Discontinuity of Value Functions of Certain Options with Barriers

Abstract

In this note we prove the following statements, conjectured by Sergei Levendorskii (private communication). Consider a first-touch digital option in a L'evy-driven model, where the underlying Levy process has finite variance and drifts away from the barrier (in other words, we assume that the drift is strictly positive in the case of a down-and-in option, and strictly negative in the case of an up-and-in option). The value function of this option has a discontinuity at the barrier. A similar result is valid for a knock-out barrier option under certain assumptions on the terminal payoff function (these assumptions hold in all examples that arise in practice). Both perpetual and finite-lived options are considered in this article.