Optimal selling time in stock market over a finite time horizon

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 ≤ τ ≤ T 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”.