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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, 2018. "Limit order books, diffusion approximations and reflected SPDEs: from microscopic to macroscopic models," Papers 1808.07107, arXiv.org, revised Jun 2019.
    • Handle: RePEc:arx:papers:1808.07107
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    References listed on IDEAS

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    Cited by:

    1. Rama Cont & Marvin S. Mueller, 2019. "A stochastic partial differential equation model for limit order book dynamics," Papers 1904.03058, arXiv.org, revised May 2021.
    2. Rama Cont & Marvin Muller, 2019. "A Stochastic Pde Model For Limit Order Book Dynamics," Working Papers hal-02090449, HAL.

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