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Two Rationales Behind the ‘Buy-And-Hold or Sell-At-Once’ Strategy

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

S. C. P. Yam ,
S. P. Yung  and
W. Zhou
S. C. P. Yam*
Affiliation:
The Hong Kong Polytechnic University
S. P. Yung*
Affiliation:
The University of Hong Kong
W. Zhou*
Affiliation:
The University of Hong Kong
*
∗Postal address: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.
∗∗Postal address: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.
∗∗Postal address: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.
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Abstract

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The trading strategy of ‘buy-and-hold for superior stock and sell-at-once for inferior stock’, as suggested by conventional wisdom, has long been prevalent in Wall Street. In this paper, two rationales are provided to support this trading strategy from a purely mathematical standpoint. Adopting the standard binomial tree model (or CRR model for short, as first introduced in Cox, Ross and Rubinstein (1979)) to model the stock price dynamics, we look for the optimal stock selling rule(s) so as to maximize (i) the chance that an investor can sell a stock precisely at its ultimate highest price over a fixed investment horizon [0,T]; and (ii) the expected ratio of the selling price of a stock to its ultimate highest price over [0,T]. We show that both problems have exactly the same optimal solution which can literally be interpreted as ‘buy-and-hold or sell-at-once’ depending on the value of p (the going-up probability of the stock price at each step): when p›½, selling the stock at the last time step N is the optimal selling strategy; when p=½, a selling time is optimal if the stock is sold either at the last time step or at the time step when the stock price reaches its running maximum price; and when p‹½, time 0, i.e. selling the stock at once, is the unique optimal selling time.

Keywords

p-random walkbinomial tree modeloptimal stoppingmartingales

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 651 - 668
Copyright
Copyright © Applied Probability Trust 2009 

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