Selling a stock at the ultimate maximum
Assuming that the stock price $Z=(Z_t)_{0\leq t\leq T}$ follows a geometric Brownian motion with drift $μ\in\mathbb{R}$ and volatility $σ>0$, and letting $M_t=\max_{0\leq s\leq t}Z_s$ for $t\in[0,T]$, we consider the optimal prediction problems \[V_1=\inf_{0\leqτ\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_τ}\biggr)\quadand\quad V_2=\sup_{0\leqτ\leq T}\mathsf{E}\biggl(\frac{Z_τ}{M_T}\biggr),\] 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 $μ\leq0$ stop immediately; if $μ\in (0,σ^2)$ stop as soon as $M_t/Z_t$ hits a specified function of time; and if $μ\geqσ^2$ wait until the final time $T$. By contrast we show that the following strategy is optimal in the second problem: if $μ\leqσ^2/2$ stop immediately, and if $μ>σ^2/2$ 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.