# Pricing American options time-capped by a drawdown event in a L\'evy market

## TL;DR

This paper derives an explicit formula for pricing perpetual American put options with a drawdown cap in a Le9vy market, using martingale and fluctuation theory, and supports it with numerical analysis.

## Contribution

It introduces a novel explicit pricing formula for American options with drawdown constraints in Le9vy markets, extending existing models.

## Key findings

- Explicit pricing formula derived for the option.
- Optimal stopping rule identified as first crossing below a threshold.
- Numerical analysis confirms theoretical results.

## 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 L\'evy 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 L\'evy processes. We also provide a numerical analysis.