Optimal exercise of American puts with transaction costs under utility maximization

American option pricing plays an essential role in quantitative finance and has been extensively studied in the past. However, how transaction costs affect the American option price, particularly the most important feature of American options, the optimal exercise price, is much less investigated. It is primarily because such a study must be conducted under an incomplete market, which presents additional difficulties on top of an already difficult nonlinear mathematical problem. This paper attempts to provide a supplement study in this area by analyzing the optimal exercise price of an American option in addition to the option price itself in the presence of transaction costs through a utility-based approach. With a computationally efficient numerical scheme, we are able to demonstrate clearly how the optimal exercise price should be calculated and consequently how the option prices for the buyer and writer as well as the early exercise decision are affected by the inclusion of transaction cost.