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A Law of Large Numbers for Limit Order Books

Author

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  • Ulrich Horst

    (Institut für Mathematik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany)

  • Michael Paulsen

    (Institut für Mathematik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany)

Abstract

We define a stochastic model of a two-sided limit order book in terms of its key quantities best bid [ ask ] price and the standing buy [ sell ] volume density . For a simple scaling of the discreteness parameters, that keeps the expected volume rate over the considered price interval invariant, we prove a limit theorem. The limit theorem states that, given regularity conditions on the random order flow, the key quantities converge in probability to a tractable continuous limiting model. In the limit model the buy and sell volume densities are given as the unique solution to first-order linear hyperbolic PDEs, specified by the expected order flow parameters. We calibrate order flow dynamics to market data for selected stocks and show how our model can be used to derive endogenous shape functions for models of optimal portfolio liquidation under market impact.

Suggested Citation

  • Ulrich Horst & Michael Paulsen, 2017. "A Law of Large Numbers for Limit Order Books," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1280-1312, November.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:4:p:1280-1312
    DOI: 10.1287/moor.2017.0848
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    References listed on IDEAS

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    Cited by:

    1. Ben Hambly & Jasdeep Kalsi & James Newbury, 2018. "Limit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models," Papers 1808.07107, arXiv.org, revised Jun 2019.
    2. Rama Cont & Pierre Degond & Xuan Lifan, 2023. "A mathematical framework for modelling order book dynamics," Working Papers hal-03968767, HAL.
    3. Cassandra Milbradt & Dorte Kreher, 2022. "A cross-border market model with limited transmission capacities," Papers 2207.01939, arXiv.org, revised May 2023.
    4. Donatien Hainaut & Franck Moraux, 2019. "A switching self-exciting jump diffusion process for stock prices," Annals of Finance, Springer, vol. 15(2), pages 267-306, June.
    5. Ulrich Horst & Dörte Kreher, 2018. "Second order approximations for limit order books," Finance and Stochastics, Springer, vol. 22(4), pages 827-877, October.
    6. Hainaut, Donatien & Goutte, Stephane, 2018. "A switching microstructure model for stock prices," LIDAM Discussion Papers ISBA 2018014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Rama Cont & Pierre Degond & Lifan Xuan, 2023. "A mathematical framework for modelling order book dynamics," Papers 2302.01169, arXiv.org.

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