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

2025-08-28

AI Summary

This paper studies the pricing of a perpetual American put option subject to a drawdown-triggered time constraint: the option automatically expires when the underlying asset price falls below its historical maximum by a predetermined threshold. The asset price follows a geometric Lévy process incorporating downward exponential jumps. Using martingale methods, fluctuation theory for Lévy processes, and first-passage analysis, we derive an explicit analytical solution for the option price and rigorously prove that the optimal stopping time coincides with the first passage time below a critical boundary. Our key contribution lies in the novel coupling of the American optimal stopping problem with dynamic drawdown events, introducing a path-dependent time constraint that extends classical Lévy-based option pricing theory. Numerical experiments confirm the accuracy of the analytical solution and demonstrate the robustness of the associated exercise strategy.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• 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 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.

Problem

• Pricing perpetual American put options with drawdown time caps
• Deriving explicit prices in geometric Lévy process markets
• Identifying optimal stopping rules using martingale and fluctuation theory

Innovation

• Pricing perpetual American put options
• Using geometric Lévy process modeling
• Applying martingale arguments and fluctuation theory