Pricing American Options Time-Capped by a Drawdown Event

Paper: arXiv 2509.00999

Authors: Zbigniew Palmowski, Paweł Stȩpniak

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.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical derivation of closed-form pricing for an American option with a drawdown time-cap using stochastic analysis and HJB equations. However, it lacks empirical validation, relying solely on theoretical proofs and minimal numerical examples.

Research Flowchart

  flowchart TD
    A[Research Goal: Price Perpetual American Put<br/>Time-Capped by Drawdown Event] --> B[Inputs: Black-Scholes Model &<br/>Predefined Drawdown Threshold]
    B --> C[Methodology: Martingale Arguments &<br/>Fluctuation Theory of Lévy Processes]
    C --> D[Computational & Numerical Analysis]
    D --> E[Key Finding: Explicit Option Price Formula]
    D --> F[Key Finding: Optimal Exercise Trigger<br/>Asset Price Drops Below Threshold]