Selling a stock at the ultimate maximum

Abstract

Assuming that the stock price $Z=(Z_t{)}_{0\le t\le T}$ follows a geometric Brownian motion with drift $\mu \in \mathbb{R}$ and volatility $\sigma >0$, and letting $M_t={\max }_{0\le s\le t}{Z}_s$ for $t\in [0,T]$, we consider the optimal prediction problems \[V_1=\inf_{0\leqτ\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_τ}\biggr)\quadand\quad V_2=\sup_{0\leqτ\leq T}\mathsf{E}\biggl(\frac{Z_τ}{M_T}\biggr),\] 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 $\mu \le 0$ stop immediately; if $\mu \in (0,{\sigma }^2)$ stop as soon as $M_t/Z_t$ hits a specified function of time; and if $\mu \ge {\sigma }^2$ wait until the final time $T$. By contrast we show that the following strategy is optimal in the second problem: if $\mu \le {\sigma }^2/2$ 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.

Published in at http://dx.doi.org/10.1214/08-AAP566 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• The trap of complacency in predicting the maximum