Theoretical Base of the PUCK-Model with Application to Foreign Exchange Markets

  • Misako Takayasu
  • Kota Watanabe
  • Takayuki Mizuno
  • Hideki Takayasu


We analyze statistical properties of a random walker in a randomly changing potential function called the PUCK model both theoretically and numerically. In this model the center of the potential function moves with the moving average of the random walker’s trace, and the potential function is given by a quadratic function with its curvature slowly changing around zero. By tuning several parameters the basic statistical properties fit nicely with those of real financial market prices, such as power law price change distribution, very short decay of autocorrelation of price changes, long tails in autocorrelation of the square of price changes and abnormal diffusion in short time scale.


Random Walk Market Price Price Change Random Walk Model Potential Force 
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.



The authors acknowledge Professors Takatoshi Ito and Tsutomu Watanabe for helpful discussions.


  1. 1.
    Bachelier L (1900) Théorie de la Spéculation, Doctoral disseration. Annales Scientifiques de l’Ecole Normale Superieure. Translation: Cootner PH (ed) (1964) The Random Character of Stock Market Prices. MIT Press, Cambridge, MA, pp 21–86Google Scholar
  2. 2.
    Einstein A (1905) Analen der Physik 17:549–560ADSMATHCrossRefGoogle Scholar
  3. 3.
    Mantegna TN, Stanley HE (2000) An introduction to econophysics: correlation and complexity in finance. Cambridge University Press, CambridgeGoogle Scholar
  4. 4.
    Mandelbrot BB (1963) J Business 36:394–419CrossRefGoogle Scholar
  5. 5.
    Gopikrishnan P, Meyer M, Amaral LAN, Stanley HE (1998) Eur Phys J B 3:139–140ADSCrossRefGoogle Scholar
  6. 6.
    Vandewalle N, Ausloos M (1998) Int J Mod Phys C 9:711–719ADSCrossRefGoogle Scholar
  7. 7.
    Engle R (1982) Econometrica 50:987–1008MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Takayasu M, Mizuno T, Ohnishi T, Takayasu H (2005) In: Takayasu H (ed) Proceedings of practical fruits of econophysics. Springer, Tokyo, pp 29–32Google Scholar
  9. 9.
    Takayasu M, Mizuno T, Takayasu H (2006) Physica A 370:96–97MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Takayasu M, Mizuno T, Takayasu H (2007) Physica A 383:115–119MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Takayasu H, Sato A-H, Takayasu M (1997) Phys Rev Lett 79:966–969ADSMATHCrossRefGoogle Scholar
  12. 12.
    Ohnishi T, Mizuno T, Aihara K, Takayasu M, Takayasu H (2004) Physica A 344:207–210ADSCrossRefGoogle Scholar
  13. 13.
    Takayasu M, Watanabe K, Takayasu H (2010) Phys Rev Lett (submitted)Google Scholar
  14. 14.
    Yamada K, Takayasu H, Ito T, Takayasu M (2009) Phys Rev E 79:051120MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Watanabe K, Takayasu H, Takayasu M (2010) Phys Rev E 80:056110ADSCrossRefGoogle Scholar
  16. 16.
    Takayasu M, Takayasu H (2009) Prog Theor Phys 179(suppl):1–7MATHGoogle Scholar
  17. 17.
    Aiba Y, Hatano N, Takayasu H, Marumo K, Shimizu T (2003) Physica A 324:253–257MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  • Misako Takayasu
    • 1
  • Kota Watanabe
    • 1
  • Takayuki Mizuno
    • 2
  • Hideki Takayasu
    • 3
  1. 1.Department of Computational Intelligence and Systems ScienceInterdisciplinary Graduate School of Science and Engineering, Tokyo Institute of TechnologyMidori-kuJapan
  2. 2.The Institute of Economic ResearchHitotsubashi UniversityKunitachiJapan
  3. 3.Fundamental Research GroupSony Computer Science LaboratoriesShinagawa-kuJapan

Personalised recommendations