Valuation of American put option with transaction cost and dividend using logarithmic front fixing transformation

Abstract

Due to their early exercise features, American-style options are the most traded in the financial markets. Black-Scholes derive the well-known option pricing Partial Differential Equation (PDE) with some assumptions. Black-Scholes assumes no transaction costs in his option pricing PDE. However, considering transaction costs makes the holder's and writer's option prices not unique, indicating the real-world complexities of trading. This study illustrates the understanding of how transaction costs impact American option pricing. The American-style option pricing problem is a moving boundary problem due to the early exercise feature. The front-fixing transformation technique can be employed to convert a moving boundary problem into a fixed boundary problem. In this work, we used a discrete hedging strategy to derive the option pricing PDEs to evaluate the holder's and writer's option prices and then apply the Logarithmic Front Fixing Transformation (Landau Transformation) to transform them into fixed boundary PDE systems. Then the explicit finite difference scheme is used to obtain numerical solutions. Numerical results confirm that the holder's and writer's option prices are not unique due to the inclusion of the transaction cost. Holders will choose to buy an option at a lower price, in contrast, writers will choose to sell an option at a higher price. The optimal exercise price of the American put option price is higher with transaction costs suggesting that option holders may exercise early to minimize losses, adding practical insight into trading strategies. Additionally, the faster execution time due to the front-fixing technique contributes to more efficient numerical methods in financial modeling.