The Analysis Of Optimal Stock Buying And Selling Time With Reference To The Ultimate Average

When an investor trades a stock he naturally hopes to "buy low and sell high (BLSH)". Yet one could never be able to "buy at the lowest and sell at the highest" over a given time period. There have been different interpretations on the maxim BLSH, depending on the meanings of the "low" and the "high".We aim to determine the optimal stock buying and selling time over a given horizon with reference to the ultimate arithmetic and geometric average price of the stock. Assuming stock price to satisfy the geometric brownian motion, we formulate the problem as an optimal stopping time problem. We provide a measure transformation approach to characterize the resulting free boundary. It turns out that the optimal buying strategy is bang-bang, whereas the optimal selling strategy can be a feedback one subject to the type of averaging and parameter values. This result coheres with that of Min Dai, Yifei Zhong[7], where the ratio of the selling or buying price to the ultimate average price is regarded as the decision-making goal.