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A mathematical framework for modelling order book dynamics

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  • Rama Cont
  • Pierre Degond
  • Lifan Xuan

Abstract

We present a general framework for modelling the dynamics of limit order books, built on the combination of two modelling ingredients: the order flow, modelled as a general spatial point process, and market clearing, modelled via a deterministic mass transport operator acting on distributions of buy and sell orders. At the mathematical level, this corresponds to a natural decomposition of the infinitesimal generator describing the evolution of the limit order book into two operators: the generator of the order flow and the clearing operator. Our model provides a flexible framework for modelling and simulating order book dynamics and studying various scaling limits of discrete order book models. We show that our framework includes previous models as special cases and yields insights into the interplay between order flow and price dynamics.

Suggested Citation

  • Rama Cont & Pierre Degond & Lifan Xuan, 2023. "A mathematical framework for modelling order book dynamics," Papers 2302.01169, arXiv.org.
    • Handle: RePEc:arx:papers:2302.01169
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    References listed on IDEAS

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    1. Rama Cont & Adrien De Larrard, 2012. "Order book dynamics in liquid markets: limit theorems and diffusion approximations," Papers 1202.6412, arXiv.org.
    2. Jean-Philippe Bouchaud & Marc Mezard & Marc Potters, 2002. "Statistical properties of stock order books: empirical results and models," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 251-256.
    3. Ben Hambly & Jasdeep Kalsi & James Newbury, 2020. "Limit Order Books, Diffusion Approximations and Reflected SPDEs: From Microscopic to Macroscopic Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(1-2), pages 132-170, July.
    4. Rama Cont & Sasha Stoikov & Rishi Talreja, 2010. "A Stochastic Model for Order Book Dynamics," Operations Research, INFORMS, vol. 58(3), pages 549-563, June.
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    6. Hugh Luckock, 2003. "A steady-state model of the continuous double auction," Quantitative Finance, Taylor & Francis Journals, vol. 3(5), pages 385-404.
    7. 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.
    8. Jean-Philippe Bouchaud & Marc Mezard & Marc Potters, 2002. "Statistical properties of stock order books: empirical results and models," Science & Finance (CFM) working paper archive 0203511, Science & Finance, Capital Fund Management.
    9. Weibing Huang & Charles-Albert Lehalle & Mathieu Rosenbaum, 2015. "Simulating and Analyzing Order Book Data: The Queue-Reactive Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 107-122, March.
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