Abstract
In this manuscript, we analytically and numerically study statistical properties of an heteroskedastic process based on the celebrated ARCH generator of random variables whose variance is defined by a memory of qm-exponencial, form (eqm=1 x=ex). Specifically, we inspect the self-correlation function of squared random variables as well as the kurtosis. In addition, by numerical procedures, we infer the stationary probability density function of both of the heteroskedastic random variables and the variance, the multiscaling properties, the first-passage times distribution, and the dependence degree. Finally, we introduce an asymmetric variance version of the model that enables us to reproduce the so-called leverage effect in financial markets.
Similar content being viewed by others
References
J.-P. Bouchaud, M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, 2000); R.N. Mantegna, H.E. Stanley, An introduction to Econophysics: correlations and Complexity in Finance (Cambridge University Press, Cambrigde, 1999); J. Voit, The Statistical Mechanics of Financial Markets (Springer-Verlag, Berlin, 2003); M. Dacorogna, R. Gençay, U. Müller, R. Olsen, O. Pictet, An Introduction to High-Frequency Finance (Academic Press, London, 2001)
S. Campbell, F.X. Diebold, J. Am. Stat. Ass. 100, 6 (2005)
J.D. Martin-Guerrero, G. Camps-Valls, E. Soria-Olivas, A.J. Serrano-Lopez, J.J. Perez-Ruixo, N.V. Jimenez-Torres, IEEE Trans. Biomed. Eng. 50, 1136 (2003)
P. Gronke, J. Brehm, Elect. Stud. 21, 425 (2002)
T.G. Andersen, T. Bollerslev, P.F. Christoffersen, F.X. Diebold, Volatility Forecasting, PIER working paper 05-011, 2005
R.F. Engle, Econometrica 50, 987 (1982)
S.M. Duarte Queirós, Europhys. Lett. 80, 30005 (2007)
B. Pobodnik, P.C. Ivanov, Y. Lee, A. Cheesa, H.E. Stanley, Europhys. Lett. 50, 711 (2000); S.M. Duarte Queirós, C. Tsallis, Europhys. Lett. 69, 893 (2005)
M. Porto, H.E. Roman, Phys. Rev. E 63, 036128 (2001)
T. Bollerslev, R.Y. Chou, K.F. Kroner, J. Econometrics 52, 5 (1992); T.G. Andersen, T. Bollerslev, F.X. Diebold, in Handbook of Financial Econometrics, edited by Y. Aït-Sahalia (Elsevier, Amsterdam, 2006)
Z. Zing, C.W.J. Granger, R.F. Engle, J. Emp. Fin. 1, 83 (1983)
C. Gourieroux, A. Montfort, Statistics and Econometric Models (Cambridge University Press, Cambridge, 1996)
C. Tsallis, J. Stat. Phys. 52, 479 (1988); C. Tsallis, Braz. J. Phys. 29, 1 (1999)
C.W.J. Granger, Z. Ding, J. Econometrics 73, 61 (1996); H.E. Roman, M. Porto, Int. J. Mod. Phys. C (to be published)
C. Dose, M. Porto, H.E. Roman, Phys. Rev. E 67, 067103 (2003)
G.M. Schütz, S. Trimper, Phys. Rev. E 70, 045101(R) (2004)
http://functions.wolfram.com
http://functions.wolfram.com/07.24.26.0272.01
D.C. Boes, F.A. Graybill, A.M. Mood, Introduction to the Theory of Statistics, 3rd edn. (McGraw-Hill, New York, 1974); H.R. Neave, Statistics Tables for mathematicians, engineers, economists and the behavioural and management sciences (Routledge, London, 1999)
I.S. Gradshteyn, I.M. Ryzhik, Tables of integrals, series and products (Academic Press, London, 1965)
C. Beck, E.G.D. Cohen, Physica A 322, 267 (2003)
