Bridging stylized facts in finance and data non-stationarities

  • Sabrina Camargo
  • Sílvio M. Duarte Queirós
  • Celia Anteneodo
Regular Article

Abstract

Employing a recent technique which allows the representation of nonstationary data by means of a juxtaposition of locally stationary paths of different length, we introduce a comprehensive analysis of the key observables in a financial market: the trading volume and the price fluctuations. From the segmentation procedure we are able to introduce a quantitative description of statistical features of these two quantities, which are often named stylized facts, namely the tails of the distribution of trading volume and price fluctuations and a dynamics compatible with the U-shaped profile of the volume in a trading section and the slow decay of the autocorrelation function. The segmentation of the trading volume series provides evidence of slow evolution of the fluctuating parameters of each patch, pointing to the mixing scenario. Assuming that long-term features are the outcome of a statistical mixture of simple local forms, we test and compare different probability density functions to provide the long-term distribution of the trading volume, concluding that the log-normal gives the best agreement with the empirical distribution. Moreover, the segmentation of the magnitude price fluctuations are quite different from the results for the trading volume, indicating that changes in the statistics of price fluctuations occur at a faster scale than in the case of trading volume.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    R.N. Mantegna, H.E. Stanley, An introduction to Econophysics: correlations and Complexity in Finance (Cambridge University Press, Cambrigde, 1999)Google Scholar
  2. 2.
    J.-P. Bouchaud, M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, 2000)Google Scholar
  3. 3.
    M. Dacorogna, R. Gençay, U. Müller, R. Olsen, O. Pictet, An Introduction to High-Frequency Finance (Academic Press, London, 2001)Google Scholar
  4. 4.
    J. Voit, The Statistical Mechanics of Financial Markets (Springer-Verlag, Berlin, 2003)Google Scholar
  5. 5.
    W. Feller, An Introduction to Probability Theory and Its Applications (John Wiley & Sons, New York, 1971), Vol. 2Google Scholar
  6. 6.
    T. Lux, M. Marchesi, Nature 397, 498 (1999)ADSCrossRefGoogle Scholar
  7. 7.
    T. Lux, M. Marchesi, Int. J. Theor. Appl. Finance 3, 675 (2000)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    W.K. Bertram, Physica A 341, 533 (2004)ADSCrossRefGoogle Scholar
  9. 9.
    E. Scalas, Chaos Sol. Frac. 34, 33 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    M. Marsili, G. Raffaelli, B. Ponsot, J. Econ. Dyn. Control 33, 1170 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    G. Livan, J. Inoue, E. Scalas, J. Stat. Mech. 12, 07025 (2012)CrossRefGoogle Scholar
  12. 12.
    P. Gopikrishnan, V. Plerou, X. Gabaix, H.E. Stanley, Phys. Rev. E 62, R4493 (2000)ADSCrossRefGoogle Scholar
  13. 13.
    R. Osorio, L. Borland, C. Tsallis, in Distributions of High-Frequency Stock-Market Observablesin Nonextensive Entropy – Interdisciplinary Applications, edited by M. Gell-Mann, C. Tsallis (Oxford University Press, New York, 2004), pp. 321–334Google Scholar
  14. 14.
    G.-H. Mu, W. Chen, J. Kertész, W.-X. Zhou, Eur. Phys. J. B 68, 145 (2009)ADSCrossRefGoogle Scholar
  15. 15.
    G.-F. Gua, F. Rena, X.-H. Nia, W. Chene, W.-X. Zhou, Physica A 389, 278 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    S.M. Duarte Queirós, Europhys. Lett. 71, 339 (2005)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    A.A.G. Cortines, R. Riera, C. Anteneodo, Europhys. Lett. 83, 30003 (2008)ADSCrossRefGoogle Scholar
  18. 18.
    A.R. Gallant, P.E. Rossi, G. Tauchen, Rev. Financ. Stud. 5, 199 (1992)CrossRefGoogle Scholar
  19. 19.
