IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-02090449.html
   My bibliography  Save this paper

A Stochastic Pde Model For Limit Order Book Dynamics

Author

Listed:
  • Rama Cont

    (Department of Mathematics [Imperial College London] - Imperial College London, LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - UPD7 - Université Paris Diderot - Paris 7 - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique)

  • Marvin Muller

    (ETH Zürich - Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich])

Abstract

We propose an analytically tractable class of models for the dynamics of a limit order book, described as the solution of a stochastic partial differential equation (SPDE) with multiplicative noise. We provide conditions under which the model admits a finite dimensional realization driven by a (low-dimensional) Markov process, leading to efficient methods for estimation and computation. We study two examples of parsimonious models in this class: a two-factor model and a model in which the order book depth is mean-reverting. For each model we perform a detailed analysis of the role of different parameters, study the dynamics of the price, order book depth, volume and order imbalance, provide an intuitive financial interpretation of the variables involved and show how the model reproduces statistical properties of price changes, market depth and order flow in limit order markets.

Suggested Citation

  • Rama Cont & Marvin Muller, 2019. "A Stochastic Pde Model For Limit Order Book Dynamics," Working Papers hal-02090449, HAL.
    • Handle: RePEc:hal:wpaper:hal-02090449
    Note: View the original document on HAL open archive server: https://hal.science/hal-02090449v1
    as

    Download full text from publisher

    File URL: https://hal.science/hal-02090449v1/document
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Rama Cont & Arseniy Kukanov & Sasha Stoikov, 2014. "The Price Impact of Order Book Events," Journal of Financial Econometrics, Oxford University Press, vol. 12(1), pages 47-88.
    2. Julie Lyng Forman & Michael Sørensen, 2008. "The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 438-465, September.
    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. Ulrich Horst & Dorte Kreher, 2017. "Second order approximations for limit order books," Papers 1708.07394, arXiv.org, revised Mar 2018.
    5. Hugh Luckock, 2003. "A steady-state model of the continuous double auction," Quantitative Finance, Taylor & Francis Journals, vol. 3(5), pages 385-404.
    6. 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.
    7. 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.
    8. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
    9. Rama Cont & Adrien De Larrard, 2012. "Order book dynamics in liquid markets: limit theorems and diffusion approximations," Papers 1202.6412, arXiv.org.
    10. Andrei Kirilenko & Albert S. Kyle & Mehrdad Samadi & Tugkan Tuzun, 2017. "The Flash Crash: High-Frequency Trading in an Electronic Market," Journal of Finance, American Finance Association, vol. 72(3), pages 967-998, June.
    11. Rama Cont & Adrien de Larrard, 2013. "Price Dynamics in a Markovian Limit Order Market," Post-Print hal-00552252, HAL.
    12. Frank Kelly & Elena Yudovina, 2018. "A Markov Model of a Limit Order Book: Thresholds, Recurrence, and Trading Strategies," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 181-203, February.
    13. Álvaro Cartea & Ryan Donnelly & Sebastian Jaimungal, 2018. "Enhancing trading strategies with order book signals," Applied Mathematical Finance, Taylor & Francis Journals, vol. 25(1), pages 1-35, January.
    14. Ulrich Horst & Dörte Kreher, 2018. "Second order approximations for limit order books," Finance and Stochastics, Springer, vol. 22(4), pages 827-877, October.
    15. Chávez-Casillas, Jonathan A. & Figueroa-López, José E., 2017. "A one-level limit order book model with memory and variable spread," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2447-2481.
    16. Rene Carmona & Kevin Webster, 2013. "The Self-Financing Equation in High Frequency Markets," Papers 1312.2302, arXiv.org.
    17. Rama Cont, 2005. "Modeling Term Structure Dynamics: An Infinite Dimensional Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(03), pages 357-380.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. Cassandra Milbradt & Dorte Kreher, 2022. "A cross-border market model with limited transmission capacities," Papers 2207.01939, arXiv.org, revised Nov 2024.
    3. Konark Jain & Nick Firoozye & Jonathan Kochems & Philip Treleaven, 2024. "Limit Order Book Simulations: A Review," Papers 2402.17359, arXiv.org, revised Mar 2024.
    4. Philippe Bergault & Enzo Cogn'eville, 2024. "Simulating and analyzing a sparse order book: an application to intraday electricity markets," Papers 2410.06839, arXiv.org.
    5. Chávez-Casillas, Jonathan A. & Figueroa-López, José E., 2017. "A one-level limit order book model with memory and variable spread," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2447-2481.
    6. Zijian Shi & John Cartlidge, 2024. "Neural stochastic agent‐based limit order book simulation with neural point process and diffusion probabilistic model," Intelligent Systems in Accounting, Finance and Management, John Wiley & Sons, Ltd., vol. 31(2), June.
    7. Johannes Bleher & Michael Bleher & Thomas Dimpfl, 2020. "From orders to prices: A stochastic description of the limit order book to forecast intraday returns," Papers 2004.11953, arXiv.org, revised May 2021.
    8. Anatoliy Swishchuk & Nelson Vadori, 2016. "A Semi-Markovian Modeling of Limit Order Markets," Papers 1601.01710, arXiv.org.
    9. Mohammad Zare & Omid Naghshineh Arjmand & Erfan Salavati & Adel Mohammadpour, 2021. "An Agent‐Based model for Limit Order Book: Estimation and simulation," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(1), pages 1112-1121, January.
    10. 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.
    11. Justin Sirignano & Rama Cont, 2018. "Universal features of price formation in financial markets: perspectives from Deep Learning," Papers 1803.06917, arXiv.org.
    12. Antoine Jacquier & Hao Liu, 2017. "Optimal liquidation in a Level-I limit order book for large tick stocks," Papers 1701.01327, arXiv.org, revised Nov 2017.
    13. Justin Sirignano & Rama Cont, 2018. "Universal features of price formation in financial markets: perspectives from Deep Learning," Working Papers hal-01754054, HAL.
    14. Nicolas Baradel & Bruno Bouchard & David Evangelista & Othmane Mounjid, 2019. "Optimal inventory management and order book modeling," Post-Print hal-01710301, HAL.
    15. M. Derksen & B. Kleijn & R. de Vilder, 2019. "Clearing price distributions in call auctions," Papers 1904.07583, arXiv.org, revised Nov 2019.
    16. Marcello Rambaldi & Emmanuel Bacry & Jean-Franc{c}ois Muzy, 2018. "Disentangling and quantifying market participant volatility contributions," Papers 1807.07036, arXiv.org.
    17. Ulrich Horst & Michael Paulsen, 2017. "A Law of Large Numbers for Limit Order Books," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1280-1312, November.
    18. Antonio Briola & Silvia Bartolucci & Tomaso Aste, 2024. "Deep Limit Order Book Forecasting," Papers 2403.09267, arXiv.org, revised Jun 2024.
    19. Matthew F Dixon, 2017. "A High Frequency Trade Execution Model for Supervised Learning," Papers 1710.03870, arXiv.org, revised Dec 2017.
    20. 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.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:wpaper:hal-02090449. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.