Pricing American options time-capped by a drawdown event in a Lévy market

Paper: arXiv 2508.20677

Authors: Zbigniew Palmowski, Paweł Stȩpniak

Abstract

This paper presents a derivation of the explicit price for the perpetual American put option time-capped by the first drawdown epoch beyond a predefined level. We consider the market in which an asset price is described by geometric Lévy process with downward exponential jumps. We show that the optimal stopping rule is the first time when the asset price gets below a special value. The proof relies on martingale arguments and the fluctuation theory of Lévy processes. We also provide a numerical analysis.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical derivation for pricing perpetual American options time-capped by a drawdown event under a Lévy market with exponential jumps using optimal stopping theory and fluctuation analysis. However, it is a purely theoretical paper with minimal empirical validation or real-market data application, focusing instead on analytical proofs and numerical verification.

Research Flowchart

  flowchart TD
    A[Research Goal: Price Perpetual American Put<br/>Time-Capped by Drawdown Event] --> B[Inputs: Asset Price as Geometric Lévy Process<br/>with Downward Exponential Jumps]
    B --> C[Methodology: Apply Fluctuation Theory &<br/>Martingale Arguments]
    C --> D[Determine Optimal Stopping Rule:<br/>First Exit Below Special Value Threshold]
    D --> E[Derive Explicit Option Pricing Formula]
    E --> F[Perform Numerical Analysis & Validation]
    F --> G[Outcome: Explicit Price for Time-Capped<br/>American Option in Lévy Market]