IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v42y2017i4p1280-1312.html
   My bibliography  Save this article

A Law of Large Numbers for Limit Order Books

Author

Listed:
  • Ulrich Horst

    (Institut für Mathematik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany)

  • Michael Paulsen

    (Institut für Mathematik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany)

Abstract

We define a stochastic model of a two-sided limit order book in terms of its key quantities best bid [ ask ] price and the standing buy [ sell ] volume density . For a simple scaling of the discreteness parameters, that keeps the expected volume rate over the considered price interval invariant, we prove a limit theorem. The limit theorem states that, given regularity conditions on the random order flow, the key quantities converge in probability to a tractable continuous limiting model. In the limit model the buy and sell volume densities are given as the unique solution to first-order linear hyperbolic PDEs, specified by the expected order flow parameters. We calibrate order flow dynamics to market data for selected stocks and show how our model can be used to derive endogenous shape functions for models of optimal portfolio liquidation under market impact.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:4:p:1280-1312
    DOI: 10.1287/moor.2017.0848
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/moor.2017.0848
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.2017.0848?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Thierry Foucault & Ohad Kadan & Eugene Kandel, 2005. "Limit Order Book as a Market for Liquidity," The Review of Financial Studies, Society for Financial Studies, vol. 18(4), pages 1171-1217.
    2. Horst, Ulrich & Rothe, Christian, 2008. "Queuing, Social Interactions, And The Microstructure Of Financial Markets," Macroeconomic Dynamics, Cambridge University Press, vol. 12(2), pages 211-233, April.
    3. Glosten, Lawrence R. & Milgrom, Paul R., 1985. "Bid, ask and transaction prices in a specialist market with heterogeneously informed traders," Journal of Financial Economics, Elsevier, vol. 14(1), pages 71-100, March.
    4. Rama Cont & Adrien De Larrard, 2012. "Order book dynamics in liquid markets: limit theorems and diffusion approximations," Papers 1202.6412, arXiv.org.
    5. Rama Cont & Adrien de Larrard, 2013. "Price Dynamics in a Markovian Limit Order Market," Post-Print hal-00552252, HAL.
    6. Ulrich Horst & Dorte Kreher, 2016. "A diffusion approximation for limit order book models," Papers 1608.01795, arXiv.org, revised Aug 2017.
    7. Aur'elien Alfonsi & Antje Fruth & Alexander Schied, 2007. "Optimal execution strategies in limit order books with general shape functions," Papers 0708.1756, arXiv.org, revised Feb 2010.
    8. Aurelien Alfonsi & Antje Fruth & Alexander Schied, 2010. "Optimal execution strategies in limit order books with general shape functions," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 143-157.
    9. Easley, David & O'Hara, Maureen, 1987. "Price, trade size, and information in securities markets," Journal of Financial Economics, Elsevier, vol. 19(1), pages 69-90, September.
    10. Rama Cont & Sasha Stoikov & Rishi Talreja, 2010. "A Stochastic Model for Order Book Dynamics," Operations Research, INFORMS, vol. 58(3), pages 549-563, June.
    11. 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.
    12. 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.
    13. Garman, Mark B., 1976. "Market microstructure," Journal of Financial Economics, Elsevier, vol. 3(3), pages 257-275, June.
    14. Lo, Andrew W. & MacKinlay, A. Craig & Zhang, June, 2002. "Econometric models of limit-order executions," Journal of Financial Economics, Elsevier, vol. 65(1), pages 31-71, July.
    15. Ioanid Rosu, 2009. "A Dynamic Model of the Limit Order Book," Post-Print hal-00515873, HAL.
    16. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "Queueing Theoretic Approaches to Financial Price Fluctuations," Papers math/0703832, arXiv.org.
    17. 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.
    18. Frédéric Abergel & Aymen Jedidi, 2013. "A Mathematical Approach To Order Book Modeling," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(05), pages 1-40.
    19. Łukasz Kruk, 2003. "Functional Limit Theorems for a Simple Auction," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 716-751, November.
    20. Ioanid Rosu, 2009. "A Dynamic Model of the Limit Order Book," The Review of Financial Studies, Society for Financial Studies, vol. 22(11), pages 4601-4641, November.
    21. Frederic Abergel & Aymen Jedidi, 2010. "A Mathematical Approach to Order Book Modeling," Papers 1010.5136, arXiv.org, revised Mar 2013.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. Rama Cont & Pierre Degond & Xuan Lifan, 2023. "A mathematical framework for modelling order book dynamics," Working Papers hal-03968767, HAL.
    3. Cassandra Milbradt & Dorte Kreher, 2022. "A cross-border market model with limited transmission capacities," Papers 2207.01939, arXiv.org, revised May 2023.
    4. Michael Giegrich & Roel Oomen & Christoph Reisinger, 2024. "Limit Order Book Simulation and Trade Evaluation with $K$-Nearest-Neighbor Resampling," Papers 2409.06514, arXiv.org.
    5. Donatien Hainaut & Franck Moraux, 2019. "A switching self-exciting jump diffusion process for stock prices," Annals of Finance, Springer, vol. 15(2), pages 267-306, June.
    6. Ulrich Horst & Dörte Kreher, 2018. "Second order approximations for limit order books," Finance and Stochastics, Springer, vol. 22(4), pages 827-877, October.
    7. Hainaut, Donatien & Goutte, Stephane, 2018. "A switching microstructure model for stock prices," LIDAM Discussion Papers ISBA 2018014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. Philippe Bergault & Enzo Cogn'eville, 2024. "Simulating and analyzing a sparse order book: an application to intraday electricity markets," Papers 2410.06839, arXiv.org.
    9. Konark Jain & Nick Firoozye & Jonathan Kochems & Philip Treleaven, 2024. "Limit Order Book Simulations: A Review," Papers 2402.17359, arXiv.org, revised Mar 2024.
    10. Rama Cont & Pierre Degond & Lifan Xuan, 2023. "A mathematical framework for modelling order book dynamics," Papers 2302.01169, arXiv.org.

