IDEAS home Printed from https://ideas.repec.org/p/arx/papers/0908.1014.html
   My bibliography  Save this paper

Selling a stock at the ultimate maximum

Author

Listed:
  • Jacques du Toit
  • Goran Peskir

Abstract

Assuming that the stock price $Z=(Z_t)_{0\leq t\leq T}$ follows a geometric Brownian motion with drift $\mu\in\mathbb{R}$ and volatility $\sigma>0$, and letting $M_t=\max_{0\leq s\leq t}Z_s$ for $t\in[0,T]$, we consider the optimal prediction problems \[V_1=\inf_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_{\tau}}\biggr)\quadand\quad V_2=\sup_{0\leq\tau\leq T}\mathsf{E}\biggl(\frac{Z_{\tau}}{M_T}\biggr),\] where the infimum and supremum are taken over all stopping times $\tau$ of $Z$. We show that the following strategy is optimal in the first problem: if $\mu\leq0$ stop immediately; if $\mu\in (0,\sigma^2)$ stop as soon as $M_t/Z_t$ hits a specified function of time; and if $\mu\geq\sigma^2$ wait until the final time $T$. By contrast we show that the following strategy is optimal in the second problem: if $\mu\leq\sigma^2/2$ stop immediately, and if $\mu>\sigma^2/2$ 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.

Suggested Citation

  • Jacques du Toit & Goran Peskir, 2009. "Selling a stock at the ultimate maximum," Papers 0908.1014, arXiv.org.
  • Handle: RePEc:arx:papers:0908.1014
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/0908.1014
    File Function: Latest version
    Download Restriction: no
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Arcand, Jean-Louis & Hongler, Max-Olivier & Rinaldo, Daniele, 2020. "Increasing risk: Dynamic mean-preserving spreads," Journal of Mathematical Economics, Elsevier, vol. 86(C), pages 69-82.
    2. Luluwah Al-Fagih, 2015. "The British Knock-Out Put Option," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(02), pages 1-32.
    3. Lempa, Jukka & Mordecki, Ernesto & Salminen, Paavo, 2024. "Diffusion spiders: Green kernel, excessive functions and optimal stopping," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    4. Liu, Yue & Sun, Huaping & Meng, Bo & Jin, Shunlin & Chen, Bin, 2023. "How to purchase carbon emission right optimally for energy-consuming enterprises? Analysis based on optimal stopping model," Energy Economics, Elsevier, vol. 124(C).
    5. Peskir, Goran, 2012. "Optimal detection of a hidden target: The median rule," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2249-2263.
    6. Johnson, P. & Pedersen, J.L. & Peskir, G. & Zucca, C., 2022. "Detecting the presence of a random drift in Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1068-1090.
    7. Cohen, Albert, 2010. "Examples of optimal prediction in the infinite horizon case," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 950-957, June.
    8. Buonaguidi, B., 2023. "An optimal sequential procedure for determining the drift of a Brownian motion among three values," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 320-349.
    9. Christensen, Sören & Crocce, Fabián & Mordecki, Ernesto & Salminen, Paavo, 2019. "On optimal stopping of multidimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2561-2581.
    10. Aleksandar Mijatovic & Martijn R. Pistorius, 2011. "On the drawdown of completely asymmetric Levy processes," Papers 1103.1460, arXiv.org, revised Sep 2012.
    11. Yang, Aijun & Liu, Yue & Xiang, Ju & Yang, Hongqiang, 2016. "Optimal buying at the global minimum in a regime switching model," Mathematical Social Sciences, Elsevier, vol. 84(C), pages 50-55.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:0908.1014. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.