On the continuous and smooth fit principle for optimal stopping problems in spectrally negative Levy models

TL;DR

This paper analyzes optimal stopping problems for spectrally negative Levy processes, establishing conditions for optimal thresholds and applying results to classic American option problems with explicit formulas.

Contribution

It provides explicit expressions for expected payoffs, proves the equivalence of continuous/smooth fit and first-order conditions, and applies these to classical American option problems.

Findings

Explicit formulas for expected payoffs using scale functions

Equivalence of fit conditions and first-order optimality

Application to perpetual American put option problem

Abstract

We consider a class of infinite-time horizon optimal stopping problems for spectrally negative Levy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshold levels. We obtain and show the equivalence of the continuous/smooth fit condition and the first-order condition for maximization over threshold levels. As examples of its applications, we give a short proof of the McKean optimal stopping problem (perpetual American put option) and solve an extension to Egami and Yamazaki (2013).