Optimal Pairs Trading with Transaction Costs and Stop-Loss Constraints

Summary

This paper formulates pairs trading as an optimal stopping problem. It considers two historically correlated securities whose price difference is modeled as a mean-reverting process. When their relative prices diverge, the strategy shorts the outperforming security and buys the underperforming one, aiming to profit when the spread returns toward convergence. The objective is to maximize overall trading return while accounting for a fixed commission on each transaction and a stop-loss imposed as a state constraint.

The authors characterize the value functions through Hamilton–Jacobi–Bellman equations expressed as quasi-variational inequalities. They show that the solution can be determined by solving a set of quasi-algebraic equations and provide sufficient conditions for verification. Numerical examples illustrate the proposed results. The document describes a mathematical framework rather than evidence from live trading or a broad empirical comparison; its conclusions depend on the mean-reverting model and specified cost and stop-loss assumptions.

Key ideas

• The strategy trades divergences between two historically correlated securities in anticipation of spread convergence.
• The spread is represented by a mean-reverting process.
• A fixed per-transaction commission and a stop-loss state constraint enter the optimization problem.
• HJB quasi-variational inequalities characterize the value functions and optimal stopping decisions.
• Numerical examples illustrate the method, whose conclusions depend on the model assumptions.

Tags

# An Optimal Pairs-Trading Rule


# An Optimal Pairs-Trading Rule









This paper is concerned with a pairs trading rule. The idea is to monitor two historically correlated securities. When divergence is underway, i.e., one stock moves up while the other moves down, a pairs trade is entered which consists of a pair to short the outperforming stock and to long the underperforming one. Such a strategy bets the "spread" between the two would eventually converge. In this paper, a difference of the pair is governed by a mean-reverting model. The objective is to trade the pair so as to maximize an overall return. A fixed commission cost is charged with each transaction. In addition, a stop-loss limit is imposed as a state constraint. The associated HJB equations (quasi-variational inequalities) are used to characterize the value functions. It is shown that the solution to the optimal stopping problem can be obtained by solving a number of quasi-algebraic equations. We provide a set of sufficient conditions in terms of a verification theorem. Numerical examples are reported to demonstrate the results.

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