Trading securities using trailing stops

In financial markets traders often protect their position from a significant decline by using a trailing stop. Assume the trader is long the market (owns the security). A trailing stop is an order to sell the security at the market, if the price of the security drops to the stop price. The stop price is always less than the market price when the stop is entered. As the price fluctuates, the stop is raised to remain a fixed distance from the maximum price at which the security trades. In this paper the authors consider two models for the price process: a discrete time random walk and continuous time Brownian motion, both with positive drift. For these price processes they compute the distribution, mean, and variance of the gain to the trader as well as the duration of the trade when a trailing stop strategy is used. Also discussed is the question of optimizing the distance from the current price to the stop.