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Optimal selling time in stock market over a finite time horizon

  • S. C. P. YamEmail author
  • S. P. Yung
  • W. Zhou
Article

Abstract

In this paper, we examine the best time to sell a stock at a price being as close as possible to its highest price over a finite time horizon [0, T], where the stock price is modelled by a geometric Brownian motion and the ‘closeness’ is measured by the relative error of the stock price to its highest price over [0, T]. More precisely, we want to optimize the expression:
$$V^* = \mathop {\sup }\limits_{0 \leqslant \tau \leqslant T} \mathbb{E}[\tfrac{{V_\tau }} {{M_T }}] $$
where (V t ) t≥0 is a geometric Brownian motion with constant drift α and constant volatility σ > 0, \(M_t = \mathop {\max }\limits_{0 \leqslant s \leqslant t} V_s \) is the running maximum of the stock price, and the supremum is taken over all possible stopping times 0 ≤ τT adapted to the natural filtration (F t ) t≥0 of the stock price. The above problem has been considered by Shiryaev, Xu and Zhou (2008) and Du Toit and Peskir (2009). In this paper we provide an independent proof that when \(\alpha = \tfrac{1} {2}\sigma ^2 \) , a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the moment when the stock price hits its maximum price so far. Besides, when \(\alpha > \tfrac{1} {2}\sigma ^2 \) , selling the stock at the terminal time T is the unique optimal selling strategy. Our approach to the problem is purely probabilistic and has been inspired by relating the notion of dominant stopping ρ τ of a stopping time τ to the optimal stopping strategy arisen in the classical “Secretary Problem”.

Keywords

optimal Stopping stock selling buy and hold local time 

2000 MR Subject Classification

60G40 

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Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.The Chinese University of Hong KongHong KongChina
  2. 2.The University of Hong KongHong KongChina

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