A Mean-Field Model of Financial Markets: Reproducing Long Tailed Distributions and Volatility Correlations

  • S. V. Vikram
  • Sitabhra Sinha
Conference paper
Part of the New Economic Windows book series (NEW)


A model for financial market activity should reproduce the several stylized facts that have been observed to be invariant across different markets and periods. Here we present a mean-field model of agents trading a particular asset, where their decisions (to buy or to sell or to hold) is based exclusively on the price history. As there are no direct interactions between agents, the price (computed as a function of the net demand, i.e., the difference between the numbers of buyers and sellers at a given time) is the sole mediating signal driving market activity. We observe that this simple model reproduces the long-tailed distribution of price fluctuations (measured by logarithmic returns) and trading volume (measured in terms of the number of agents trading at a given instant), that has been seen in most markets across the world. By using a quenched random distribution of a model parameter that governs the probability of an agent to trade, we obtain quantitatively accurate exponents for the two distributions. In addition, the model exhibits volatility clustering, i.e., correlation between periods with large fluctuations, remarkably similar to that seen in reality. To the best of our knowledge, this is the simplest model that gives a quantitatively accurate description of financial market behavior.


Financial Market Asset Price Trading Volume Stylize Fact Coloured Version 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Farmer J D, Shubik M Smith E (2005) Is economics the next physical science? Physics Today 58(9): 37–42CrossRefGoogle Scholar
  2. 2.
    Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues, Quant. Fin. 1: 223:236CrossRefGoogle Scholar
  3. 3.
    Lux T (1996) The stable Paretian hypothesis and the frequency of large returns: An examination of major German stocks, Appl. Fin. Econ. 6: 463–475CrossRefGoogle Scholar
  4. 4.
    Gopikrishnan P, Meyer M, Amaral L A N, Stanley H E (1998) Inverse cubic law for the probability distribution of stock price variations, Eur. Phys. J. B 3: 139–140CrossRefGoogle Scholar
  5. 5.
    Gopikrishnan P, Plerou V, Amaral L A N, Meyer M, Stanley H E (1999) Scaling of fluctuations of financial market indices, Phys. Rev. E 60: 5305–5316CrossRefGoogle Scholar
  6. 6.
    Pan R K, Sinha S (2007) Self-organization of price fluctuation distribution in evolving markets, Europhys. Lett. 77: 58004CrossRefGoogle Scholar
  7. 7.
    Pan, R K, Sinha S (2008) Inverse-cubic law of index fluctuation distribution in Indian markets, Physica A 387: 2055–2065CrossRefGoogle Scholar
  8. 8.
    Gopikrishnan P, Plerou V, Gabaix X, Stanley H E (2000) Statistical properties of share volume traded in financial markets. Phys. Rev. E 62: R4493–R4496CrossRefGoogle Scholar
  9. 9.
    Liu Y, Gopikrishnan P, Cizeau P, Meyer M, Peng C-K, Stanley H E (1999) The statistical properties of the volatility of price fluctuations. Phys. Rev. E 60: 1390–1400CrossRefGoogle Scholar
  10. 10.
    Lux T, Marchesi M (1999) Scaling and criticality in a stochastic multi-agent model of a financial market, Nature 397: 498–500CrossRefGoogle Scholar
  11. 11.
    Chowdhury D, Stauffer D (1999) A generalised spin model of financial markets, Eur. Phys. J. B 8: 477–482CrossRefGoogle Scholar
  12. 12.
    Cont R, Bouchad J P (2000) Herd behavior and aggregate fluctuations in financial markets, Macroecon. Dyn. 4: 170–196CrossRefGoogle Scholar
  13. 13.
    Bornholdt S (2001) Expectation bubbles in a spin model of markets: Intermittency from frustration across scales, Int. J. Mod. Phys. C 12: 667–674CrossRefGoogle Scholar
  14. 14.
    Iori G (2002) A microsimulation of traders activity in the stock market: the role of heterogeneity, agents’ interaction and trade frictions, J. Econ. Behav. Organ. 49:269–285CrossRefGoogle Scholar
  15. 15.
    Bachelier L (1900) Théorie de la spéculation, Ann. Sci. École Norm. Sup. Sér 3 17: 21–86Google Scholar
  16. 16.
    Potters M, Bouchaud J-P (2003) More statistical properties of order books and price impact, Physica A 324: 133–140CrossRefGoogle Scholar
  17. 17.
    Alfi V, Coccetti F, Marotta M, Pietronero L, Takayasu M(2006) Hidden forces and fluctuations from moving averages: A test study, Physica A 370: 30–37CrossRefGoogle Scholar
  18. 18.
    Newman M E J (2005) Power laws, Pareto distributions and Zipf’s law, Contemp. Phys. 46: 323–351CrossRefGoogle Scholar
  19. 19.
    Hill B M (1975) A simple approach to inference about tail of a distribution, Ann. Stat. 3: 1163–1174CrossRefGoogle Scholar
  20. 20.
    Fama E (1970) Efficient capital markets: A review of theory and empirical work, J. Finance 25: 383–417CrossRefGoogle Scholar
  21. 21.
    Kaizoji T (2006) A precursor of market crashes: Empirical laws of Japan’s internet bubble, Eur. Phys. J. B 50: 123–127CrossRefGoogle Scholar
  22. 22.
    Yang J S, Chae S, Jung W S, Moon H T (2006) Microscopic spin model for the dynamics of the return distribution of the Korean stock market index, Physica A 363: 377–382CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2010

Authors and Affiliations

  • S. V. Vikram
    • 1
  • Sitabhra Sinha
    • 2
  1. 1.Department of PhysicsIndian Institute of Technology MadrasChennaiIndia
  2. 2.The Institute of Mathematical SciencesChennaiIndia

Personalised recommendations