    C. Jones, K. Gautam, M.L. Lipson, Rev. Financ. Stud. 7, 631 (1994)CrossRefGoogle Scholar
  20. 20.
    X. Gabaix, P. Gopikrishnan, V. Plerou, H.E. Stanley, Nature 423, 267 (2003)ADSCrossRefGoogle Scholar
  21. 21.
    J.M. Karpoff, J. Financ. Quant. Anal. 22, 109 (1987)CrossRefGoogle Scholar
  22. 22.
    E. Wienman, Principles of Multiscale Modeling (Cambridge University Press, Cambridge, 2011)Google Scholar
  23. 23.
    J.-P. Fouque, G. Papanicolaou, R. Sircar, K. Sølna, Multiscale Stochastic Volatility for Equity, Interest Rate and Credit Derivatives (Cambridge University Press, Cambridge, 2011)Google Scholar
  24. 24.
    C. Beck, Philos. Trans. R. Soc. A 369, 453 (2011)ADSMATHCrossRefGoogle Scholar
  25. 25.
    E. Van der Straeten, C. Beck, Phys. Rev. E 80, 036108 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    C. Beck, E.G.D. Cohen, Physica A 322, 267 (2003)MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    S. Camargo, S.M. Duarte Queirós, C. Anteneodo, Phys. Rev. E 84, 046702 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    E. Moro, J. Vicente, L.G. Moyano, A. Gerig, J. Doyne Farmer, G. Vaglica, F. Lillo, R.N. Mantegna, Phys. Rev. E 80, 066102 (2009)ADSCrossRefGoogle Scholar
  29. 29.
    P. Bernaola-Galván, I. Grosse, P. Carpena, J.L. Oliver, R. Román-Roldán, H.E. Stanley, Phys. Rev. Lett. 85, 1342 (2000)ADSCrossRefGoogle Scholar
  30. 30.
    W. Li, Phys. Rev. Lett. 86, 5815 (2001)ADSCrossRefGoogle Scholar
  31. 31.
    G. Shafer, J. Am. Stat. Assoc. 77, 325 (1982)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    A.E. Raftery, Biometrika 83, 251 (1995)MathSciNetCrossRefGoogle Scholar
  33. 33.
    B. Tóth, F. Lillo, J.D. Farmer, Eur. Phys. J. B 78, 235 (2010)ADSCrossRefGoogle Scholar
  34. 34.
    M.F.M. Osborne, Oper. Res. 7, 145 (1959)MathSciNetCrossRefGoogle Scholar
  35. 35.
    P.K. Clark, Econometrica 41, 135 (1973)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    G. Tauchen, M. Pitts, Econometrica 51, 485 (1983)MATHCrossRefGoogle Scholar
  37. 37.
    J. Ruseckas, B. Kaulakys, Phys. Rev. E 84, 051125 (2011)ADSCrossRefGoogle Scholar
  38. 38.
    J. Ruseckas, V. Gontis, B. Kaulakys, Adv. Complex Syst. 15, 1250073 (2012)MathSciNetCrossRefGoogle Scholar
  39. 39.
    J. de Souza, L.G. Moyano, S.M. Duarte Queirós, Eur. Phys. J. B 50, 165 (2006)ADSCrossRefGoogle Scholar
  40. 40.
    A. Admati, P. Pfleiderer, Rev. Financ. Stud. 1, 3 (1988)CrossRefGoogle Scholar
  41. 41.
    T. Andersen, T. Bollerslev, J. Empir. Financ. 4, 115 (1997)CrossRefGoogle Scholar
  42. 42.
    R. Allez, J.-P. Bouchaud, New J. Phys. 13, 025010 (2011)ADSCrossRefGoogle Scholar
  43. 43.
    C. Anteneodo, S.M. Duarte Queirós, J. Stat. Mech. P10023 (2010)Google Scholar
  44. 44.
