  We characterize the set of market models when there are a finite number of
traded Vanilla and Barrier options with maturity $T$ written on the asset $S$.
From a probabilistic perspective, our result describes the set of joint
distributions for $(S_T, \sup_{u \leq T} S_u)$ when a finite number of marginal
law constraints on both $S_T$ and $\sup_{u \leq T} S_u$ is imposed. An
extension to the case of multiple maturities is obtained.
  Our characterization requires a decomposition of the call price function and
once it is obtained, we can explicitly express certain joint probabilities in
this model. In order to obtain a fully specified joint distribution we discuss
interpolation methods.
