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Return Intervals Approach to Financial Fluctuations

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Complex Sciences (Complex 2009)

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Abstract

Financial fluctuations play a key role for financial markets studies. A new approach focusing on properties of return intervals can help to get better understanding of the fluctuations. A return interval is defined as the time between two successive volatilities above a given threshold. We review recent studies and analyze the 1000 most traded stocks in the US stock markets. We find that the distribution of the return intervals has a well approximated scaling over a wide range of thresholds. The scaling is also valid for various time windows from one minute up to one trading day. Moreover, these results are universal for stocks of different countries, commodities, interest rates as well as currencies. Further analysis shows some systematic deviations from a scaling law, which are due to the nonlinear correlations in the volatility sequence. We also examine the memory in return intervals for different time scales, which are related to the long-term correlations in the volatility. Furthermore, we test two popular models, FIGARCH and fractional Brownian motion (fBm). Both models can catch the memory effect but only fBm shows a good scaling in the return interval distribution.

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References

  1. Pareto, V.: Cours d’Economie Politique, Lausanne and Paris (1897)

    Google Scholar 

  2. Bachelier, L.: Théorie de la spéculation [Ph.D. thesis in mathematics]. Annales Scientifiques de l’Ecole Normale Supérieure III-17, 21 (1900); translated and Reprinted in, Cootner, P. (ed.). The Random Character of Stock Market Prices, p. 17. MIT Press, Cambridge (1967)

    MathSciNet  MATH  Google Scholar 

  3. Lévy, P.: Théorie de l’Addition des Variables Aléatoires, Gauthier-Villars, Paris (1937)

    Google Scholar 

  4. Fama, E.F.: J. Business 36, 420 (1963)

    Google Scholar 

  5. Officer, R.R.: J. Amer. Statistical Assoc. 67, 807 (1972)

    Google Scholar 

  6. Clark, P.K.: Econometrica 41, 135 (1973)

    Google Scholar 

  7. Wood, R.A., McInish, T.H., Ord, J.K.: J. Finance 40, 723 (1985)

    Google Scholar 

  8. Harris, L.: J. Financ. Econ. 16, 99 (1986)

    Google Scholar 

  9. Admati, A., Pfleiderer, P.: Rev. Financ. Stud. 1, 3 (1988)

    Google Scholar 

  10. Schwert, G.W.: J. Finance 44, 1115 (1989); Chan, K., Chan, K.C., Karolyi, G.A.: Rev. Financ. Stud. 4, 657 (1991); Bollerslev, T., Chou, R.Y., Kroner, K.F.: J. Econometr. 52, 5 (1992); Gallant, A.R., Rossi, P.E., Tauchen, G.: Rev. Financ. Stud. 5, 199 (1992); Le Baron, B.: J. Business 65, 199 (1992)

    Google Scholar 

  11. Ding, Z., Granger, C.W.J., Engle, R.F.: J. Empirical Finance 1, 83 (1993)

    Google Scholar 

  12. Dacorogna, M.M., Muller, U.A., Nagler, R.J., Olsen, R.B., Pictet, O.V.: J. Int. Money Finance 12, 413 (1993)

    Google Scholar 

  13. Loretan, M., Phillips, P.C.B.: J. Empirical Finance 1, 211 (1994)

    Google Scholar 

  14. Pagan, A.: J. Empirical Finance 3, 15 (1996)

    Google Scholar 

  15. Mantegna, R.N., Stanley, H.E.: Nature 383, 587 (1996)

    Google Scholar 

  16. Cizeau, P., Liu, Y., Meyer, M., Peng, C.-K., Stanley, H.E.: Physica A 245, 441 (1997)

    Google Scholar 

  17. Cont, R.: Ph.D. thesis, Universite de Paris XI (1998) (unpublished); see also e-print cond-mat/9705075

    Google Scholar 

  18. Pasquini, M., Serva, M.: Econ. Lett. 65, 275 (1999)

    Google Scholar 

  19. Lux, T., Marchesi, M.: Int. J. Theor. Appl. Finance 3, 675 (2000); Giardina, I., Bouchaud, J.-P.: Physica A 299, 28 (2001); Lux, T., Ausloos, M.: In: Bunde, A., Kropp, J., Schellnhuber, H.J. (eds.) The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes, p. 373. Springer, Berlin (2002)

    Google Scholar 

  20. Rosenow, B.: Int. J. Mod. Phys. C 13, 419 (2002)

    Google Scholar 

  21. Gabaix, X., Gopikrishnan, P., Plerou, V., Stanley, H.E.: Nature 423, 267 (2003); Gabaix, X., Gopikrishnan, P., Plerou, V., Stanley, H.E.: Quart. J. Econ. 121, 461 (2006)

