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

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 $(S_T,{\sup }_{u\le T}{S}_u)$ when a finite number of marginal law constraints on both $S_T$ and ${\sup }_{u\le T}{S}_u$ 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.