framework thus interpolates between
model-independent and model-specific settings and allows to quantify the impact
of making assumptions or gaining information. We obtain a general
pricing-hedging duality result: the infimum over superhedging prices is equal
to supremum over calibrated martingale measures. In presence of non-trivial
beliefs, the equality is between limiting values of perturbed problems. In
particular, our results include the martingale optimal transport duality of
Dolinsky and Soner (2013) and extend it to multiple dimensions and multiple
maturities.
