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

Characterization of Market Models in the Presence of Traded Vanilla and Barrier Options