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.
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References
M Gallegati, S Keen, T Lux and Paul Ormerod, Physica A370, 1 (2006)
B LeBaron, Nature (London) 408, 290 (2000)
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)
F Baldovin and A L Stella, Proc. Natl Acad. Sci. 104, 19741 (2007)
R Cont, Quant. Finance 1, 223 (2001); in Fractals in engineering edited by E Lutton and J Levy Véhel (Springer-Verlag, New York, 2005)
J-P Bouchaud and M Potters, Theory of financial risks (Cambridge University Press, Cambridge, UK, 2000)
R N Mantegna and H E Stanley, An introduction to econophysics (Cambridge University Press, Cambridge, UK, 2000)
K Yamasaki, L Muchnik, S Havlin, A Bunde and H E Stanley, Proc. Natl Acad. Sci. USA 102, 9424 (2005)
T Lux, in Power laws in the social sciences edited by C Cioffi-Revilla (Cambridge University Press, Cambridge, UK) (in press)
L Bachelier, Ann. Sci. Ecole Norm. Sup. 17, 21 (1900)
G Jona-Lasinio, Phys. Rep. 352, 439 (2001)
L P Kadanoff, Statistical physics, statics, dynamics and renormalization (World Scientific, Singapore, 2005)
F Baldovin and A L Stella, Phys. Rev. E75, 020101(R) (2007)
T Di Matteo, T Aste and M M Dacorogna, J. Bank. & Fin. 29, 827 (2005)
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)
See, e.g., B V Gnedenko and A N Kolmogorov, Limit distributions for sums of independent random variables (Addison Wesley, Reading, MA, 1954)
F Baldovin and A L Stella, to be published (2008)
I M Sokolov, A V Chechkin and J Klafter, Physica A336, 245 (2004)
W Feller, An introduction to probability theory and its applications (John Wiley & Sons, 1971) 2nd edition, Vol. 2
K E Bassler, J L McCauley and G H Gunaratne, Proc. Natl Acad. Sci. 104, 17287 (2007)
R Engle, J. Money, Credit and Banking 15, 286 (1983)
T Bollerslev, J. Econom. 31, 307 (1986)
N Goldenfeld and L P Kadanoff, Science 284, 87 (1999)
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Stella, A.L., Baldovin, F. Role of scaling in the statistical modelling of finance. Pramana - J Phys 71, 341–352 (2008). https://doi.org/10.1007/s12043-008-0167-0
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DOI: https://doi.org/10.1007/s12043-008-0167-0