On a free boundary problem for an American put option under the CEV process

TL;DR

This paper studies a free boundary problem for American put options under the CEV process, deriving a nonlinear integral equation for the boundary and analyzing its behavior in the small $ ho$ limit using perturbation methods.

Contribution

It introduces a nonlinear integral equation for the free boundary and analyzes its asymptotic behavior across different time ranges, advancing understanding of American options under CEV.

Findings

Free boundary satisfies a nonlinear integral equation.

Boundary behavior varies across five time-to-expiry ranges.

Perturbation methods reveal different asymptotic behaviors.

Abstract

We consider an American put option under the CEV process. This corresponds to a free boundary problem for a PDE. We show that this free bondary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho$ = $2r/ \sigma^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We use perturbation methods to find that the free boundary behaves differently for five ranges of time to expiry.