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Limit Order Books, Diffusion Approximations and Reflected SPDEs: From Microscopic to Macroscopic Models

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  • Ben Hambly
  • Jasdeep Kalsi
  • James Newbury

Abstract

Motivated by a zero-intelligence approach, the aim of this paper is to connect the microscopic (discrete price and volume), mesoscopic (discrete price and continuous volume) and macroscopic (continuous price and volume) frameworks for the modelling of limit order books, with a view to providing a natural probabilistic description of their behaviour in a high- to ultra high-frequency setting. Starting with a microscopic framework, we first examine the limiting behaviour of the order book process when order arrival and cancellation rates are sent to infinity and when volumes are considered to be of infinitesimal size. We then consider the transition between this mesoscopic model and a macroscopic model for the limit order book, obtained by letting the tick size tend to zero. The macroscopic limit can then be described using reflected SPDEs which typically arise in stochastic interface models. We then use financial data to discuss a possible calibration procedure for the model and illustrate numerically how it can reproduce observed behaviour of prices. This could then be used as a market simulator for short-term price prediction or for testing optimal execution strategies.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:apmtfi:v:27:y:2020:i:1-2:p:132-170
    DOI: 10.1080/1350486X.2020.1758176
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    Citations

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    Cited by:

    1. Rama Cont & Pierre Degond & Xuan Lifan, 2023. "A mathematical framework for modelling order book dynamics," Working Papers hal-03968767, HAL.
    2. Aleksejus Kononovicius & Rytis Kazakeviv{c}ius & Bronislovas Kaulakys, 2022. "Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processes," Papers 2205.07563, arXiv.org, revised Jul 2022.
    3. Kononovicius, Aleksejus & Kazakevičius, Rytis & Kaulakys, Bronislovas, 2022. "Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processes," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    4. Zacharia Issa & Blanka Horvath & Maud Lemercier & Cristopher Salvi, 2023. "Non-adversarial training of Neural SDEs with signature kernel scores," Papers 2305.16274, arXiv.org.
    5. Johannes Muhle-Karbe & Eyal Neuman & Yonatan Shadmi, 2024. "Fluid-Limits of Fragmented Limit-Order Markets," Papers 2407.04354, arXiv.org.
    6. Ulrich Horst & Dorte Kreher & Konstantins Starovoitovs, 2023. "Second-Order Approximation of Limit Order Books in a Single-Scale Regime," Papers 2308.00805, arXiv.org, revised Sep 2024.
    7. Konark Jain & Nick Firoozye & Jonathan Kochems & Philip Treleaven, 2024. "Limit Order Book Simulations: A Review," Papers 2402.17359, arXiv.org, revised Mar 2024.
    8. Rama Cont & Pierre Degond & Lifan Xuan, 2023. "A mathematical framework for modelling order book dynamics," Papers 2302.01169, arXiv.org.

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