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Second-Order Approximation of Limit Order Books in a Single-Scale Regime

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  • Ulrich Horst
  • Dorte Kreher
  • Konstantins Starovoitovs

Abstract

We establish a first and second-order approximation for an infinite dimensional limit order book model (LOB) in a single (''critical'') scaling regime where market and limit orders arrive at a common time scale. With our choice of scaling we obtain non-degenerate first-order and second-order approximations for the price and volume dynamics. While the first-order approximation is given by a standard coupled ODE-PDE system, the second-order approximation is non-standard and described in terms of an infinite-dimensional stochastic evolution equation driven by a cylindrical Brownian motion. The driving noise processes exhibit a non-trivial correlation in terms of the model parameters. We prove that the evolution equation has a unique solution and that the sequence of standardized LOB models converges weakly to the solution of the evolution equation. The proof uses a non-standard martingale problem. We calibrate a simplified version of our model to market data and show that the model accurately captures correlations between price and volume fluctuations.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:2308.00805
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    References listed on IDEAS

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    1. Jose Blanchet & Xinyun Chen, 2013. "Continuous-time Modeling of Bid-Ask Spread and Price Dynamics in Limit Order Books," Papers 1310.1103, arXiv.org.
    2. 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.
    3. Rama Cont & Sasha Stoikov & Rishi Talreja, 2010. "A Stochastic Model for Order Book Dynamics," Operations Research, INFORMS, vol. 58(3), pages 549-563, June.
    4. Cebiroğlu, Gökhan & Horst, Ulrich, 2015. "Optimal order display in limit order markets with liquidity competition," Journal of Economic Dynamics and Control, Elsevier, vol. 58(C), pages 81-100.
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