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 an...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^＊=sup 0≤τ≤T E[Vτ/MT],where （Vt）t≥0 is a geometric Drownian motion with constant drift α and constant volatility σ 〉 0, Mt = max Vs 0≤a〈t 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 （Ft）t≥0 of the stock price. The above problem has been considered by Shiryacv, Xu and Zhou （2008） and Du Toit and Peskir （2009）. In this paper we provide an independent proof that when α=1/ 2 σ^2 , a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the 1 2 moment when the stock price hits its maximum price so far. Besides, when α 〉1/2σ^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 pr of a stopping time τ to the optimal stopping strategy arisen in the classical ＂Secretary Problem＂.展开更多
基金supported by the Hong Kong RGC GRF 502909The Hong Kong Polytechnic University Internal Grant APC0D+1 种基金The Hong Kong Polytechnic University Collaborative Research Grant G-YH96supported by an internal grant of code 201109176016 from the University of Hong Kong
文摘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^＊=sup 0≤τ≤T E[Vτ/MT],where （Vt）t≥0 is a geometric Drownian motion with constant drift α and constant volatility σ 〉 0, Mt = max Vs 0≤a〈t 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 （Ft）t≥0 of the stock price. The above problem has been considered by Shiryacv, Xu and Zhou （2008） and Du Toit and Peskir （2009）. In this paper we provide an independent proof that when α=1/ 2 σ^2 , a selling strategy is optimal if and only if it sells the stock either at the terminal time T or at the 1 2 moment when the stock price hits its maximum price so far. Besides, when α 〉1/2σ^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 pr of a stopping time τ to the optimal stopping strategy arisen in the classical ＂Secretary Problem＂.
