  Optimal trading strategies for pairs trading have been studied by models that
try to find either optimal shares of stocks by assuming no transaction costs or
optimal timing of trading fixed numbers of shares of stocks with transaction
costs. To find optimal strategies which determine optimally both trade times
and number of shares in pairs trading process, we use a singular stochastic
control approach to study an optimal pairs trading problem with proportional
transaction costs. Assuming a cointegrated relationship for a pair of stock
log-prices, we consider a portfolio optimization problem which involves dynamic
trading strategies with proportional transaction costs. We show that the value
function of the control problem is the unique viscosity solution of a nonlinear
quasi-variational inequality, which is equivalent to a free boundary problem
for the singular stochastic control value function. We then develop a discrete
time dynamic programming algorithm to compute the transaction regions, and show
the convergence of the discretization scheme. We illustrate our approach with
numerical examples and discuss the impact of different parameters on
transaction regions. We study the out-of-sample performance in an empirical
study that consists of six pairs of U.S. stocks selected from different
industry sectors, and demonstrate the efficiency of the optimal strategy.
