Annals of Applied Probability

Selling a stock at the ultimate maximum

Jacques du Toit and Goran Peskir

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 3 (2009), 983-1014.

Dates
First available in Project Euclid: 15 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1245071016

Digital Object Identifier
doi:10.1214/08-AAP566

Mathematical Reviews number (MathSciNet)
MR2537196

Zentralblatt MATH identifier
1201.60037

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 35R35: Free boundary problems 60J65: Brownian motion [See also 58J65]
Secondary: 91B28 60G25: Prediction theory [See also 62M20] 45G10: Other nonlinear integral equations

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

Citation

du Toit, Jacques; Peskir, Goran. Selling a stock at the ultimate maximum. Ann. Appl. Probab. 19 (2009), no. 3, 983--1014. doi:10.1214/08-AAP566. https://projecteuclid.org/euclid.aoap/1245071016


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