On the regularity of American options with regime-switching uncertainty

TL;DR

This paper investigates the regularity properties of American option value functions in regime-switching models, establishing their continuity and smoothness, and identifying minimal payoff scenarios under uncertain transition rates.

Contribution

It introduces new techniques to prove regularity of value functions in regime-switching models and characterizes minimal payoff scenarios with uncertain transition rates.

Findings

Proved continuity and smoothness of the value function.

Identified minimal payoff scenarios under regime-switching uncertainty.

Applied coupling and time-change techniques to stochastic representations.

Abstract

We study the regularity of the stochastic representation of the solution of a class of initial-boundary value problems related to a regime-switching diffusion. This representation is related to the value function of a finite-horizon optimal stopping problem such as the price of an American-style option in finance. We show continuity and smoothness of the value function using coupling and time-change techniques. As an application, we find the minimal payoff scenario for the holder of an American-style option in the presence of regime-switching uncertainty under the assumption that the transition rates are known to lie within level-dependent compact sets.