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Long-Time Behavior of a Hawkes Process--Based Limit Order Book

Author

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  • Frédéric Abergel

    (FiQuant - Chaire de finance quantitative - MICS - Mathématiques et Informatique pour la Complexité et les Systèmes - CentraleSupélec, MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris)

  • Aymen Jedidi

    (FiQuant - Chaire de finance quantitative - MICS - Mathématiques et Informatique pour la Complexité et les Systèmes - CentraleSupélec)

Abstract

Hawkes processes provide a natural framework to model dependencies between the intensities of point processes. In the context of order-driven financial markets, the relevance of such dependencies has been amply demonstrated from an empirical, as well as theoretical, standpoint. In this work, we build on previous empirical and numerical studies and introduce a mathematical model of limit order books based on Hawkes processes with exponential kernels. After proving a general stationarity result, we focus on the long-time behaviour of the limit order book and the corresponding dynamics of the suitably rescaled price. A formula for the asymptotic (in time) volatility of the price dynamics induced by that of the order book is obtained, involving the average of functions of the various order book events under the stationary distribution.

Suggested Citation

  • Frédéric Abergel & Aymen Jedidi, 2015. "Long-Time Behavior of a Hawkes Process--Based Limit Order Book," Post-Print hal-01121711, HAL.
    • Handle: RePEc:hal:journl:hal-01121711
    DOI: 10.1137/15m1011469
    Note: View the original document on HAL open archive server: https://hal.science/hal-01121711v5
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    References listed on IDEAS

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    13. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Post-Print hal-01313995, HAL.
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    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Maxime Morariu-Patrichi & Mikko S. Pakkanen, 2018. "State-dependent Hawkes processes and their application to limit order book modelling," Papers 1809.08060, arXiv.org, revised Sep 2021.
    2. Nakahiro Yoshida, 2022. "Quasi-likelihood analysis and its applications," Statistical Inference for Stochastic Processes, Springer, vol. 25(1), pages 43-60, April.
    3. Xiaofei Lu & Frédéric Abergel, 2017. "Limit order book modelling with high dimensional Hawkes processes," Working Papers hal-01512430, HAL.
    4. Peng Wu & Marcello Rambaldi & Jean-Franc{c}ois Muzy & Emmanuel Bacry, 2019. "Queue-reactive Hawkes models for the order flow," Papers 1901.08938, arXiv.org.
    5. Nicolas Baradel & Bruno Bouchard & David Evangelista & Othmane Mounjid, 2018. "Optimal inventory management and order book modeling," Working Papers hal-01710301, HAL.
    6. Xiaofei Lu & Frédéric Abergel, 2018. "High dimensional Hawkes processes for limit order books Modelling, empirical analysis and numerical calibration," Post-Print hal-01686122, HAL.
    7. Clinet, Simon & Yoshida, Nakahiro, 2017. "Statistical inference for ergodic point processes and application to Limit Order Book," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1800-1839.
    8. Simon Clinet, 2020. "Quasi-likelihood analysis for marked point processes and application to marked Hawkes processes," Papers 2001.11624, arXiv.org, revised Aug 2021.
    9. Fan Fang & Waichung Chung & Carmine Ventre & Michail Basios & Leslie Kanthan & Lingbo Li & Fan Wu, 2020. "Ascertaining price formation in cryptocurrency markets with DeepLearning," Papers 2003.00803, arXiv.org.
    10. Peng Wu & Marcello Rambaldi & Jean-François Muzy & Emmanuel Bacry, 2023. "A single queue-reactive Hawkes model for the order flow," Post-Print hal-02409073, HAL.
    11. Simon Clinet, 2022. "Quasi-likelihood analysis for marked point processes and application to marked Hawkes processes," Statistical Inference for Stochastic Processes, Springer, vol. 25(2), pages 189-225, July.
    12. Li, Zhicheng & Chen, Xinyun & Xing, Haipeng, 2023. "A multifactor regime-switching model for inter-trade durations in the high-frequency limit order market," Economic Modelling, Elsevier, vol. 118(C).
    13. Xiaofei Lu & Fr'ed'eric Abergel, 2018. "Order-book modelling and market making strategies," Papers 1806.05101, arXiv.org.
    14. Ulrich Horst & Dörte Kreher, 2018. "Second order approximations for limit order books," Finance and Stochastics, Springer, vol. 22(4), pages 827-877, October.
    15. Xuefeng Gao & Xiang Zhou & Lingjiong Zhu, 2017. "Transform Analysis for Hawkes Processes with Applications in Dark Pool Trading," Papers 1710.01452, arXiv.org.
    16. Fabrizio Lillo, 2021. "Order flow and price formation," Papers 2105.00521, arXiv.org.
    17. Kim, Gunhee & Choe, Geon Ho, 2019. "Limit properties of continuous self-exciting processes," Statistics & Probability Letters, Elsevier, vol. 155(C), pages 1-1.
    18. Eduard Silantyev, 2019. "Order flow analysis of cryptocurrency markets," Digital Finance, Springer, vol. 1(1), pages 191-218, November.
    19. Zhicheng Li & Haipeng Xing & Xinyun Chen, 2019. "A multifactor regime-switching model for inter-trade durations in the limit order market," Papers 1912.00764, arXiv.org.
    20. Nicolas Baradel & Bruno Bouchard & David Evangelista & Othmane Mounjid, 2018. "Optimal inventory management and order book modeling," Papers 1802.08135, arXiv.org, revised Nov 2018.
    21. Peng Wu & Marcello Rambaldi & Jean-François Muzy & Emmanuel Bacry, 2021. "Queue-reactive Hawkes models for the order flow," Working Papers hal-02409073, HAL.
    22. Ingemar Kaj & Mine Caglar, 2017. "A buffer Hawkes process for limit order books," Papers 1710.03506, arXiv.org.
    23. Ulrich Horst & Wei Xu, 2017. "A Scaling Limit for Limit Order Books Driven by Hawkes Processes," Papers 1709.01292, arXiv.org, revised Aug 2018.
    24. Maxime Morariu-Patrichi & Mikko Pakkanen, 2018. "State-dependent Hawkes processes and their application to limit order book modelling," CREATES Research Papers 2018-26, Department of Economics and Business Economics, Aarhus University.

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