# Selling a stock at the ultimate maximum

## Author Info

• Jacques du Toit
• Goran Peskir
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## 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.

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File URL: http://arxiv.org/pdf/0908.1014

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 0908.1014.

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Publication status: Published in Annals of Applied Probability 2009, Vol. 19, No. 3, 983-1014
Handle: RePEc:arx:papers:0908.1014

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Web page: http://arxiv.org/

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## Citations

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Cited by:
1. Peskir, Goran, 2012. "Optimal detection of a hidden target: The median rule," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2249-2263.
2. Cohen, Albert, 2010. "Examples of optimal prediction in the infinite horizon case," Statistics & Probability Letters, Elsevier, vol. 80(11-12), pages 950-957, June.

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