Open Access
June 2009 Selling a stock at the ultimate maximum
Jacques du Toit, Goran Peskir
Ann. Appl. Probab. 19(3): 983-1014 (June 2009). DOI: 10.1214/08-AAP566


Assuming that the stock price Z=(Zt)0≤tT follows a geometric Brownian motion with drift μ∈ℝ and volatility σ>0, and letting Mt=max 0≤stZs 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 Mt/Zt hits a specified function of time; and if μσ2 wait until the final time T. By contrast we show that the following strategy is optimal in the second problem: if μσ2/2 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.


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Jacques du Toit. Goran Peskir. "Selling a stock at the ultimate maximum." Ann. Appl. Probab. 19 (3) 983 - 1014, June 2009.


Published: June 2009
First available in Project Euclid: 15 June 2009

zbMATH: 1201.60037
MathSciNet: MR2537196
Digital Object Identifier: 10.1214/08-AAP566

Primary: 35R35 , 60G40 , 60J65
Secondary: 45G10 , 60G25 , 91B28

Keywords: curved boundary , diffusion process , Geometric Brownian motion , local time–space calculus , Markov process , nonlinear Volterra integral equation , normal reflection , optimal prediction , Optimal stopping , parabolic free-boundary problem , smooth fit , ultimate maximum

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.19 • No. 3 • June 2009
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