# Variational inequality for perpetual American option price and convergence to the solution of the difference equation

## TL;DR

This paper establishes a variational inequality framework for pricing perpetual American options, proves solution properties, and demonstrates convergence of difference equation solutions to the continuous model, linking finite-maturity and perpetual options.

## Contribution

It introduces a variational inequality approach for perpetual American options and proves convergence of difference equation solutions to the continuous model's viscosity solution.

## Key findings

- Maximum principle and uniqueness for the variational inequality
- Existence and uniqueness of solutions to the difference equation
- Convergence of difference equation solutions to the viscosity solution

## Abstract

A variational inequality for pricing the perpetual American option and the corresponding difference equation are considered. First, the maximum principle and uniqueness of the solution to variational inequality for pricing the perpetual American option are proved. Then the maximum principle, the existence and uniqueness of the solution to the difference equation corresponding to the variational inequality for pricing the perpetual American option and the solution representation are provided and the fact that the solution to the difference equation converges to the viscosity solution to the variational inequality is proved. It is shown that the limits of the prices of variational inequality and BTM models for American Option when the maturity goes to infinity do not depend on time and they become the prices of the perpetual American option.