Singular Stochastic Control for Pairs Trading with Transaction Costs

Summary

This study formulates pairs trading as a portfolio optimization problem for two cointegrated stock log-prices. A singular stochastic control model jointly determines when to trade and how many shares to adjust, while incorporating proportional transaction costs. The value function is characterized as the unique viscosity solution of a nonlinear quasi-variational inequality, which can also be expressed as a free-boundary problem.

For computation, the authors develop a discrete-time dynamic programming method to identify transaction regions and report convergence of the discretization scheme. Numerical examples examine how parameter choices affect those regions. An empirical out-of-sample study uses six U.S. stock pairs from different sectors and is presented as evidence of strategy efficiency. The document provides no performance figures or detailed assumptions about execution, and its results depend on the cointegration model and the selected pairs; they do not by themselves establish broad or live-trading performance.

Key ideas

• The model assumes a cointegrated relationship between two stock log-prices.
• Singular stochastic control jointly optimizes trade timing and share adjustments under proportional costs.
• The control problem is characterized by a nonlinear quasi-variational inequality and a free-boundary formulation.
• A discrete-time dynamic programming scheme computes transaction regions, with convergence reported by the authors.
• The empirical evaluation uses six out-of-sample U.S. stock pairs from different sectors.

Tags

# A singular stochastic control approach for optimal pairs trading with proportional transaction costs


# A singular stochastic control approach for optimal pairs trading with proportional transaction costs









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.

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