In this paper, we examine the problem of finding the reservation option prices and corresponding exercise policies of American options in a market with proportional transaction costs using the utility-based approach proposed by Davis and Zariphopoulou (1995). We present a model in which the option holder has a constant absolute risk-aversion. We discuss the numerical algorithm and propose a new characterization of the option holder’s value function. We suggest original discretization schemes for computing reservation prices and exercise policies of American options. The discretization schemes are implemented for the cases of American put and call options. We present a study of the optimal transaction policy of the option holder. We examine the effects on the reservation option prices and the corresponding exercise policies of varying the levels of absolute risk-aversion and transaction costs.