Optimal Pair-Trade Exit Rules with Trading Constraints

Summary

This paper studies when to close a stock pairs trade, which holds one stock long and another short. It formulates the exit decision as an optimal selling and repurchasing problem under trading constraints. The model assumes that the two stock prices follow a two-dimensional geometric Brownian motion, while whether trading is permitted is governed by a two-state Markov chain.

The proposed optimal policy is characterized by a threshold curve derived by solving the associated Hamilton-Jacobi-Bellman quasi-variational inequalities. The paper reports a closed-form solution and provides a verification theorem to support the claimed optimality. Numerical experiments illustrate the resulting policies and value functions. These conclusions depend on the specified price dynamics and permission process; the abstract gives no empirical evaluation on market data or accounting for transaction costs, and it does not establish that the rule outperforms other exits in live trading.

Key ideas

• The problem is to choose when to close a long-short stock pair under constraints on trading availability.
• The model represents the two stock prices with a two-dimensional geometric Brownian motion.
• A two-state Markov chain determines whether trading is currently allowed.
• The optimal exit policy is described by a threshold curve obtained from HJB quasi-variational inequalities.
• The paper reports a closed-form solution, a verification theorem, and numerical illustrations.

Tags

# Pairs Trading: An Optimal Selling Rule with Constraints


# Pairs Trading: An Optimal Selling Rule with Constraints









The focus of this paper is on identifying the most effective selling strategy for pairs trading of stocks. In pairs trading, a long position is held in one stock while a short position is held in another. The goal is to determine the optimal time to sell the long position and repurchase the short position in order to close the pairs position. The paper presents an optimal pairs-trading selling rule with trading constraints. In particular, the underlying stock prices evolve according to a two dimensional geometric Brownian motion and the trading permission process is given in terms of a two-state {trading allowed, trading not allowed} Markov chain. It is shown that the optimal policy can be determined by a threshold curve which is obtained by solving the associated HJB equations (quasi-variational inequalities). A closed form solution is obtained. A verification theorem is provided. Numerical experiments are also reported to demonstrate the optimal policies and value functions.

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