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. More precisely, we aim to determine an optimal selling (buying) time 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 problem gives rise to a free boundary that corresponds to the optimal selling (buying) strategy. We provide a partial differential equation approach to characterize the free boundary (or equivalently, the optimal selling (buying) region). It turns out that the optimal selling strategy is bang-bang, which is the same as that obtained by Shiryaev, Xu, and Zhou taking the ultimate maximum of the stock price as benchmark, whereas the optimal buying strategy can be a feedback one subject to the type of averaging and parameter values. Moreover, by a thorough characterization of free boundary, we reveal that the bang-bang optimal selling strategy heavily depends on the assumption that no time-vesting restrictions are imposed. If a time-vested stock is considered, then the optimal selling strategy can also be a feedback one. In terms of a similar analysis developed by the present paper, the same phenomenon can be proved when taking the ultimate maximum as benchmark.