    L.G. Moyano, J. de Souza, S.M. Duarte Queirós, Physica A 371, 118 (2006)ADSCrossRefGoogle Scholar
  45. 45.
    Z. Eisler, J. Kertész, Europhys. Lett. 77, 28001 (2007)ADSCrossRefGoogle Scholar
  46. 46.
    A.R. Krommer, C.W. Ueberhuber, Computational Integration (SIAM Publications, Philadelphia, 1998)Google Scholar
  47. 47.
    C.W.J. Granger, O. Morgenstern, Kyklos 16, 1 (1963)CrossRefGoogle Scholar
  48. 48.
    C.M. Jones, G. Kaul, M.L. Lipson, Rev. Financ. Stud. 7, 631 (1994)CrossRefGoogle Scholar
  49. 49.
    K. Chan, W.-M. Fong, J. Financ. Econ. 57, 247 (2000)CrossRefGoogle Scholar
  50. 50.
    T.G. Andersen, J. Financ. 51, 169 (1996)CrossRefGoogle Scholar
  51. 51.
    H. Bessembinder, P.J. Seguin, J. Financ. Quant. Anal. 28, 21 (1993)CrossRefGoogle Scholar
  52. 52.
    T. Ané, L. Ureche-Rangau, Int. Financ. Markets, Inst. Money 18, 216 (2008)CrossRefGoogle Scholar
  53. 53.
    B. Cornell, J. Futures Markets 1, 303 (1981)CrossRefGoogle Scholar
  54. 54.
    T.F. Martell, A.S. Wolf, J. Futures Markets 7, 233 (1987)CrossRefGoogle Scholar
  55. 55.
    R.T. Daigler, M.K. Wiley, J. Financ. 54, 2297 (1999)CrossRefGoogle Scholar
  56. 56.
    H.-G. Fung, G.A. Patterson, Int. Financ. Market Inst. Money 9, 33 (2009)CrossRefGoogle Scholar
  57. 57.
    J.-P. Bouchaud, J.D. Farmer, F. Lillo, How Markets Slowly Digest Changes in Supply and Demand, in Handbook of Financial Markets: Dynamics and Evolution, edited by T. Hens, K. Schenk-Hoppe (Elsevier: Academic Press, New York, 2008), pp. 57 − 156Google Scholar
  58. 58.
    J.D. Farmer, F. Lillo, Quantitative Finance 4, 7 (2004)CrossRefGoogle Scholar
  59. 59.
    J.D. Farmer, L. Gillemot, F. Lillo, S. Mike, A. Sen, Quantitative Finance 4, 383 (2004)CrossRefGoogle Scholar
  60. 60.
    P. Weber, B. Rosenow, Quantitative Finance 6, 7 (2006)MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    A.A. Christie, On Information Arrival and Hypothesis Testing in Events Studies, University of Rochester Report number MERC/83-13 (1983) http://hdl.handle.net/1802/4856 [Last retrieved 7th August 2012]
  62. 62.
    R.J. Rogalski, Rev. Econ. Stat. 36, 268 (1978)CrossRefGoogle Scholar
  63. 63.
    C.C. Ying, Econometrica 34, 676 (1966)CrossRefGoogle Scholar
  64. 64.
    F. Lillo, J.D. Farmer, R.N. Mantegna, Nature 421, 129 (2003)ADSCrossRefGoogle Scholar
  65. 65.
    J.D. Farmer, A. Gerig, F. Lillo, S. Mike, Quantitative Finance 6, 107 (2006)MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    J.D. Farmer, N. Zamani, Eur. Phys. J. B 55, 1899 (2007)MathSciNetGoogle Scholar
  67. 67.
    P. Weber, B. Rosenow, Quantitative Finance 5, 357 (2005)MATHCrossRefGoogle Scholar
  68. 68.
