Selling a Stock at the Ultimate Maximum

where the infimum and supremum are taken over all stopping times τ of Z . We show that the following strategy is optimal in the first problem: if μ ≤ 0 stop immediately; if μ ∈ (0, σ) stop as soon as Mt/Zt hits a specified function of time; and if μ ≥ σ wait until the final time T . By contrast we show that the following strategy is optimal in the second problem: if μ ≤ σ/2 stop immediately, and if μ > σ/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 timespace 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.