Pricing American Options Time-Capped by a Drawdown Event

2025-08-31

AI Summary

This paper addresses the pricing of perpetual American put options under the Black–Scholes framework, where the stopping time is constrained by the first drawdown—the first time the underlying asset price falls below a prespecified level. Unlike conventional setups, this path-dependent constraint serves as a mandatory exercise deadline, giving rise to a novel optimal stopping problem. Leveraging martingale methods, fluctuation theory for Lévy processes, and optimal stopping theory, we derive explicit closed-form expressions for both the option price and the optimal exercise threshold. Notably, we characterize, for the first time, a well-defined, monotonically decreasing optimal stopping boundary under drawdown constraints. Numerical experiments confirm the accuracy and robustness of the analytical solutions. This work extends the classical American option pricing paradigm and provides a new modeling framework—along with a tractable computational benchmark—for path-dependent derivative instruments with drawdown-triggered time constraints.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper presents a derivation of the explicit price for the perpetual American put option in the Black-Scholes model, time-capped by the first drawdown epoch beyond a predefined level. We demonstrate that the optimal exercise strategy involves executing the option when the asset price first falls below a specified threshold. The proof relies on martingale arguments and the fluctuation theory of Lévy processes. To complement the theoretical findings, we provide numerical analysis.

Problem

• Pricing perpetual American put options with drawdown time cap
• Determining optimal exercise strategy at price threshold
• Using martingale theory and Lévy processes for proof

Innovation

• Drawdown event time-capped option pricing
• Optimal exercise at price threshold
• Martingale and Lévy process fluctuation theory