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Analytic solutions for optimal statistical arbitrage trading

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  • Bertram, William K.

Abstract

In this paper we derive analytic formulae for statistical arbitrage trading where the security price follows an Ornstein–Uhlenbeck process. By framing the problem in terms of the first-passage time of the process, we derive expressions for the mean and variance of the trade length and the return. We examine the problem of choosing an optimal strategy under two different objective functions: the expected return, and the Sharpe ratio. An exact analytic solution is obtained for the case of maximising the expected return.

Suggested Citation

  • Bertram, William K., 2010. "Analytic solutions for optimal statistical arbitrage trading," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(11), pages 2234-2243.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:11:p:2234-2243
    DOI: 10.1016/j.physa.2010.01.045
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    References listed on IDEAS

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    1. Gençay, Ramazan & Dacorogna, Michel & Muller, Ulrich A. & Pictet, Olivier & Olsen, Richard, 2001. "An Introduction to High-Frequency Finance," Elsevier Monographs, Elsevier, edition 1, number 9780122796715.
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    Cited by:

    1. Endres, Sylvia & Stübinger, Johannes, 2017. "Optimal trading strategies for Lévy-driven Ornstein-Uhlenbeck processes," FAU Discussion Papers in Economics 17/2017, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    2. Matthew Clegg & Christopher Krauss, 2018. "Pairs trading with partial cointegration," Quantitative Finance, Taylor & Francis Journals, vol. 18(1), pages 121-138, January.
    3. Ahmet Göncü & Erdinc Akyildirim, 2016. "A stochastic model for commodity pairs trading," Quantitative Finance, Taylor & Francis Journals, vol. 16(12), pages 1843-1857, December.
    4. Stübinger, Johannes & Endres, Sylvia, 2017. "Pairs trading with a mean-reverting jump-diffusion model on high-frequency data," FAU Discussion Papers in Economics 10/2017, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    5. Yaoyuan Zhang & Dewen Xiong, 2023. "Optimal Strategy of the Dynamic Mean-Variance Problem for Pairs Trading under a Fast Mean-Reverting Stochastic Volatility Model," Mathematics, MDPI, vol. 11(9), pages 1-19, May.
    6. Viviana Fanelli & Claudio Fontana & Francesco Rotondi, 2023. "A hidden Markov model for statistical arbitrage in international crude oil futures markets," Papers 2309.00875, arXiv.org, revised Sep 2024.
    7. Alexander Lipton & Marcos Lopez de Prado, 2020. "A closed-form solution for optimal mean-reverting trading strategies," Papers 2003.10502, arXiv.org.
    8. Erdinc Akyildirim & Ahmet Goncu & Alper Hekimoglu & Duc Khuong Nguyen & Ahmet Sensoy, 2023. "Statistical arbitrage: factor investing approach," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 45(4), pages 1295-1331, December.
    9. Tim Leung & Xin Li, 2015. "Optimal Mean Reversion Trading With Transaction Costs And Stop-Loss Exit," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(03), pages 1-31.
    10. Haican Diao & Guoshan Liu & Zhuangming Zhu, 2020. "Research on a stock-matching trading strategy based on bi-objective optimization," Frontiers of Business Research in China, Springer, vol. 14(1), pages 1-14, December.
    11. Kasper Johansson & Thomas Schmelzer & Stephen Boyd, 2024. "Finding Moving-Band Statistical Arbitrages via Convex-Concave Optimization," Papers 2402.08108, arXiv.org.
    12. Marianna Brunetti & Roberta De Luca, 2022. "Sensitivity of Profitability in Cointegration-Based Pairs Trading," CEIS Research Paper 540, Tor Vergata University, CEIS, revised 11 Apr 2022.
    13. Johannes Stübinger & Lucas Schneider, 2019. "Statistical Arbitrage with Mean-Reverting Overnight Price Gaps on High-Frequency Data of the S&P 500," JRFM, MDPI, vol. 12(2), pages 1-19, April.
    14. 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.
    15. 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.
    16. Vladimír Holý & Michal Černý, 2022. "Bertram’s pairs trading strategy with bounded risk," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 30(2), pages 667-682, June.
    17. Jeff Stephenson & Bruce Vanstone & Tobias Hahn, 2021. "A Unifying Model for Statistical Arbitrage: Model Assumptions and Empirical Failure," Computational Economics, Springer;Society for Computational Economics, vol. 58(4), pages 943-964, December.
    18. Masood Tadi & Jiří Witzany, 2025. "Copula-based trading of cointegrated cryptocurrency Pairs," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 11(1), pages 1-32, December.
    19. Wang, Xi & Bao, Si & Chen, Jingchao, 2017. "High-frequency stock linkage and multi-dimensional stationary processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 70-83.
    20. Endres, Sylvia & Stübinger, Johannes, 2018. "A flexible regime switching model with pairs trading application to the S&P 500 high-frequency stock returns," FAU Discussion Papers in Economics 07/2018, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    21. Johannes Stübinger & Sylvia Endres, 2018. "Pairs trading with a mean-reverting jump–diffusion model on high-frequency data," Quantitative Finance, Taylor & Francis Journals, vol. 18(10), pages 1735-1751, October.
    22. Kim, Min Jae & Kim, Sehyun & Jo, Yong Hwan & Kim, Soo Yong, 2011. "Dependence structure of the commodity and stock markets, and relevant multi-spread strategy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(21), pages 3842-3854.
    23. Zhengqin Zeng & Chi-Guhn Lee, 2014. "Pairs trading: optimal thresholds and profitability," Quantitative Finance, Taylor & Francis Journals, vol. 14(11), pages 1881-1893, November.
    24. Roy Cerqueti & Viviana Fanelli, 2021. "Long memory and crude oil’s price predictability," Annals of Operations Research, Springer, vol. 299(1), pages 895-906, April.
    25. Vladim'ir Hol'y & Petra Tomanov'a, 2018. "Estimation of Ornstein-Uhlenbeck Process Using Ultra-High-Frequency Data with Application to Intraday Pairs Trading Strategy," Papers 1811.09312, arXiv.org, revised Jul 2022.

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