Optimal stopping problems in diffusion-type models with running maxima and drawdowns

TL;DR

This paper analyzes optimal stopping problems for perpetual American options within an extended Black-Merton-Scholes model where dividend and volatility rates depend on the asset's maximum and maximum drawdown, providing explicit solutions.

Contribution

It introduces a novel model incorporating running maxima and drawdowns into option pricing, deriving explicit free-boundary solutions and differential equations for optimal exercise boundaries.

Findings

Closed-form solutions for value functions with smooth fit.

Explicit characterization of optimal stopping regions.

Derivation of differential equations for exercise boundaries.

Abstract

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.