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Fractality and Multifractality in a Stock Market’s Nonstationary Financial Time Series

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Abstract

A financial time series, such as a stock market index, foreign exchange rate, or a commodity price, fluctuates heavily and shows scaling behaviors. Scaling and multi-scaling behaviors are measured for a nonstationary time series, such as stock market indices, high-frequency stock prices of individual stocks, or the volatility time series of a stock index. We review the fractality, multi-scaling, and multifractality of the financial time series of a stock market. We introduce a detrended fluctuation analysis of the financial time series to extract fluctuation patterns. Multifractality is measured using various methods, such as generalized Hurst exponents, the generalized partition function method, a detrended fluctuation analysis, the detrended moving average method, and a wavelet transformation.

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References

  1. B. B. Mandelbrot, Int. Econ. Rev. 1, 79 (1960).

    Google Scholar 

  2. B. B. Mandelbrot, Int. Econ. Rev. 4, 111 (1963).

    Google Scholar 

  3. B. B. Mandelbrot, J. Bus. 36, 394 (1963).

    Google Scholar 

  4. B. B. Mandelbrot, Econometrica 39, 68 (1971).

    Google Scholar 

  5. R. N. Mantegna, Physica A 179, 232 (1991).

    ADS  Google Scholar 

  6. R. N. Mantegna and H. E. Stanley, Nature 376, 46 (1995).

    ADS  Google Scholar 

  7. R. N. Mantegna and H. E. Stanley, Nature 383, 587 (1996).

    ADS  Google Scholar 

  8. S. Ghashghaie et al., Nature 381, 767 (1996).

    ADS  Google Scholar 

  9. J. Laherrère and D. Sornette, Eur. Phys. J. B 2, 525 (1998).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  11. P. Gopikrishnan et al., Phys. Rev. E 60, 5305 (1999).

    ADS  Google Scholar 

  12. E. Peters, Fractal Market Analysis-Applying Chaos Theory to Investment and Analysis (John Wiley & Sons, Inc., New York, 1994).

    Google Scholar 

  13. R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics Correlations and Complexity in Finance (Cambridge University Press, New York, 2007).

    MATH  Google Scholar 

  14. R. Cont, Quant. Finance 1, 223 (2001).

    Google Scholar 

  15. V. M. Yakovenko and J. B. Rosser, Rev. Mod. Phys. 81, 1703 (2009).

    ADS  Google Scholar 

  16. E. A. Novikov, J. Appl. Math. Mech. 35, 231 (1971).

    Google Scholar 

  17. B. B. Mandelbrot, J. Fluid Mech. 62, 331 (1974).

    ADS  Google Scholar 

  18. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, New York, 1983).

