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Order book dynamics in liquid markets: limit theorems and diffusion approximations


  • Rama Cont
  • Adrien De Larrard


We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the underlying order flow. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family.

Suggested Citation

  • Rama Cont & Adrien De Larrard, 2012. "Order book dynamics in liquid markets: limit theorems and diffusion approximations," Papers 1202.6412,
  • Handle: RePEc:arx:papers:1202.6412

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    References listed on IDEAS

    1. Eric Smith & J Doyne Farmer & Laszlo Gillemot & Supriya Krishnamurthy, 2003. "Statistical theory of the continuous double auction," Quantitative Finance, Taylor & Francis Journals, vol. 3(6), pages 481-514.
    2. Ghysels Eric & Jasiak Joanna, 1998. "GARCH for Irregularly Spaced Financial Data: The ACD-GARCH Model," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 2(4), pages 1-19, January.
    3. Parameswaran Gopikrishnan & Vasiliki Plerou & Xavier Gabaix & H. Eugene Stanley, 2000. "Statistical Properties of Share Volume Traded in Financial Markets," Papers cond-mat/0008113,
    4. 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.
    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. 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.
    7. Zhou, Chunsheng, 2001. "An Analysis of Default Correlations and Multiple Defaults," Review of Financial Studies, Society for Financial Studies, vol. 14(2), pages 555-576.
    8. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2006. "A Limit Theorem for Financial Markets with Inert Investors," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 789-810, November.
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    Cited by:

    1. Xin Liu & Qi Gong & Vidyadhar G. Kulkarni, 2015. "A Stochastic Model of Order Book Dynamics using Bouncing Geometric Brownian Motions," Papers 1511.04096,, revised Mar 2016.
    2. repec:eee:phsmap:v:485:y:2017:i:c:p:61-72 is not listed on IDEAS
    3. Ulrich Horst & Dorte Kreher, 2017. "Second order approximations for limit order books," Papers 1708.07394,, revised Mar 2018.
    4. Xin Guo & Zhao Ruan & Lingjiong Zhu, 2015. "Dynamics of Order Positions and Related Queues in a Limit Order Book," Papers 1505.04810,, revised Oct 2015.
    5. Vadim Kaushansky & Alexander Lipton & Christoph Reisinger, 2017. "Transition probability of Brownian motion in the octant and its application to default modeling," Papers 1801.00362,
    6. repec:eee:spapps:v:127:y:2017:i:8:p:2447-2481 is not listed on IDEAS
    7. Nicolas Baradel & Bruno Bouchard & David Evangelista & Othmane Mounjid, 2018. "Optimal inventory management and order book modeling," Working Papers hal-01710301, HAL.
    8. Tzu-Wei Yang & Lingjiong Zhu, 2015. "A reduced-form model for level-1 limit order books," Papers 1508.07891,, revised Nov 2016.
    9. Alexander Lykov & Stepan Muzychka & Kirill Vaninsky, 2016. "Investor'S Sentiment In Multi-Agent Model Of The Continuous Double Auction," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(06), pages 1-29, September.
    10. Nicolas Baradel & Bruno Bouchard & David Evangelista & Othmane Mounjid, 2018. "Optimal inventory management and order book modeling," Papers 1802.08135,
    11. Justin Sirignano, 2016. "Deep Learning for Limit Order Books," Papers 1601.01987,, revised Jul 2016.
    12. Ioane Muni Toke, 2015. "Stationary distribution of the volume at the best quote in a Poisson order book model," Papers 1502.03871,
    13. repec:eee:spapps:v:127:y:2017:i:6:p:1800-1839 is not listed on IDEAS

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