Annals of Applied Probability

Selling a stock at the ultimate maximum

Jacques du Toit and Goran Peskir

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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

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

First available in Project Euclid: 15 June 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • [1] Doob, J. L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Statist. 20 393–403.
  • [2] du Toit, J. and Peskir, G. (2007). The trap of complacency in predicting the maximum. Ann. Probab. 35 340–365.
  • [3] Du Toit, J. and Peskir, G. (2008). Predicting the time of the ultimate maximum for Brownian motion with drift. In Proc. Math. Control Theory Finance (Lisbon 2007) 95–112. Springer, Berlin.
  • [4] du Toit, J., Peskir, G. and Shiryaev, A. N. (2008). Predicting the last zero of Brownian motion with drift. Stochastics 80 229–245.
  • [5] Graversen, S. E., Peskir, G. and Shiryaev, A. N. (2000). Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Theory Probab. Appl. 45 41–50.
  • [6] Graversen, S. E. and Shiryaev, A. N. (2000). An extension of P. Lévy’s distributional properties to the case of a Brownian motion with drift. Bernoulli 6 615–620.
  • [7] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Applications of Mathematics (New York) 39. Springer, New York.
  • [8] Malmquist, S. (1954). On certain confidence contours for distribution functions. Ann. Math. Statist. 25 523–533.
  • [9] Pedersen, J. L. (2003). Optimal prediction of the ultimate maximum of Brownian motion. Stoch. Stoch. Rep. 75 205–219.
  • [10] Peskir, G. (2005). A change-of-variable formula with local time on curves. J. Theoret. Probab. 18 499–535.
  • [11] Peskir, G. (2006). On reflecting Brownian motion with drift. In Proc. Symp. Stoch. Syst. (Osaka 2005) 1–5. ISCIE, Kyoto, Japan.
  • [12] Peskir, G. and Shiryaev, A.N. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.
  • [13] Shiryaev, A. N. (2002). Quickest detection problems in the technical analysis of the financial data. In Mathematical Finance—Bachelier Congress, 2000 (Paris) 487–521. Springer, Berlin.
  • [14] Shiryaev, A. N. (2007). On the conditionally extremal problems of the quickest detection of non-predictable times for the observable Brownian motion. Theory Probab. Appl. To appear.
  • [15] Shiryaev, A. N., Xu, Z. and Zhou, X. Y. (2008). Thou shalt buy and hold. Working Paper 21/5/2008, University of Oxford.
  • [16] Urusov, M. A. (2004). On a property of the time of attaining the maximum by Brownian motion and some optimal stopping problems. Theory Probab. Appl. 49 169–176.