Threshold-Based Round-Trip Pairs Trading with Geometric Brownian Prices

Summary

This paper formulates a pairs-trading strategy for two stocks whose prices follow general geometric Brownian motions, rather than assuming their price difference is mean-reverting. The trader compares the stocks’ relative strength, opens a position by shorting the stronger stock and buying the weaker one, then seeks to close the pair as its relative performance reverses. Each transaction incurs a fixed commission, and the objective is to maximize overall return across opening and closing trades.

The analysis characterizes optimal decisions through threshold curves derived from Hamilton–Jacobi–Bellman equations. It considers two starting-position settings: initially long or flat, and initially long, flat, or short. The provided description explains the model and solution approach but gives no numerical results, data tests, or details about parameter estimation. Its conclusions therefore concern the stated stochastic model; practical performance will depend on how well its price and cost assumptions fit a real market.

Key ideas

• The model allows each stock price to follow a geometric Brownian motion without requiring the price difference to mean-revert.
• A trade pairs a short position in the stronger stock with a long position in the weaker stock.
• A fixed commission is charged for each transaction in the round trip.
• The optimal opening and closing policy is represented by threshold curves obtained from HJB equations.
• The paper analyzes both long-or-flat and long-flat-or-short initial position choices.

Tags

# Optimal Strategies for Round-Trip Pairs Trading Under Geometric Brownian Motions


# Optimal Strategies for Round-Trip Pairs Trading Under Geometric Brownian Motions









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.

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