An optimal stock liquidation rule

Trading in stock markets consists of three major steps: select a stock, purchase a number of shares, and eventually sell them to make a profit. The timing to buy and sell is extremely crucial. A selling rule can be specified by two pre-selected levels: a target price and a stop-loss limit. The paper is concerned with an optimal selling rule based on the model characterized by a number of geometric Brownian motions coupled by a finite-state Markov chain. Such policy can be obtained by solving a set of two-point boundary value differential equations. Moreover, the corresponding expected target period and probability of making money and that of losing money are derived. A numerical example is considered to demonstrate the effectiveness of our method