  This paper is concerned with an optimal strategy for simultaneously trading a
pair of stocks. The idea of pairs trading is to monitor their price movements
and compare their relative strength over time. A pairs trade is triggered by
the divergence of their prices and consists of a pair of positions to short the
strong stock and to long the weak one. Such a strategy bets on the reversal of
their price strengths. A round-trip trading strategy refers to opening and
closing such a pair of security positions. Typical pairs-trading models usually
assume a difference of the stock prices satisfies a mean-reversion equation.
However, we consider the optimal pairs-trading problem by allowing the stock
prices to follow general geometric Brownian motions. The objective is to trade
the pairs over time to maximize an overall return with a fixed commission cost
for each transaction. Initially, we allow the initial pairs position to be
either long or flat. We then consider the problem when the initial pairs
position may be long, flat, or short. In each case, the optimal policy is
characterized by threshold curves obtained by solving the associated HJB
equations.