    M. Wyart, J.-P. Bouchaud, J. Kockelkoren, M. Potters, M. Vettorazzo, arXiv:physics/0603084v3 [physics.data-an] (2006)Google Scholar
  69. 69.
    W.-X. Zhou, New J. Phys. 14, 023055 (2012)ADSCrossRefGoogle Scholar
  70. 70.
    B. Tóth, Y. Lempérière, C. Deremble, J. de Lataillade, J. Kockelkoren, J.-P. Bouchaud, Phys. Rev. X 1, 021006 (2011)CrossRefGoogle Scholar
  71. 71.
    A.S. Kyle, Econometrica 53, 1315 (1985)MATHCrossRefGoogle Scholar
  72. 72.
    M. Potters, J.-P. Bouchaud, Physica A 324, 133 (2003)ADSMATHCrossRefGoogle Scholar
  73. 73.
    C. Hopman, Quantitative Finance 7, 37 (2007)MATHCrossRefGoogle Scholar
  74. 74.
    L. Gillemot, J.D. Farmer, F. Lillo, Quantitative Finance 6, 371 (2006)MATHCrossRefGoogle Scholar
  75. 75.
    S. Mike, J. Farmer, J. Econ. Dyn. Control, 32, 200 (2008)CrossRefGoogle Scholar
  76. 76.
    A. Joulin, A. Lefevre, D. Grunberg, J.-P. Bouchaud, arXiv:0803.1769v1 [physics-soc-ph] (2008)Google Scholar
  77. 77.
    Micciche et al., Physica A 314, 756761 (2002)Google Scholar
  78. 78.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, 2nd edn. (Cambridge University Press, Cambridge, 1994)Google Scholar
  79. 79.
    C.M. Jarque, A.K. Bera, Econ. Lett. 6, 255 (1980)MathSciNetCrossRefGoogle Scholar
  80. 80.
    R.F. Engle, Econometrica 50, 987 (1982)MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    S.L. Heston, Rev. Financ. Stud. 6, 327 (1993)CrossRefGoogle Scholar
  82. 82.
    A.A. Drǎgulescu, V.M. Yakovenko, Quantitative Finance 2, 443 (2002)MathSciNetGoogle Scholar
  83. 83.
    M. Porto, H.E. Roman, Phys. Rev. E 65, 046149 (2002)ADSCrossRefGoogle Scholar
  84. 84.
    S.M. Duarte Queirós, C. Anteneodo, C. Tsallis, Power-law distributions in economics: a nonextensive statistical approach, in Noise and Fluctuations in Econophysics and Finance, Proc. of SPIE, edited by D. Abbott, J.P. Bouchaud, X. Gabaix, J.L. McCauley (SPIE, Bellingham -WA, 2005), Vol. 5848, pp. 151 − 164Google Scholar
  85. 85.
    T.G. Andersen, T. Bollerslev, F.X. Diebold, Parametric and Nonparametric Volatility Measurement, in Handbook of Financial Econometrics, edited by Y. Aït-Sahalia, L.P. Hansen (Elsevier, Amsterdam, 2006), pp. 67−139Google Scholar
  86. 86.
    G. Vaglica, F. Lillo, E. Moro, R.N. Mantegna, Phys. Rev. E 77, 036110 (2008)ADSCrossRefGoogle Scholar
  87. 87.
    W.S. Cleveland, J. Am. Stat. Assoc. 74, 829 (1979)MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    W.S. Cleveland, S.J. Devlin, J, Am. Stat. Assoc. 83, 596 (1988)MATHCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sabrina Camargo
    • 1
  • Sílvio M. Duarte Queirós
    • 2
  • Celia Anteneodo
    • 3
  1. 1.Department of PhysicsPUC-RioRio de JaneiroBrazil
  2. 2.Istituto dei Sistemi Complessi - CNRRomaItaly
  3. 3.Department of PhysicsPUC-Rio and National Institute of Science and Technology for Complex SystemsRio de JaneiroBrazil

Personalised recommendations