SELLING A STOCK AT THE ULTIMATE MAXIMUM

• 成果类型: Article
• 署名作者: du Toit, Jacques; Peskir, Goran
• 署名单位: University of Manchester
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/08-AAP566
• 发表日期: 2009
• 页码: 983-1014
• 关键词: brownian-motion

摘要

Assuming that the stock price Z = (Z(t))(0 <= t <= T) follows a geometric Brownian motion with drift mu is an element of R and volatility sigma > 0, and letting M-t = max(0 <= s <= t) Z(s) for t is an element of [0, T], we consider the optimal prediction problems V-1 = inf(0 <=tau <= T) E(M-T/Z(tau)) and V-2 = sup(0 <=tau <= T) E(Z(tau)/M-T), 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 <= 0 stop immediately; if mu is an element of (0, sigma(2)) stop as soon as M-t/Z(t), hits a specified function of time; and if mu >= sigma(2) wait until the final time T. By contrast we show that the following strategy is optimal in the second problem: if mu <= 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 freeboundary 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.