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

TL;DR

This paper characterizes the set of possible market models with finite traded vanilla and barrier options, describing joint distributions of the asset price and its maximum, and extends the results to multiple maturities.

Contribution

It provides a probabilistic characterization of market models with traded vanilla and barrier options, including explicit joint distribution expressions and interpolation methods.

Findings

Explicit joint distribution formulas derived

Extension to multiple maturities achieved

Decomposition of call price functions developed

Abstract

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.