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Ergodicity and diffusivity of Markovian order book models: a general framework

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  • Weibing Huang
  • Mathieu Rosenbaum

Abstract

We present a general Markovian framework for order book modeling. Through our approach, we aim at providing a tool enabling to get a better understanding of the price formation process and of the link between microscopic and macroscopic features of financial assets. To do so, we propose a new method of order book representation, and decompose the problem of order book modeling into two sub-problems: dynamics of a continuous-time double auction system with a fixed reference price; interactions between the double auction system and the reference price movements. State dependency is included in our framework by allowing the order flow intensities to depend on the order book state. Furthermore, contrary to most existing models, the impact of the order book updates on the reference price dynamics is not assumed to be instantaneous. We first prove that under general assumptions, our system is ergodic. Then we deduce the convergence towards a Brownian motion of the rescaled price process.

Suggested Citation

  • Weibing Huang & Mathieu Rosenbaum, 2015. "Ergodicity and diffusivity of Markovian order book models: a general framework," Papers 1505.04936, arXiv.org.
    • Handle: RePEc:arx:papers:1505.04936
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    References listed on IDEAS

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    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. Frédéric Abergel & Anirban Chakraborti & B.K. Chakrabarti & M. Mitra, 2011. "Econophysics of order-driven markets," Post-Print hal-00872396, HAL.
    3. Foucault, Thierry, 1999. "Order flow composition and trading costs in a dynamic limit order market1," Journal of Financial Markets, Elsevier, vol. 2(2), pages 99-134, May.
    4. Rama Cont & Adrien de Larrard, 2013. "Price Dynamics in a Markovian Limit Order Market," Post-Print hal-00552252, HAL.
    5. Aim'e Lachapelle & Jean-Michel Lasry & Charles-Albert Lehalle & Pierre-Louis Lions, 2013. "Efficiency of the Price Formation Process in Presence of High Frequency Participants: a Mean Field Game analysis," Papers 1305.6323, arXiv.org, revised Aug 2015.
    6. Rama Cont & Sasha Stoikov & Rishi Talreja, 2010. "A Stochastic Model for Order Book Dynamics," Operations Research, INFORMS, vol. 58(3), pages 549-563, June.
    7. Parlour, Christine A, 1998. "Price Dynamics in Limit Order Markets," The Review of Financial Studies, Society for Financial Studies, vol. 11(4), pages 789-816.
    8. Christian Bayer & Ulrich Horst & Jinniao Qiu, 2014. "A Functional Limit Theorem for Limit Order Books with State Dependent Price Dynamics," Papers 1405.5230, arXiv.org, revised Aug 2016.
    9. Weibing Huang & Charles-Albert Lehalle & Mathieu Rosenbaum, 2015. "Simulating and Analyzing Order Book Data: The Queue-Reactive Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 107-122, March.
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    Citations

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    Cited by:

    1. Ulrich Horst & Dorte Kreher, 2017. "Second order approximations for limit order books," Papers 1708.07394, arXiv.org, revised Mar 2018.
    2. Clinet, Simon & Yoshida, Nakahiro, 2017. "Statistical inference for ergodic point processes and application to Limit Order Book," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1800-1839.
    3. Nicolas Baradel & Bruno Bouchard & David Evangelista & Othmane Mounjid, 2018. "Optimal inventory management and order book modeling," Working Papers hal-01710301, HAL.
    4. Maxime Morariu-Patrichi & Mikko S. Pakkanen, 2017. "Hybrid marked point processes: characterisation, existence and uniqueness," Papers 1707.06970, arXiv.org, revised Oct 2018.
    5. Ioane Muni Toke & Nakahiro Yoshida, 2017. "Modelling intensities of order flows in a limit order book," Quantitative Finance, Taylor & Francis Journals, vol. 17(5), pages 683-701, May.
    6. Nicolas Baradel & Bruno Bouchard & David Evangelista & Othmane Mounjid, 2018. "Optimal inventory management and order book modeling," Papers 1802.08135, arXiv.org, revised Nov 2018.
    7. Ioane Muni Toke & Nakahiro Yoshida, 2017. "Modelling intensities of order flows in a limit order book," Post-Print hal-01705080, HAL.

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