Advertisement

Pramana

, Volume 71, Issue 2, pp 341–352 | Cite as

Role of scaling in the statistical modelling of finance

  • Attilio L. StellaEmail author
  • Fulvio Baldovin
Article

Abstract

Modelling the evolution of a financial index as a stochastic process is a problem awaiting a full, satisfactory solution since it was first formulated by Bachelier in 1900. Here it is shown that the scaling with time of the return probability density function sampled from the historical series suggests a successful model. The resulting stochastic process is a heteroskedastic, non-Markovian martingale, which can be used to simulate index evolution on the basis of an autoregressive strategy. Results are fully consistent with volatility clustering and with the multiscaling properties of the return distribution. The idea of basing the process construction on scaling, and the construction itself, are closely inspired by the probabilistic renormalization group approach of statistical mechanics and by a recent formulation of the central limit theorem for sums of strongly correlated random variables.

Keywords

Scaling stochastic processes renormalization group volatility clustering 

PACS Nos

02.50.-r 05.10.Cc 05.40.Jc 89.75.Da 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M Gallegati, S Keen, T Lux and Paul Ormerod, Physica A370, 1 (2006)ADSMathSciNetGoogle Scholar
  2. [2]
    B LeBaron, Nature (London) 408, 290 (2000)ADSCrossRefGoogle Scholar
  3. [3]
    H E Stanley, L A N Amaral, S V Buldyrev, P Gopikrishnan, V Plerou and M A Salinger, Proc. Natl Acad. Sci. 99, 2561 (2002)ADSCrossRefGoogle Scholar
  4. [4]
    F Baldovin and A L Stella, Proc. Natl Acad. Sci. 104, 19741 (2007)ADSCrossRefGoogle Scholar
  5. [5]
    R Cont, Quant. Finance 1, 223 (2001); in Fractals in engineering edited by E Lutton and J Levy Véhel (Springer-Verlag, New York, 2005)CrossRefGoogle Scholar
  6. [6]
    J-P Bouchaud and M Potters, Theory of financial risks (Cambridge University Press, Cambridge, UK, 2000)Google Scholar
  7. [7]
    R N Mantegna and H E Stanley, An introduction to econophysics (Cambridge University Press, Cambridge, UK, 2000)Google Scholar
  8. [8]
    K Yamasaki, L Muchnik, S Havlin, A Bunde and H E Stanley, Proc. Natl Acad. Sci. USA 102, 9424 (2005)ADSCrossRefGoogle Scholar
  9. [9]
    T Lux, in Power laws in the social sciences edited by C Cioffi-Revilla (Cambridge University Press, Cambridge, UK) (in press)Google Scholar
  10. [10]
    L Bachelier, Ann. Sci. Ecole Norm. Sup. 17, 21 (1900)MathSciNetGoogle Scholar
  11. [11]
    G Jona-Lasinio, Phys. Rep. 352, 439 (2001)zbMATHADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    L P Kadanoff, Statistical physics, statics, dynamics and renormalization (World Scientific, Singapore, 2005)Google Scholar
  13. [13]
    F Baldovin and A L Stella, Phys. Rev. E75, 020101(R) (2007)Google Scholar
  14. [14]
    T Di Matteo, T Aste and M M Dacorogna, J. Bank. & Fin. 29, 827 (2005)CrossRefGoogle Scholar
  15. [15]
    M P Nightingale, Physica A83, 561 (1976) See also T W Burkhardt and J M J van Leeuwen (eds) Real-space renormalization (Springer-Verlag, Berlin, Heidelberg, 1982)Google Scholar
  16. [16]
    See, e.g., B V Gnedenko and A N Kolmogorov, Limit distributions for sums of independent random variables (Addison Wesley, Reading, MA, 1954)zbMATHGoogle Scholar
  17. [17]
    F Baldovin and A L Stella, to be published (2008)Google Scholar
  18. [18]
    I M Sokolov, A V Chechkin and J Klafter, Physica A336, 245 (2004)ADSGoogle Scholar
  19. [19]
    W Feller, An introduction to probability theory and its applications (John Wiley & Sons, 1971) 2nd edition, Vol. 2Google Scholar
  20. [20]
    K E Bassler, J L McCauley and G H Gunaratne, Proc. Natl Acad. Sci. 104, 17287 (2007)ADSCrossRefGoogle Scholar
  21. [21]
    R Engle, J. Money, Credit and Banking 15, 286 (1983)CrossRefGoogle Scholar
  22. [22]
    T Bollerslev, J. Econom. 31, 307 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    N Goldenfeld and L P Kadanoff, Science 284, 87 (1999)ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2008

Authors and Affiliations

  1. 1.Dipartimento di Fisica and Sezione INFNUniversità di PadovaPadovaItaly

Personalised recommendations