We aim to determine an optimal stock selling time to minimize the　expectation of the square error between the selling price and the　global maximum price over a given period. Assuming that stock price follows the　geometric Brownian motion, we formulate the problem as an optimal stopping　time problem, or equivalently, a variational inequality problem. We provide　a partial differential equation (PDE) approach to characterize the resulting　free boundary that corresponds to the optimal selling strategy. The　monotonicity and smoothness of the free boundary are addressed as well.