Last passage American cancellable option in L\'evy models

TL;DR

This paper derives an explicit formula for the price of a perpetual American put option that can be canceled at the last passage time, using spectrally negative Lévy processes and fluctuation theory.

Contribution

It introduces a novel pricing formula for last passage American cancellable options within Lévy models, extending classical methods to more complex stochastic processes.

Findings

Explicit pricing formula derived for the option.

Optimal exercise threshold identified.

Numerical analysis includes Black-Scholes and jump models.

Abstract

We derive the explicit price of the perpetual American put option cancelled at the last passage time of the underlying above some fixed level. We assume the asset process is governed by a geometric spectrally negative L\'evy process. We show that the optimal exercise time is the first epoch when asset price process drops below an optimal threshold. We perform numerical analysis as well considering classical Black-Scholes models and the model where logarithm of the asset price has additional exponential downward shocks. The proof is based on some martingale arguments and fluctuation theory of L\'evy processes.