Analysis of the optimal exercise boundary of American options for jump diffusions

TL;DR

This paper proves the continuous differentiability of the optimal exercise boundary for American put options under jump diffusions, extending previous results and establishing infinite differentiability under certain conditions.

Contribution

It extends the differentiability results of the exercise boundary to cases where previous conditions do not hold, using a unified approach, and shows infinite differentiability under regular jump distribution assumptions.

Findings

Optimal exercise boundary is continuously differentiable except at maturity.

Boundary is infinitely differentiable under regular jump distribution.

Unified approach applies to cases with and without previous conditions.

Abstract

In this paper we show that the optimal exercise boundary / free boundary of the American put option pricing problem for jump diffusions is continuously differentiable (except at the maturity). This differentiability result has been established by Yang et al. (European Journal of Applied Mathematics, 17(1):95-127, 2006) in the case where the condition $r\geq q+ \lambda \int_{\R_+} (e^z-1) \nu(dz)$ is satisfied. We extend the result to the case where the condition fails using a unified approach that treats both cases simultaneously. We also show that the boundary is infinitely differentiable under a regularity assumption on the jump distribution.