Some remarks on first passage of Levy processes, the American put and pasting principles

TL;DR

This paper establishes a generic link between first passage times of Levy processes and optimal stopping problems for American perpetual puts, providing new conditions for smooth pasting and connecting various known identities.

Contribution

It introduces a fluctuation identity that unifies existing results on Levy process first passage times and applies it to American perpetual put options, clarifying conditions for smooth pasting.

Findings

Unified framework for first passage time identities

Necessary and sufficient conditions for smooth pasting

Application to American perpetual put optimal stopping problem

Abstract

The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Levy process and the solution of Gerber and Shiu [Astin Bull. 24 (1994) 195-220], Boyarchenko and Levendorskii [Working paper series EERS 98/02 (1998), Unpublished manuscript (1999), SIAM J. Control Optim. 40 (2002) 1663-1696], Chan [Original unpublished manuscript (2000)], Avram, Chan and Usabel [Stochastic Process. Appl. 100 (2002) 75-107], Mordecki [Finance Stoch. 6 (2002) 473-493], Asmussen, Avram and Pistorius [Stochastic Process. Appl. 109 (2004) 79-111] and Chesney and Jeanblanc [Appl. Math. Fin. 11 (2004) 207-225] to the American perpetual put optimal stopping problem. Furthermore, we make folklore precise and give necessary and sufficient conditions for smooth pasting to occur in the considered problem.