Selling a stock at the ultimate maximum

Abstract

Assuming that the stock price Z=(Z_t)_0≤t≤T follows a geometric Brownian motion with drift μ∈ℝ and volatility σ>0, and letting M_t=max _0≤s≤tZ_s for t∈[0, T], we consider the optimal prediction problems $$V_{1}=\inf_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{M_{T}}{Z_{\tau}}\biggr) \quad \mathrm{and} \quad V_{2}=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_{T}}\biggr),$$ 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≤0 stop immediately; if μ∈(0, σ²) stop as soon as M_t/Z_t hits a specified function of time; and if μ≥σ² wait until the final time T. By contrast we show that the following strategy is optimal in the second problem: if μ≤σ²/2 stop immediately, and if μ>σ²/2 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.