    Google Scholar 

  22. Farmer, J.D., Gillemot, L., Lillo, F., Mike, S., Sen, A.: Quant. Finance 4, 383 (2004); Farmer, J.D., Lillo, F.: ibid 4, C8, (2004); Plerou, V., Gopikrishnan, P., Gabaix, X., Stanley, H.E.: ibid 4, C11 (2004)

    Google Scholar 

  23. Lillo, F., Farmer, J.D., Mantegna, R.N.: Nature 421, 129 (2003)

    Google Scholar 

  24. Weber, P., Wang, F., Vodenska-Chitkushev, I., Havlin, S., Stanley, H.E.: Phys. Rev. E 76, 016109 (2007)

    Google Scholar 

  25. Eisler, Z., Bartos, I., Kertész, J.: Adv. Phys. 57, 89 (2008)

    Google Scholar 

  26. Jung, W.-S., Kwon, O., Wang, F., Kaizoji, T., Moon, H.-T., Stanley, H.E.: Physica A 387, 537 (2008)

    Google Scholar 

  27. Lux, T.: Appl. Finan. Econ. 6, 463 (1996)

    Google Scholar 

  28. Gopikrishnan, P., Meyer, M., Amaral, L.A.N., Stanley, H.E.: Eur. Phys. J. B 3, 139 (1998)

    Google Scholar 

  29. Muller, U.A., Dacorogna, M.M., Pictet, O.V.: Heavy Tails in High-Frequency Financial Data. In: Adler, R.J., Feldman, R.E., Taqqu, M.S. (eds.) A Practical Guide to Heavy Tails, p. 83. Birkhäuser Publishers, Basel (1998)

    Google Scholar 

  30. Plerou, V., Gopikrishnan, P., Amaral, L.A.N., Meyer, M., Stanley, H.E.: Phys. Rev. E 60, 6519 (1999); Plerou, V., Gopikrishnan, P., Amaral, L.A.N., Gabaix, X., Stanley, H.E.: ibid 62, 3023 (2000); Gopikrishnan, P., Plerou, V., Gabaix, X., Stanley, H.E.: ibid 62, 4493 (2000)

    Google Scholar 

  31. Liu, Y., Gopikrishnan, P., Cizeau, P., Meyer, M., Peng, C.-K., Stanley, H.E.: Phys. Rev. E 60, 1390, (1999); Plerou, V., Gopikrishnan, P., Gabaix, X., Amaral, L.A.N., Stanley, H.E.: Quant. Finance 1, 262, (2001); Plerou, V., Gopikrishnan, P., Stanley, H.E.: Phys. Rev. E 71, 046131 (2005); for application to heartbeat intervals see, Ashkenazy, Y., Ivanov, P.C., Havlin, S., Peng, C.-K., Goldberger, A.L., Stanley, H.E.: Phys. Rev. Lett. 86, 1900 (2001)

    Google Scholar 

  32. Plerou, V., Stanley, H.E.: Phys. Rev. E 76, 046109 (2007)

    Google Scholar 

  33. Clauset, A., Shalizi, C.R., Newman, M.E.J.: http://arxiv.org/abs/0706.1062v1

  34. Mantegna, R., Stanley, H.E.: Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge Univ. Press, Cambridge (2000)

    MATH  Google Scholar 

  35. Mandelbrot, B.B.: J. Business 36, 394 (1963)

    Google Scholar 

  36. Bunde, A., Havlin, S. (eds.): Fractals in Science. Springer, Heidelberg (1994)

    MATH  Google Scholar 

  37. Mantegna, R.N., Stanley, H.E.: Nature 376, 46 (1995)

    Google Scholar 

  38. Stanley, H.E.: Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, Oxford (1971)

    Google Scholar 

  39. Stanley, H.E.: Rev. Mod. Phys. 71, 358 (1999)

    Google Scholar 

  40. Peng, C.K., Buldyrev, S., Goldberger, A., Havlin, S., Sciortino, F., Simons, M., Stanley, H.E.: Nature 356, 168 (1992); Buldyrev, S.V., Goldberger, A.L., Havlin, S., Peng, C.-K., Stanley, H.E., Stanley, M.H.R., Simons, M.: Biophys. J. 65, 2673 (1993); Buldyrev, S.V., Goldberger, A.L., Havlin, S., Peng, C.-K., Simons, M., Stanley, H.E.: Phys. Rev. E 47, 4514 (1993); Mantegna, R.N., Buldyrev, S.V., Goldberger, A.L., Havlin, S., Peng, C.-K., Simons, M., Stanley, H.E.: Phys. Rev. E 52, 2939 (1995)

