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An Optimal Pairs-Trading Rule

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  • Qingshuo Song
  • Qing Zhang

Abstract

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.

Suggested Citation

  • Qingshuo Song & Qing Zhang, 2013. "An Optimal Pairs-Trading Rule," Papers 1302.6120, arXiv.org.
    • Handle: RePEc:arx:papers:1302.6120
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    References listed on IDEAS

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    1. Liam A. Gallagher & Mark P. Taylor, 2002. "Permanent and Temporary Components of Stock Prices: Evidence from Assessing Macroeconomic Shocks," Southern Economic Journal, John Wiley & Sons, vol. 69(2), pages 345-362, October.
    2. Fama, Eugene F & French, Kenneth R, 1988. "Permanent and Temporary Components of Stock Prices," Journal of Political Economy, University of Chicago Press, vol. 96(2), pages 246-273, April.
    3. Hafner, Christian M. & Herwartz, Helmut, 2001. "Option pricing under linear autoregressive dynamics, heteroskedasticity, and conditional leptokurtosis," Journal of Empirical Finance, Elsevier, vol. 8(1), pages 1-34, March.
    4. Evan Gatev & William N. Goetzmann & K. Geert Rouwenhorst, 2006. "Pairs Trading: Performance of a Relative-Value Arbitrage Rule," The Review of Financial Studies, Society for Financial Studies, vol. 19(3), pages 797-827.
    5. Liam A. Gallagher & Mark P. Taylor, 2002. "Permanent and Temporary Components of Stock Prices: Evidence from Assessing Macroeconomic Shocks," Southern Economic Journal, John Wiley & Sons, vol. 69(2), pages 345-362, October.
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    Citations

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    Cited by:

    1. Yuji Yamada & James A. Primbs, 2018. "Model Predictive Control for Optimal Pairs Trading Portfolio with Gross Exposure and Transaction Cost Constraints," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 25(1), pages 1-21, March.
    2. Haipeng Xing, 2019. "A singular stochastic control approach for optimal pairs trading with proportional transaction costs," Papers 1911.10450, arXiv.org.
    3. Fenghui Yu & Wai-Ki Ching & Chufang Wu & Jia-Wen Gu, 2023. "Optimal Pairs Trading Strategies: A Stochastic Mean–Variance Approach," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 36-55, January.
    4. Haipeng Xing, 2022. "A Singular Stochastic Control Approach for Optimal Pairs Trading with Proportional Transaction Costs," JRFM, MDPI, vol. 15(4), pages 1-23, March.
    5. Kiyoshi Suzuki, 2018. "Optimal pair-trading strategy over long/short/square positions—empirical study," Quantitative Finance, Taylor & Francis Journals, vol. 18(1), pages 97-119, January.
    6. Minh Man Ngo & Huyen Pham, 2014. "Optimal switching for pairs trading rule: a viscosity solutions approach," Papers 1412.7649, arXiv.org.
    7. Kiyoshi Suzuki, 2021. "Infinite-Horizon Optimal Switching Regions for a Pair-Trading Strategy with Quadratic Risk Aversion Considering Simultaneous Multiple Switchings: A Viscosity Solution Approach," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 336-360, February.
    8. Yerkin Kitapbayev & Tim Leung, 2017. "Optimal mean-reverting spread trading: nonlinear integral equation approach," Annals of Finance, Springer, vol. 13(2), pages 181-203, May.
    9. Krauss, Christopher, 2015. "Statistical arbitrage pairs trading strategies: Review and outlook," FAU Discussion Papers in Economics 09/2015, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    10. Ruyi Liu & Jingzhi Tie & Zhen Wu & Qing Zhang, 2023. "Pairs Trading: An Optimal Selling Rule with Constraints," Papers 2307.15300, arXiv.org.
    11. Dong-Mei Zhu & Jia-Wen Gu & Feng-Hui Yu & Tak-Kuen Siu & Wai-Ki Ching, 2021. "Optimal pairs trading with dynamic mean-variance objective," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(1), pages 145-168, August.
    12. Atul Deshpande & John A Gubner & B. Ross Barmish, 2020. "On Simultaneous Long-Short Stock Trading Controllers with Cross-Coupling," Papers 2011.09109, arXiv.org.
    13. Jingzhi Tie & Hanqin Zhang & Qing Zhang, 2018. "An Optimal Strategy for Pairs Trading Under Geometric Brownian Motions," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 654-675, November.

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