Optimal Stock Selling/Buying Strategy with Reference to the Ultimate Average

We are concerned with the optimal decision to sell or buy a stock in a given period with reference to the ultimate average of the stock price. Strictly speaking, we aim to determine an optimal selling (buying) time so as to maximize (minimize) the expectation of the ratio of the selling (buying) price to the ultimate average price over the period. This is an optimal stopping time problem which can be formulated as a variational inequality problem. The associated stopping region corresponds to the optimal selling (buying) strategy. We provide a partial differential equation approach to study the optimal strategy. It turns out that the optimal selling strategy is bang-bang, which is the same as that obtained by Shiryaev, Xu and Zhou (2008) taking the ultimate maximum of the stock price as the benchmark. However, the optimal buying strategy can be a feedback one subject to the type of average and parameter values