    Google Scholar 

  41. Suki, B., Barabási, A.-L., Hantos, Z., Peták, F., Stanley, H.E.: Nature 368, 615 (1994)

    Google Scholar 

  42. Peng, C.K., Mietus, J., Hausdorff, J., Havlin, S., Stanley, H.E., Goldberger, A.L.: Phys. Rev. Lett. 70, 1343 (1993)

    Google Scholar 

  43. Makse, H.A., Havlin, S., Stanley, H.E.: Nature 377, 608 (1995); Makse, H.A., Andrade, J.S., Batty, M., Havlin, S., Stanley, H.E.: Phys. Rev. E 58, 7054 (1998)

    Google Scholar 

  44. Plerou, V., Amaral, L.A.N., Gopikrishnan, P., Meyer, M., Stanley, H.E.: Nature 400, 433 (1999)

    Google Scholar 

  45. Keitt, T.H., Stanley, H.E.: Nature 393, 257 (1998); Keitt, T.H., Amaral, L.A.N., Buldyrev, S.V., Stanley, H.E.: Scaling in the Growth of Geographically Subdivided Populations: Scale-Invariant Patterns from a Continent-Wide Biological Survey. Focus issue: The Biosphere as a Complex Adaptive System, Phil. Trans. Royal Soc. B: Biological Sciences 357, 627 (2002)

    Google Scholar 

  46. Gutenberg, B., Richter, C.F.: Seismicity of the Earth and Associated Phenomenon, 2nd edn. Princeton University Press, Princeton (1954)

    Google Scholar 

  47. Turcotte, D.L.: Fractals and Chaos in Geology and Geophysics. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  48. Bunde, A., Eichner, J.F., Havlin, S., Kantelhardt, J.W.: Physica A, 342, 308 (2004)

    Google Scholar 

  49. Bunde, A., Eichner, J.F., Kantelhardt, J.W., Havlin, S.: Phys. Rev. Lett. 94, 048701 (2005)

    Google Scholar 

  50. Livina, V.N., Havlin, S., Bunde, A.: Phys. Rev. Lett. 95, 208501 (2005)

    Google Scholar 

  51. Eichner, J.F., Kantelhardt, J.W., Bunde, A., Havlin, S.: Phys. Rev. E 75, 011128 (2007)

    Google Scholar 

  52. Yamasaki, K., Muchnik, L., Havlin, S., Bunde, A., Stanley, H.E.: Proc. Natl. Acad. Sci. U.S.A. 102, 9424, (2005); Yamasaki, K., Muchnik, L., Havlin, S., Bunde, A., Stanley, H.E.: In: Takayasu, H. (ed.) Proceedings of the Third Nikkei Econophysics Research Workshop and Symposium, The Fruits of Econophysics, Tokyo, p. 43. Springer, Berlin (2005)

    Google Scholar 

  53. Wang, F., Yamasaki, K., Havlin, S., Stanley, H.E.: Phys. Rev. E 73, 026117 (2006)

    Google Scholar 

  54. Wang, F., Weber, P., Yamasaki, K., Havlin, S., Stanley, H.E.: Eur. Phys. J. B 55, 123 (2007)

    Google Scholar 

  55. Jung, W.-S., Wang, F.Z., Havlin, S., Kaizoji, T., Moon, H.-T., Stanley, H.E.: Eur. Phys. J. B 62, 113 (2008)

    Google Scholar 

  56. Qiu, T., Guo, L., Chen, G.: Physica A 387, 6812 (2008)

    Google Scholar 

  57. Ren, F., Guo, L., Zhou, W.-X.: http://arxiv.org/abs/0807.1818v1

  58. Vodenska-Chitkushev, I., Wang, F.Z., Weber, P., Yamasaki, K., Havlin, S., Stanley, H.E.: Eur. Phys. J. B 61, 217 (2008)

    Google Scholar 

  59. Bogachev, M.I., Eichner, J.F., Bunde, A.: Phys. Rev. Lett. 99, 240601 (2007)

    Google Scholar 

  60. Wang, F., Yamasaki, K., Havlin, S., Stanley, H.E.: Phys. Rev. E 77, 016109 (2008)

    Google Scholar 

  61. Wang, F., Yamasaki, K., Havlin, S., Stanley, H.E.: http://arxiv.org/abs/0808.3200v1

  62. Ren, F., Zhou, W.-X.: http://arxiv.org/abs/0809.0250v1

  63. Bogachev, M.I., Bunde, A.: Phys. Rev. E 78, 036114 (2008)

    Google Scholar 

  64. Black, F., Scholes, M.: J. Polit. Econ. 81, 637 (1973)

    Google Scholar 

  65. Cox, J.C., Ross, S.A.: J. Financ. Econ. 3, 145 (1976); Cox, J.C., Ross, S.A., Rubinstein, M.: J. Financ. Econ. 7, 229 (1979)

    Google Scholar 

  66. Bouchaud, J.-P., Potters, M.: Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management. Cambridge Univ. Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  67. Johnson, N.F., Jefferies, P., Hui, P.M.: Financial Market Complexity. Oxford Univ. Press, New York (2003)

