Optimal selling time in stock market over a finite time horizon

  • S. C. P. YamEmail author
  • S. P. Yung
  • W. Zhou


In this paper, we examine the best time to sell a stock at a price being as close as possible to its highest price over a finite time horizon [0, T], where the stock price is modelled by a geometric Brownian motion and the ‘closeness’ is measured by the relative error of the stock price to its highest price over [0, T]. More precisely, we want to optimize the expression:
$$V^* = \mathop {\sup }\limits_{0 \leqslant \tau \leqslant T} \mathbb{E}[\tfrac{{V_\tau }} {{M_T }}] $$
where (V t ) t≥0 is a geometric Brownian motion with constant drift α and constant volatility σ > 0, \(M_t = \mathop {\max }\limits_{0 \leqslant s \leqslant t} V_s \) is the running maximum of the stock price, and the supremum is taken over all possible stopping times 0 ≤ τT adapted to the natural filtration (F t ) t≥0 of the stock price. The above problem has been considered by Shiryaev, Xu and Zhou (2008) and Du Toit and Peskir (2009). In this paper we provide an independent proof that when \(\alpha = \tfrac{1} {2}\sigma ^2 \) , a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the moment when the stock price hits its maximum price so far. Besides, when \(\alpha > \tfrac{1} {2}\sigma ^2 \) , selling the stock at the terminal time T is the unique optimal selling strategy. Our approach to the problem is purely probabilistic and has been inspired by relating the notion of dominant stopping ρ τ of a stopping time τ to the optimal stopping strategy arisen in the classical “Secretary Problem”.


optimal Stopping stock selling buy and hold local time 

2000 MR Subject Classification



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  1. [1]
    Allaart, P.C. A general “bang-bang” principle for predicting the maximum of a random walk. arXiv:0910.0545. (2009)Google Scholar
  2. [2]
    Allaart, P.C. A “bang-bang” principle for predicting the supremum of a random walk or Levy process. arXiv:0912.0615 (2009)Google Scholar
  3. [3]
    Dai, M., Zhong, Y.F. Optimal stock selling/buying strategy with reference to the ultimate average. to appear in Math. Finan. (2009)Google Scholar
  4. [4]
    Du.Toit, J., Peskir, G. The trap of complacency in predicting the maximum. Ann. Probab., 35 340–365 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Du.Toit, J., Peskir, G. Selling a Stock at the Ultimate Maximum. Ann. Appl. Probab., 19: 983–1014 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Graversen, S.E., Peskir, G., Shiryaev, A.N. Stopping Brownian motion without anticipationas close as possible to its ultimate maximum. Theory Probab. Appl., 45: 125–136 (2001)MathSciNetCrossRefGoogle Scholar
  7. [7]
    Graversen, S.E., Shiryaev, A.N. An extension of P. Levy’s distributional properties to the case of a Brownian motion with drift. Bernoulli, 6: 615–620 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Karatzas, I., Shreve, S.E. Methods of Mathematical Finance. Springer, 1998Google Scholar
  9. [9]
    Pedersen, J.L. Optimal prediction of the ultimate maximum of Brownian motion. Stoch. Stoch. Rep., 75: 205–219 (2003)MathSciNetzbMATHGoogle Scholar
  10. [10]
    Peskir, G. A change-of-variable formula with local time on curves. J. Theoret. Probab., 18: 499–535 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Peskir, G. On Reflecting Brownian Motion with Drift. In: Proc. Symp. Stoch. Syst., (Osaka, 2005), ISCIE Kyoto. 2006, 1–5Google Scholar
  12. [12]
    Peskir, G., Shiryaev, A.N. Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics, ETH Zürich, Birkhäuser, 2006Google Scholar
  13. [13]
    Shiryaev, A.N. Quickest detection problems in the technical analysis of the fancial data. In: Proc. Math. Finance Bachelier Congress (Paris, 2000), Springer, 2002, 487–521Google Scholar
  14. [14]
    Shiryaev, A.N., Xu, Z., Zhou, X.Y. Thou shalt buy and hold. Quantitative Finance, 8: 765–776 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Yam, S.C.P., Yung, S.P., Zhou, W. Two rationales behind ‘buy-and-hold or sell-at-once’. J. Appl. Probab., 46: 651–668 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Yam, S.C.P., Yung, S.P., Zhou, W. A unified ‘bang-bang’ principle with respect to a class of nonanticipative benchmarks. Accepted by Probability Theory and Its Application (2010)Google Scholar
  17. [17]
    Zhou X.Y. Thou Shalt Buy and Hold. A conference talk in 2008 International Conference on Mathematics of Finance and Related Applications in Hong Kong, 2008Google Scholar

Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.The Chinese University of Hong KongHong KongChina
  2. 2.The University of Hong KongHong KongChina

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