A.M. Reynolds, N. Mordant, A.M. Crawford, E. Bodenschatz, New J. Phys. 7, 58 (2005); C. Beck, Phys. Rev. Lett. 98, 064502 (2007)
K. Hlaváčková-Schlinder, M. Paruš, M. Vejmelka, J. Bhattacharya, Phys. Rep. 441, 1 (2007)
C. Tsallis, Phys. Rev. E 58, 1442 (1998)
L. Borland, A.R. Plastino, C. Tsallis, J. Math. Phys. 39, 6490 (1998); L. Borland, A.R. Plastino, C. Tsallis, J. Math. Phys. 40, 2196 (1999)
S.M. Duarte Queirós, Quantitatit. Finance 5, 475 (2005); M. Portesi, F. Pennini, A. Plastino, Physica A 373, 273 (2007); S.M. Duarte Queirós, e-print arXiv:0805.2254 [cond-mat.stat-mech] (preprint, 2008)
S.M.D. Queirós, C. Tsallis, Eur. Phys. J. B 48, 139 (2005)
H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer-Verlag, Berlin, 1989)
F. Wang, P. Weber, K. Yamasaki, S. Havlin, H.E. Stanley, Eur. Phys. J. B 55, 123 (2007); E. Scalas, Chaos Solitons & Fractals (to be published)
B. Hoskins, in Predictability of Wheather and Climate, edited by T. Palmer, R. Hagedorn (Cambridge University Press, Cambridge, 2006)
B.B. Mandelbrot, The Fractal Geometry of Nature (W.H. Freeman & Co., San Francisco – CA, 1983); J. Feder, Fractals (Plenum, New York, 1988)
B.B. Mandelbrot, Fractals and Scaling in Finance (Springer, New York, 1997)
A. Admati, P. Pfleiderer, Rev. Financial Studies 1 (1988); A. Arnéodo, J.-F. Muzy, D. Sornette, Eur. Phys. J. B 2, 277 (1998), K. Ivanova, M. Ausloos, Eur. Phys. J. 8, 665 (1999); B. Pochart, J.P. Bouchaud, e-print arXiv:cond-mat/0204047 (preprint, 2002); T. Di Matteo, Quantit. Finance 7, 21 (2007); P. Oświęcimka, J. Kwapień, S. Drożdż, Phys. Rev. E 74, 016103 (2006); L.G. Moyano, J. de Souza, S.M. Duarte Queirós, Physica A 371, 118 (2006); F. Wang, K. Yamasaki, S. Havlin, H.E. Stanley, Phys. Rev. E 77, 016109 (2008)
Z. Eisler, J. Kertész, Europhys. Lett. 77, 28001 (2007)
J.P. Bouchaud, M. Potters, M. Meyer, e-print arXiv:cond-mat/9906347 (preprint, 1999); J. de Souza, S.M. Duarte Queirós, e-print arXiv:0711.2550 [physics.data-an] (preprint, 2007); Z.-Q. Jiang, W.-X. Zhou, Physica A 387, 3605 (2008)
J.W. Kantelhardt, S.A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, H.E. Stanley, Physica A 316, 87 (2002)
R.A. Haugen, E. Talmor, W.N. Torous, J. Fin. 46, 985 (1991)
J.P. Bouchaud, A. Matacz, M. Potters, Phys. Rev. Lett. 87, 228701 (2001); J. Masoliver, J. Perelló, Int. J. Theo. Appl. Fin. 5, 541 (2002)
C. Tsallis, Physica A 340, 1 (2004)
Nonextensive Entropy – Interdisciplinary Applications, edited by M. Gell-Mann, C. Tsallis (Oxford University Press, New York, 2004); Complexity, Metastability and Nonextensivity, edited by C. Beck, G. Benedek, A. Rapisarda, C. Tsallis (World Scientific, Singapore, 2005), p. 135; Complexity, Metastability, and Nonextensivity: An International Conference edited by S. Abe, H. Herrmann, P. Quarati, A. Rapisarda, C. Tsallis, AIP Conf. Proc. 965 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Queirós, S. On discrete stochastic processes with long-lasting time dependence in the variance. Eur. Phys. J. B 66, 137–148 (2008). https://doi.org/10.1140/epjb/e2008-00387-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2008-00387-2