# Pricing American options with exogenous and endogenous transaction costs

## TL;DR

This paper develops a model for American option pricing considering both endogenous liquidity risks and exogenous proportional transaction costs, using nonlinear PDEs and numerical methods, and demonstrates improved market calibration performance.

## Contribution

It introduces a novel approach combining endogenous liquidity risk modeling with exogenous transaction costs in American option pricing, solved via nonlinear PDEs and calibrated with market data.

## Key findings

- Liquidity risks significantly affect option prices and exercise strategies.
- The proposed model outperforms the Leland model in market calibration.
- Numerical solutions effectively capture the impact of transaction costs.

## Abstract

We study an American option pricing problem with liquidity risks and transaction fees. As endogenous transaction costs, liquidity risks of the underlying asset are modeled by a mean-reverting process. Transaction fees are exogenous transaction costs and are assumed to be proportional to the trading amount, with the long-run liquidity level depending on the proportional transaction costs rate. Two nonlinear partial differential equations are established to characterize the option values for the holder and the writer, respectively. To illustrate the impact of these transaction costs on option prices and optimal exercise prices, we apply the alternating direction implicit method to solve the linear complementarity problem numerically. Finally, we conduct model calibration from market data via maximum likelihood estimation, and find that our model incorporating liquidity risks outperforms the Leland model significantly.