Some results on optimal stopping problems for one-dimensional regular diffusions

TL;DR

This paper derives closed-form solutions and characterizations for optimal stopping problems involving one-dimensional regular diffusions, focusing on specific financial options like ESOs and barrier American puts.

Contribution

It provides explicit formulas and a complete characterization of optimal stopping regions for certain diffusion-based financial options, extending previous theoretical frameworks.

Findings

Closed-form formulas for value functions of ESOs and barrier American puts

Complete characterization of optimal stopping and continuation regions

Comparison principles for critical levels and value functions

Abstract

For a type of employee stock option (ESO) and an American put option with a barrier, we obtain closed-form formulae for the value functions and provide a complete characterization for optimal stopping/continuation regions. Some comparison principles for the critical levels and the value functions are given. This work is inspired by the characterization of the value functions for general one-dimensional regular diffusion processes developed in \cite{DK03} by Dayanik and Karatzas.