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Price Dynamics on a Stock Market with Asymmetric Information

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  • Bernard De Meyer

    (Centre d'Economie de la Sorbonne, University of Paris)

Abstract

The appearance of a Brownian term in the price dynamics on a stock market was interpreted in [De Meyer, Moussa-Saley (2003)] as a consequence of the informational asymmetries between agents. To take benefit of their private information without revealing it to fast, the informed agents have to introduce a noise on their actions, and all these noises introduced in the day after day transactions for strategic reasons will aggregate in a Brownian Motion. We prove in the present paper that this kind of argument leads not only to the appearance of the Brownian motion, but it also narrows the class of the price dynamics: the price process will be, as defined in this paper, a continuous martingale of maximal variation. This class of dynamics contains in particular Black and Scholes' as well as Bachelier's dynamics. The main result in this paper is that this class is quite universal and independent of a particular model: the informed agent can choose the speed of revelation of his private information. He determines in this way the posterior martingale L, where L_{q} is the expected value of an asset at stage q given the information of the uninformed agents. The payoff of the informed agent at stage q can typically be expressed as a 1-homogeneous function M of L_{q+1}-L_{q}. In a game with n stages, the informed agent will therefore chose the martingale L? that maximizes the M-variation. Under a mere continuity hypothesis on M, we prove in this paper that L? will converge to a continuous martingale of maximal variation. This limit is independent of M.

Suggested Citation

  • Bernard De Meyer, 2007. "Price Dynamics on a Stock Market with Asymmetric Information," Cowles Foundation Discussion Papers 1604, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1604
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    References listed on IDEAS

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    1. Mertens, Jean-François & ZAMIR, Shmuel, 1976. "The normal distribution and repeated games," LIDAM Reprints CORE 312, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Bernard de Meyer & Ehud Lehrer & Dinah Rosenberg, 2009. "Evaluating information in zero-sum games with incomplete information on both sides," Post-Print halshs-00390625, HAL.
    3. DE MEYER, Bernard, 1996. "The Maximal Variation of a Bounded Martingale and the Central Limit Theorem," LIDAM Discussion Papers CORE 1996035, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Mertens,Jean-François & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107662636.
      • Mertens,Jean-François & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107030206, September.
    5. Bernard de Meyer & Hadiza Moussa Saley, 2003. "On the strategic origin of Brownian motion in Finance," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00259737, HAL.
    6. Hadiza Moussa Saley & Bernard De Meyer, 2003. "On the strategic origin of Brownian motion in finance," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(2), pages 285-319.
    7. Bernard de Meyer & Ehud Lehrer & Dinah Rosenberg, 2009. "Evaluating information in zero-sum games with incomplete information on both sides," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00390625, HAL.
    8. Bernard de Meyer & Hadiza Moussa Saley, 2003. "On the strategic origin of Brownian motion in Finance," Post-Print hal-00259737, HAL.
    9. Bernard de Meyer, 1998. "The maximal variation of a bounded martingale and the central limit theorem," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00259720, HAL.
    10. Bernard De Meyer & Alexandre Marino, 2005. "Duality and optimal strategies in the finitely repeated zero-sum games with incomplete information on both sides," Cahiers de la Maison des Sciences Economiques b05027, Université Panthéon-Sorbonne (Paris 1).
    11. Kyle, Albert S, 1985. "Continuous Auctions and Insider Trading," Econometrica, Econometric Society, vol. 53(6), pages 1315-1335, November.
    12. Bernard de Meyer, 1998. "The maximal variation of a bounded martingale and the central limit theorem," Post-Print hal-00259720, HAL.
    13. MERTENS, Jean-François & ZAMIR, Shmuel, 1977. "The maximal variation of a bounded martingale," LIDAM Reprints CORE 309, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    14. Alexandre Marino & Bernard De Meyer, 2005. "Continuous versus Discrete Market Games," Cowles Foundation Discussion Papers 1535, Cowles Foundation for Research in Economics, Yale University.
    15. Victor Domansky, 2007. "Repeated games with asymmetric information and random price fluctuations at finance markets," International Journal of Game Theory, Springer;Game Theory Society, vol. 36(2), pages 241-257, October.
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    Citations

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    Cited by:

    1. Shino Takayama, 2013. "Price Manipulation, Dynamic Informed Trading and Tame Equilibria: Theory and Computation," Discussion Papers Series 492, School of Economics, University of Queensland, Australia.
    2. Laraki, Rida & Sorin, Sylvain, 2015. "Advances in Zero-Sum Dynamic Games," Handbook of Game Theory with Economic Applications,, Elsevier.
    3. Hörner, Johannes & Lovo, Stefano & Tomala, Tristan, 2018. "Belief-free price formation," Journal of Financial Economics, Elsevier, vol. 127(2), pages 342-365.
    4. Pierre Cardaliaguet & Catherine Rainer & Dinah Rosenberg & Nicolas Vieille, 2016. "Markov Games with Frequent Actions and Incomplete Information—The Limit Case," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 49-71, February.
    5. Fabien Gensbittel & Christine Grün, 2019. "Zero-Sum Stopping Games with Asymmetric Information," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 277-302, February.
    6. Takayama, Shino, 2021. "Price manipulation, dynamic informed trading, and the uniqueness of equilibrium in sequential trading," Journal of Economic Dynamics and Control, Elsevier, vol. 125(C).
    7. Pierre Cardaliaguet & Catherine Rainer, 2012. "Games with Incomplete Information in Continuous Time and for Continuous Types," Dynamic Games and Applications, Springer, vol. 2(2), pages 206-227, June.
    8. Bernard De Meyer & Gaëtan Fournier, 2015. "Price dynamics on a risk averse market with asymmetric information," Documents de travail du Centre d'Economie de la Sorbonne 15054, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    9. Shino Takayama, 2018. "Price Manipulation, Dynamic Informed Trading and Tame Equilibria: Theory and Computation," Discussion Papers Series 603, School of Economics, University of Queensland, Australia.
    10. Fedor Sandomirskiy, 2018. "On Repeated Zero-Sum Games with Incomplete Information and Asymptotically Bounded Values," Dynamic Games and Applications, Springer, vol. 8(1), pages 180-198, March.
    11. Fabien Gensbittel, 2015. "Extensions of the Cav( u ) Theorem for Repeated Games with Incomplete Information on One Side," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 80-104, February.
    12. Fedor Sandomirskiy, 2014. "Repeated games of incomplete information with large sets of states," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(4), pages 767-789, November.
    13. Bernard de Meyer & Moussa Dabo, 2019. "The CMMV Pricing Model in Practice," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02383135, HAL.
    14. Bernard de Meyer & Moussa Dabo, 2019. "The CMMV Pricing Model in Practice," Post-Print halshs-02383135, HAL.

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    More about this item

    Keywords

    Asymmetric information; Price dynamics; Martingales of maximal variation; Repeated games;
    All these keywords.

    JEL classification:

    • G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies; Insider Trading
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions

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