    Book  Google Scholar 

  68. Altmann, E.G., Kantz, H.: Phys. Rev. E 71, 056106 (2005)

    Google Scholar 

  69. Stephens, M.A.: J. Am. Stat. Assoc. 69, 730 (1974)

    Google Scholar 

  70. Engle, R., Russel, J.: Econometrica 66, 1127 (1998)

    Google Scholar 

  71. To avoid the discreteness for small τ (Ref [51] suggested a power law function for this range) and large fluctuations for very large τ, we choose the range of 0.01 ≤ CDF ≤ 0.50 to perform the stretched exponential fit

    Google Scholar 

  72. It sounds that the ratio between two a significantly deviates from 1. However, it is not that huge if we test the sensitivity on γ for Eq. (4). For instance, when γ changes from 0.30 to 0.31, the value a from Eq. (4) increases more than 30% with a constant 〈τ〉

    Google Scholar 

  73. Hill, B.M.: Ann. Stat. 3, 1163 (1975)

    Google Scholar 

  74. Di Matteo, T.: Quant. Finan. 7, 21 (2007)

    Google Scholar 

  75. Ivanov, P.C., Yuen, A., Podobnik, B., Lee, Y.: Phys. Rev. E 69, 056107 (2004)

    Google Scholar 

  76. Eisler, Z., Kertész, J.: Phys. Rev. E 73, 046109 (2006); Eisler, Z., Kertész, J.: Eur. Phys. J. B 51, 145 (2006)

    Google Scholar 

  77. Schreiber, T., Schmitz, A.: Phys. Rev. Lett. 77, 635 (1996); Schreiber, T., Schmitz, A.: Physica D 142, 346 (2000)

    Google Scholar 

  78. Makse, H.A., Havlin, S., Schwartz, M., Stanley, H.E.: Phys. Rev. E 53, 5445 (1996)

    Google Scholar 

  79. Eichner, J.F., Kantelhardt, J.W., Bunde, A., Havlin, S.: Phys. Rev. E 73, 016130 (2006)

    Google Scholar 

  80. Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.: Phys. Rev. E 49, 1685 (1994); Peng, C.-K., Havlin, S., Stanley, H.E., Goldberger, A.L.: Chaos 5, 82 (1995)

    Google Scholar 

  81. Hu, K., Ivanov, P.C., Chen, Z., Carpena, P., Stanley, H.E.: Phys. Rev. E 64, 011114 (2001); Chen, Z., Ivanov, P.C., Hu, K., Stanley, H.E.: ibid 65, 041107 (2002); Xu, L., Ivanov, P.C., Hu, K., Chen, Z., Carbone, A., Stanley, H.E.: ibid 71, 051101 (2005); Chen, Z., Hu, K., Carpena, P., Bernaola-Galvan, P., Stanley, H.E., Ivanov, P.C.: ibid 71, 011104 (2005); Kantelhardt, J.W., Zschiegner, S., Koscielny-Bunde, E., Havlin, S., Bunde, A., Stanley, H.E.: Physica A 316, 87 (2002)

    Google Scholar 

  82. Bunde, A., Havlin, S., Kantelhardt, J.W., Penzel, T., Peter, J.-H., Voigt, K.: Phys. Rev. Lett. 85, 3736 (2000)

    Google Scholar 

  83. Baillie, R.T., Bollerslev, T., Mikkelsen, H.O.: J. Econometrics 74, 3 (1996)

    Google Scholar 

  84. Mandelbrot, B.B., Van Ness, J.W.: SIAM rev. 10, 442 (1968)

    Google Scholar 

  85. To simplify the simulation and without loss of generality, we neglect the crossover in the DFA curve and obtain H = 0.86 for the S&P 500 index

    Google Scholar 

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Authors and Affiliations

  1. Center for Polymer Studies and Department of Physics, Boston University, Boston, MA, 02215, USA

    Fengzhong Wang, Kazuko Yamasaki, Shlomo Havlin & H. Eugene Stanley

  2. Department of Environmental Sciences, Tokyo University of Information Sciences, Chiba, 265-8501, Japan

    Kazuko Yamasaki

  3. Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan, 52900, Israel

    Shlomo Havlin

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  1. Fengzhong Wang
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  1. Network Technology Research Centre, Research Techno Plaza, Nanyang Technology University, 4th story X‘Frontiers, Block 50 Nanyang Drive, 637553, Singapore

    Jie Zhou

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© 2009 ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering

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Wang, F., Yamasaki, K., Havlin, S., Stanley, H.E. (2009). Return Intervals Approach to Financial Fluctuations. In: Zhou, J. (eds) Complex Sciences. Complex 2009. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02466-5_1

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