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Exact and asymptotic solutions of the call auction problem

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  • Ioane Muni Toke

    (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris, FiQuant - Chaire de finance quantitative - MICS - Mathématiques et Informatique pour la Complexité et les Systèmes - CentraleSupélec, ERIM - Equipe de Recherche en Informatique et Mathématiques - UNC - Université de la Nouvelle-Calédonie)

Abstract

The call auction is a widely used trading mechanism, especially during the opening and closing periods of financial markets. In this paper, we study a standard call auction problem where orders are submitted according to Poisson processes, with random prices distributed according to a general distribution F, and may be cancelled at any time. We compute the analytical expressions of the distributions of the traded volume, of the lower and upper bounds of the clearing prices, and of the price range of these possible clearing prices of the call auction. Using results from the theory of orders statistics and a theorem on the limit of sequences of random variables with independent random indices, we derive the weak limits of all these distributions. In this setting, traded volume and bounds of the clearing prices are found to be asymptotically normal, while the clearing price range is asymptotically exponential. All the parameters of these distributions are explicitly derived as functions of the parameters of the incoming orders' flows.

Suggested Citation

  • Ioane Muni Toke, 2015. "Exact and asymptotic solutions of the call auction problem," Post-Print hal-01061857, HAL.
    • Handle: RePEc:hal:journl:hal-01061857
    DOI: 10.1142/s238262661550001x
    Note: View the original document on HAL open archive server: https://hal.science/hal-01061857
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    References listed on IDEAS

    as
    1. Comerton-Forde, Carole & Rydge, James, 2006. "Call auction algorithm design and market manipulation," Journal of Multinational Financial Management, Elsevier, vol. 16(2), pages 184-198, April.
    2. 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.
    3. Anirban Chakraborti & Ioane Muni Toke & Marco Patriarca & Frederic Abergel, 2011. "Econophysics review: I. Empirical facts," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 991-1012.
    4. 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.
    5. Potters, Marc & Bouchaud, Jean-Philippe, 2003. "More statistical properties of order books and price impact," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 133-140.
    6. Mike, Szabolcs & Farmer, J. Doyne, 2008. "An empirical behavioral model of liquidity and volatility," Journal of Economic Dynamics and Control, Elsevier, vol. 32(1), pages 200-234, January.
    7. Rama Cont & Sasha Stoikov & Rishi Talreja, 2010. "A Stochastic Model for Order Book Dynamics," Operations Research, INFORMS, vol. 58(3), pages 549-563, June.
    8. 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.
    9. 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.
    10. Comerton-Forde, Carole & Rydge, James, 2006. "The influence of call auction algorithm rules on market efficiency," Journal of Financial Markets, Elsevier, vol. 9(2), pages 199-222, May.
    11. Chan, K C & Christie, William G & Schultz, Paul H, 1995. "Market Structure and the Intraday Pattern of Bid-Ask Spreads for NASDAQ Securities," The Journal of Business, University of Chicago Press, vol. 68(1), pages 35-60, January.
    12. G.-F. Gu & W. Chen & W.-X. Zhou, 2007. "Quantifying bid-ask spreads in the Chinese stock market using limit-order book data," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 57(1), pages 81-87, May.
    13. 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.
    14. Anirban Chakraborti & Ioane Muni Toke & Marco Patriarca & Frédéric Abergel, 2011. "Econophysics review: I. Empirical facts," Post-Print hal-00621058, HAL.
    15. Mendelson, Haim, 1982. "Market Behavior in a Clearing House," Econometrica, Econometric Society, vol. 50(6), pages 1505-1524, November.
    16. Pagano, Michael S. & Schwartz, Robert A., 2003. "A closing call's impact on market quality at Euronext Paris," Journal of Financial Economics, Elsevier, vol. 68(3), pages 439-484, June.
    17. Domowitz, Ian & Wang, Jianxin, 1994. "Auctions as algorithms : Computerized trade execution and price discovery," Journal of Economic Dynamics and Control, Elsevier, vol. 18(1), pages 29-60, January.
    18. Frédéric Abergel & Aymen Jedidi, 2013. "A Mathematical Approach to Order Book Modelling," Post-Print hal-00621253, HAL.
    19. Frederic Abergel & Aymen Jedidi, 2010. "A Mathematical Approach to Order Book Modeling," Papers 1010.5136, arXiv.org, revised Mar 2013.
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    Cited by:

    1. Mike Derksen & Bas Kleijn & Robin de Vilder, 2020. "Effects of MiFID II on stock price formation," Papers 2003.10353, arXiv.org, revised Aug 2020.
    2. M. Derksen & B. Kleijn & R. de Vilder, 2019. "Clearing price distributions in call auctions," Papers 1904.07583, arXiv.org, revised Nov 2019.
    3. Twu, Mia & Wang, Jianxin, 2018. "Call auction frequency and market quality: Evidence from the Taiwan Stock Exchange," Journal of Asian Economics, Elsevier, vol. 57(C), pages 53-62.

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