    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. Ulrich Horst & Michael Paulsen, 2015. "A law of large numbers for limit order books," Papers 1501.00843, arXiv.org.
    2. 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.
    3. Martin D. Gould & Mason A. Porter & Stacy Williams & Mark McDonald & Daniel J. Fenn & Sam D. Howison, 2010. "Limit Order Books," Papers 1012.0349, arXiv.org, revised Apr 2013.
    4. Danny Lo, 2015. "Essays in Market Microstructure and Investor Trading," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2015, January-A.
    5. Danny Lo, 2015. "Essays in Market Microstructure and Investor Trading," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 22, July-Dece.
    6. Martin D. Gould & Mason A. Porter & Stacy Williams & Mark McDonald & Daniel J. Fenn & Sam D. Howison, 2013. "Limit order books," Quantitative Finance, Taylor & Francis Journals, vol. 13(11), pages 1709-1742, November.
    7. Dimitri Vayanos & Jiang Wang, 2012. "Market Liquidity -- Theory and Empirical Evidence," NBER Working Papers 18251, National Bureau of Economic Research, Inc.
    8. Vayanos, Dimitri & Wang, Jiang, 2013. "Market Liquidity—Theory and Empirical Evidence ," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, volume 2, chapter 0, pages 1289-1361, Elsevier.
    9. 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.
    10. Xin Guo & Zhao Ruan & Lingjiong Zhu, 2015. "Dynamics of Order Positions and Related Queues in a Limit Order Book," Papers 1505.04810, arXiv.org, revised Oct 2015.
    11. M. Derksen & B. Kleijn & R. de Vilder, 2019. "Clearing price distributions in call auctions," Papers 1904.07583, arXiv.org, revised Nov 2019.
    12. 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.
    13. Jonathan A. Ch'avez-Casillas & Jos'e E. Figueroa-L'opez, 2014. "One-level limit order book models with memory and variable spread," Papers 1407.5684, arXiv.org, revised Mar 2016.
    14. Havran, Dániel & Erb, Tamás, 2015. "Mit veszítünk a piaci súrlódásokkal?. A pénzügyi piacok mikrostruktúrája [Trading mechanisms and market frictions. Microstructure of the financial markets]," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(3), pages 229-262.
    15. Ulrich Horst & Dörte Kreher, 2018. "Second order approximations for limit order books," Finance and Stochastics, Springer, vol. 22(4), pages 827-877, October.
    16. Roberto Pascual & David Veredas, 2010. "Does the Open Limit Order Book Matter in Explaining Informational Volatility?," Journal of Financial Econometrics, Oxford University Press, vol. 8(1), pages 57-87, Winter.
    17. Xin Liu & Qi Gong & Vidyadhar G. Kulkarni, 2015. "A Stochastic Model of Order Book Dynamics using Bouncing Geometric Brownian Motions," Papers 1511.04096, arXiv.org, revised Mar 2016.
    18. Kovaleva, P. & Iori, G., 2012. "Optimal Trading Strategies in a Limit Order Market with Imperfect Liquidity," Working Papers 12/05, Department of Economics, City University London.
    19. Tseng, Yi-Heng & Chen, Shu-Heng, 2015. "Limit order book transparency and order aggressiveness at the closing call: Lessons from the TWSE 2012 new information disclosure mechanism," Pacific-Basin Finance Journal, Elsevier, vol. 35(PA), pages 241-272.
    20. Marcello Rambaldi & Emmanuel Bacry & Jean-Franc{c}ois Muzy, 2018. "Disentangling and quantifying market participant volatility contributions," Papers 1807.07036, arXiv.org.

    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:inm:ormoor:v:42:y:2017:i:4:p:1280-1312. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.