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From Minority Game to Black & Scholes pricing

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  • Matteo Ortisi
  • Valerio Zuccolo

Abstract

In this paper we study the continuum time dynamics of a stock in a market where agents behavior is modeled by a Minority Game and a Grand Canonical Minority Game. The dynamics derived is a generalized geometric Brownian motion; from the Black & Scholes formula the calibration of both the Minority Game and the Grand Canonical Minority Game, by means of their characteristic parameters, is performed. We conclude that for both games the asymmetric phase with characteristic parameters close to critical ones is coherent with options implied volatility market.

Suggested Citation

  • Matteo Ortisi & Valerio Zuccolo, 2012. "From Minority Game to Black & Scholes pricing," Papers 1205.2521, arXiv.org, revised May 2013.
    • Handle: RePEc:arx:papers:1205.2521
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    File URL: http://arxiv.org/pdf/1205.2521
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    References listed on IDEAS

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    1. Irene Giardina & Jean-Philippe Bouchaud & Marc M'ezard, 2001. "Microscopic Models for Long Ranged Volatility Correlations," Papers cond-mat/0105076, arXiv.org.
    2. D. Challet & A. Chessa & M. Marsili & Y-C. Zhang, 2001. "From Minority Games to real markets," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 168-176.
    3. Irene Giardina & Jean-Philippe Bouchaud & Marc Mezard, 2001. "Microscopic models for long ranged volatility correlations," Science & Finance (CFM) working paper archive 500024, Science & Finance, Capital Fund Management.
    4. Giardina, Irene & Bouchaud, Jean-Philippe & Mézard, Marc, 2001. "Microscopic models for long ranged volatility correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 28-39.
    5. Challet, Damien & Marsili, Matteo & Zhang, Yi-Cheng, 2013. "Minority Games: Interacting agents in financial markets," OUP Catalogue, Oxford University Press, number 9780199686698.
    6. A C C Coolen & J A F Heimel, 2001. "Dynamical Solution of the On-Line Minority Game," Papers cond-mat/0107600, arXiv.org.
    7. Challet, Damien & Marsili, Matteo & Zhang, Yi-Cheng, 2001. "Stylized facts of financial markets and market crashes in Minority Games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 294(3), pages 514-524.
    8. Challet, D. & Zhang, Y.-C., 1997. "Emergence of cooperation and organization in an evolutionary game," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 246(3), pages 407-418.
    9. Paul Jefferies & Michael Hart & Neil Johnson & P.M. Hui, 2001. "From market games to real-world markets," OFRC Working Papers Series 2001mf02, Oxford Financial Research Centre.
    10. P. Jefferies & M.L. Hart & P.M. Hui & N.F. Johnson, 2001. "From market games to real-world markets," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 20(4), pages 493-501, April.
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    Cited by:

    1. Bryce Morsky & Fuwei Zhuang & Zuojun Zhou, 2023. "Social and individual learning in the Minority Game," Papers 2307.11846, arXiv.org, revised Mar 2024.

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