A general "bang-bang" principle for predicting the maximum of a random walk

TL;DR

This paper establishes a universal 'bang-bang' optimal stopping rule for predicting the maximum of a random walk or Brownian motion with drift, confirming that one should sell bad stocks immediately and hold good ones longer.

Contribution

It generalizes existing results by proving a simple 'bang-bang' stopping rule for a broad class of processes and convex functions, providing theoretical support for a common financial intuition.

Findings

Optimal stopping time is zero if drift is negative.

Optimal stopping time is at the end if drift is positive.

Results apply to Bernoulli walks and Brownian motions with drift.

Abstract

Let $(B_t)_{0\leq t\leq T}$ be either a Bernoulli random walk or a Brownian motion with drift, and let $M_t:=\max\{B_s: 0\leq s\leq t\}$, $0\leq t\leq T$. This paper solves the general optimal prediction problem \sup_{0\leq\tau\leq T}\sE[f(M_T-B_\tau)], where the supremum is over all stopping times $\tau$ adapted to the natural filtration of $(B_t)$, and $f$ is a nonincreasing convex function. The optimal stopping time $\tau^*$ is shown to be of "bang-bang" type: $\tau^*\equiv 0$ if the drift of the underlying process $(B_t)$ is negative, and $\tau^*\equiv T$ is the drift is positive. This result generalizes recent findings by S. Yam, S. Yung and W. Zhou [{\em J. Appl. Probab.} {\bf 46} (2009), 651--668] and J. Du Toit and G. Peskir [{\em Ann. Appl. Probab.} {\bf 19} (2009), 983--1014], and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good ones as long as possible.