Discontinuity of Value Functions of Certain Options with Barriers
10 Pages Posted: 2 Jan 2009 Last revised: 5 Jan 2009
Date Written: January 1, 2009
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.
Keywords: Option pricing, barrier options, first-touch digitals, Levy processes, discontinuity
JEL Classification: G13
Suggested Citation: Suggested Citation
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