Exercise Boundary of the American Put Near Maturity in an Exponential L\'evy Model

TL;DR

This paper analyzes how the critical price of an American put option approaches the stock price near maturity in an exponential Lévy model, revealing different convergence rates based on the Lévy process's variation.

Contribution

It establishes the convergence behavior of the critical price near maturity in exponential Lévy models, distinguishing finite and infinite variation cases with precise rates.

Findings

Linear convergence rate for finite variation Lévy processes.

Convergence rate of $ heta^{1/eta}| ext{ln} heta|^{1-1/eta}$ for infinite variation with stable-like Lévy measure.

Different behaviors depending on the Lévy process's variation near maturity.

Abstract

We study the behavior of the critical price of an American put option near maturity in the exponential L\'evy model when the underlying stock pays dividends at a continuous rate. In particular, we prove that, in situations where the limit of the critical price is equal to the stock price, the rate of convergence to the limit is linear if and only if the underlying L\'evy process has finite variation. In the case of infinite variation, a variety of rates of convergence can be observed: we prove that, when the negative part of the L\'evy measure exhibits an $\alpha$-stable density near the origin, with $1<\alpha<2$, the convergence rate is ruled by $\theta^{1/\alpha}|\ln \theta|^{1-\frac{1}{\alpha}}$, where $\theta$ is time until maturity.