# Selling a Stock at the Ultimate Maximum

@article{Toit2009SellingAS,
title={Selling a Stock at the Ultimate Maximum},
author={Jacques Du Toit and Goran Peskir},
journal={Annals of Applied Probability},
year={2009},
volume={19},
pages={983-1014}
}
• Published 1 June 2009
• Mathematics
• Annals of Applied Probability
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…
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