Selling a stock at the ultimate maximum

Assuming that the stock price Z=(Zt)0.t.T follows a geometric Brownian motion with drift ... and volatility .>0, and letting Mt=max.0.s.tZs for t.[0, T], we consider the optimal prediction problems V1=inf0...TE(MTZ.)andV2=sup0...TE(Z.MT), where the infimum and supremum are taken over all stopping times . of Z. We show that the following strategy is optimal in the first problem: if ..0 stop immediately; if ..(0, .2) stop as soon as Mt/Zt hits a specified function of time; and if ...2 wait until the final time T. By contrast we show that the following strategy is optimal in the second problem: if ...2/2 stop immediately, and if .>.2/2 wait until the final time T. Both solutions support and reinforce the widely held financial view that .one should sell bad stocks and keep good ones.. The method of proof makes use of parabolic free-boundary problems and local time.space calculus techniques. The resulting inequalities are unusual and interesting in their own right as they involve the future and as such have a predictive element.