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Order Book Dynamics with Liquidity Fluctuations: Asymptotic Analysis of Highly Competitive Regime

Author

Listed:
  • Helder Rojas

    (Department of Statistical Engineering, National Engineering University (UNI), Lima 15333, Peru)

  • Artem Logachov

    (Laboratory of Probability and Mathematical Statistics, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Science, 630090 Novosibirsk, Russia
    Department of Computer Science in Economics, Novosibirsk State Technical University (NSTU), 630087 Novosibirsk, Russia
    Department of Higher Mathematics, Siberian State University of Geosystems and Technologies (SSUGT), 630108 Novosibirsk, Russia)

  • Anatoly Yambartsev

    (Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo (USP), São Paulo 05508-220, Brazil)

Abstract

We introduce a class of Markov models to describe the bid–ask price dynamics in the presence of liquidity fluctuations. In a highly competitive regime, the spread evolution belongs to a class of Markov processes known as a population process with uniform catastrophes. Our mathematical analysis focuses on establishing the law of large numbers, the central limit theorem, and large deviations for this catastrophe-based model. Large deviation theory allows us to illustrate how huge deviations in the spread and prices can occur in the model. Moreover, our research highlights how these local trends and volatility are influenced by the typical values of the bid–ask spread. We calibrated the model parameters using available high-frequency data and conducted Monte Carlo numerical simulations to demonstrate its ability to reasonably replicate key phenomena in the presence of liquidity fluctuations.

Suggested Citation

  • Helder Rojas & Artem Logachov & Anatoly Yambartsev, 2023. "Order Book Dynamics with Liquidity Fluctuations: Asymptotic Analysis of Highly Competitive Regime," Mathematics, MDPI, vol. 11(20), pages 1-24, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4235-:d:1257036
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    References listed on IDEAS

    as
    1. Rama Cont & Sasha Stoikov & Rishi Talreja, 2010. "A Stochastic Model for Order Book Dynamics," Operations Research, INFORMS, vol. 58(3), pages 549-563, June.
    2. Anton Golub & John Keane & Ser-Huang Poon, 2012. "High Frequency Trading and Mini Flash Crashes," Papers 1211.6667, arXiv.org.
    3. Biais, Bruno & Hillion, Pierre & Spatt, Chester, 1995. "An Empirical Analysis of the Limit Order Book and the Order Flow in the Paris Bourse," Journal of Finance, American Finance Association, vol. 50(5), pages 1655-1689, December.
    4. Lo, Danny K. & Hall, Anthony D., 2015. "Resiliency of the limit order book," Journal of Economic Dynamics and Control, Elsevier, vol. 61(C), pages 222-244.
    5. Avellaneda, Marco & Reed, Josh & Stoikov, Sasha, 2011. "Forecasting prices from level-I quotes in the presence of hidden liquidity," Algorithmic Finance, IOS Press, vol. 1(1), pages 35-43.
    6. Lorenzo Dall’amico & Antoine Fosset & Jean-Philippe Bouchaud & Michael Benzaquen, 2019. "How does latent liquidity get revealed in the limit order book?," Post-Print hal-02323373, HAL.
    7. Lorenzo Dall’amico & Antoine Fosset & Jean-Philippe Bouchaud & Michael Benzaquen, 2019. "How does latent liquidity get revealed in the limit order book?," Post-Print hal-02283821, HAL.
    8. Logachov, A. & Logachova, O. & Yambartsev, A., 2019. "Large deviations in a population dynamics with catastrophes," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 29-37.
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    Cited by:

    1. Vladimir Glinskiy & Artem Logachov & Olga Logachova & Helder Rojas & Lyudmila Serga & Anatoly Yambartsev, 2024. "Asymptotic Properties of a Statistical Estimator of the Jeffreys Divergence: The Case of Discrete Distributions," Mathematics, MDPI, vol. 12(21), pages 1-16, October.

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