    Google Scholar 

  19. B. B. Mandelbrot, Pure Appl. Geophys. 131, 5 (1989).

    ADS  Google Scholar 

  20. P. Grassberger, Phys. Lett. A 97, 227 (1983).

    ADS  MathSciNet  Google Scholar 

  21. H. G. E. Hentschel and I. Procaccia, Physica D 8, 435 (1983).

    ADS  MathSciNet  Google Scholar 

  22. P. Grassberger, Phys. Lett. A 107, 101 (1985).

    ADS  MathSciNet  Google Scholar 

  23. P. Grassberger and I. Procaccia, Physica D 13, 34 (1984).

    ADS  MathSciNet  Google Scholar 

  24. T. C. Halsey et al., Phys. Rev. A 33, 1141 (1986).

    ADS  MathSciNet  Google Scholar 

  25. J. Feder, Fractals (Plenum Press, New York, 1988), pp. 193–199.

    MATH  Google Scholar 

  26. B. B. Mandelbrot, Econometrica 38, 122 (1970).

    Google Scholar 

  27. C. Ellis, Physica A 375, 159 (2007).

    ADS  Google Scholar 

  28. L-J. Ji et al., Physica A 388, 3345 (2009).

    ADS  Google Scholar 

  29. A. Montanari, M. S. Taqqu and V. Teverovsky, Math. Comput. Model. 29, 217 (1999).

    Google Scholar 

  30. W. Rea, L. Oxley, M. Reale and J. Brown, Math. Comput. Simul. 93, 29 (2013).

    Google Scholar 

  31. R. Weron, Physica A 312, 285 (2002).

    ADS  MathSciNet  Google Scholar 

  32. A. V. Coronado and P. Carpena, J. Biol. Phys. 31, 121 (2005).

    Google Scholar 

  33. S. V. Muniandya, S. C. Lima and R. Murugan, Physica A 301, 407 (2001).

    ADS  Google Scholar 

  34. M. Couillard and M. Davison, Physica A 348, 404 (2005).

    ADS  Google Scholar 

  35. K. E. Lee and J. W. Lee, J. Korean Phys. Soc. 46, 726 (2005).

    Google Scholar 

  36. K. Kim, S. Y. Kim, M. Lee and M. K. Yum, Int. J. Mod. Phys. C 17, 1831 (2006).

    ADS  Google Scholar 

  37. E. O. Lungu, Econ. Comput. Econ. Cybern. Stud. Res. 44, 105 (2010).

    ADS  Google Scholar 

  38. M. C. Mariani, I. Florescu, M. P. Beccar Varela and E. Ncheuguim, Physica A 389, 1653 (2010).

    ADS  Google Scholar 

  39. J. W. Kantelhardt et al., Physica A 316, 87 (2002).

    ADS  Google Scholar 

  40. P. Grau-Carles, Physica A 287, 396 (2000).

    ADS  Google Scholar 

  41. P. Grau-Carles, Physica A 299, 521 (2001).

    ADS  Google Scholar 

  42. K. E. Lee and J. W. Lee, J. Korean Phys. Soc. 47, 185 (2005).

    Google Scholar 

  43. G-F. Gu and W-X. Zhou, Physica A 383, 497 (2007).

    ADS  Google Scholar 

  44. J. Jiang, K. Ma and X. Cai, Physica A 378, 399 (2007).

    ADS  Google Scholar 

  45. Z. Eisler and J. Kertesz, Phys. Rev. E 73, 046109 (2006).

    ADS  Google Scholar 

  46. Z. Eisler and J. Kertesz, Eur. Phys. J. B 51, 145 (2006).

    ADS  Google Scholar 

  47. Z. Eisler and J. Kertesz, EPL 77, 28001 (2007).

    ADS  Google Scholar 

  48. Z. Eisler and J. J. Kertesz, Physica A 382, 66 (2007).

    ADS  Google Scholar 

  49. R. C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford Univ. Press, Oxford, 1994).

    MATH  Google Scholar 

  50. F. Selcuk, Physica A 333, 306 (2004).

    ADS  MathSciNet  Google Scholar 

  51. T. Di Matteo, T. Aste and M. M. Dacorogna, Physica A 324, 183 (2003).

    ADS  Google Scholar 

  52. S. Bianchi and A. Pianese, Quant. Finance 7, 301 (2007).

    MathSciNet  Google Scholar 

  53. A. Sensoy, Physica A 392, 5019 (2013).

    ADS  Google Scholar 

  54. C. H. Lin, C. S. Chang and S. P. Li, Phys. Rev. E 88, 062912 (2013).

    ADS  Google Scholar 

  55. K. Ivanova and M. Ausloos, Eur. Phys. J. B 8, 665 (1999).

    ADS  Google Scholar 

  56. I. Andreadis and A. Serletis, Chaos Solitons Fractals 13, 1309 (2002).

    ADS  Google Scholar 

  57. E. Canessa, J. Phys. A 33, 3637 (2000).

    ADS  MathSciNet  Google Scholar 

  58. M. Constantin and S. D. Sarma, Phys. Rev. E 72, 051106 (2005).

    ADS  MathSciNet  Google Scholar 

  59. H. Katsuragi, Physica A 278, 275 (2000).

    ADS  Google Scholar 

  60. K. Kim and S. M. Yoon, Physica A 344, 272 (2004).

    ADS  Google Scholar 

  61. P. Norouzzadeh and G. R. Jafari, Physica A 356, 609 (2005).

    ADS  Google Scholar 

  62. M. Ausloos and K. Ivanova, Comput. Phys. Commun. 147, 582 (2002).

    ADS  Google Scholar 

  63. V. G. Kleparskii, Autom. Remote Control 62, 607 (2001).

    MathSciNet  Google Scholar 

  64. F. Selcuk and R. Gencay, Physica A 367, 375 (2006).

    ADS  Google Scholar 

  65. A. Z. Gorski, S. Drozdz and J. Speth, Physica A 316, 496 (2002).

    ADS  Google Scholar 

  66. M. E. Brachet, E. Taflin and J. M. Tcheou, Chaos Solitons Fractals 11, 2343 (2000).

    ADS  Google Scholar 

  67. J. W. Lee, K. E. Lee and P. A. Rikvold, Physica A 364, 355 (2006).

    ADS  Google Scholar 

  68. K. E. Lee and J. W. Lee, J. Korean Phys. Soc. 50, 178 (2007).

    Google Scholar 

  69. D. H. Kim, M. Y. Cha and J. W. Lee, Comput. Phys. Commun. 182, 243 (2011).

    ADS  Google Scholar 

  70. Y. Fujiwara and H. Fujisaka, Physica A 294, 439 (2001).

    ADS  Google Scholar 

  71. Z-Q. Jiang and W-X. Zhou, Chin. Phys. Lett. 26, 028901 (2009).

    ADS  Google Scholar 

  72. J. Alvarez-Ramirez, J. Alvarez, E. Rodriguez and G. Fernandez-Anaya, Physica A 387, 6159 (2008).

    ADS  Google Scholar 

  73. A. Carbone, G. Castelli and H. Stanley, Physica A 344, 267 (2004).

    ADS  MathSciNet  Google Scholar 

  74. L. Czarnecki, D. Grech and G. Pamula, Physica A 387, 6801 (2008).

    ADS  Google Scholar 

  75. T. Di Matteo, Quant. Finance 7, 21 (2007).

    MathSciNet  Google Scholar 

  76. T. Di Matteo, T. Aste and M. Dacorogna, J. Bank. Finance 29, 827 (2005).

    Google Scholar 

  77. D. Grech and Z. Mazur, Physica A 336, 133 (2004).

    ADS  Google Scholar 

  78. J. Matos et al., Physica A 387, 3910 (2008).

    ADS  MathSciNet  Google Scholar 

  79. S. A. R. Rizvi, G. Dewandaru, O. I. Bacha and M. Masih, Physica A 407, 86 (2014).

    ADS  Google Scholar 

  80. P. Ferreira, A. Dionísio and S. M. S. Movahed, Physica A 486, 730 (2017).

    ADS  Google Scholar 

  81. D. Stosic, D. Stosic, T. Stosic and H. E. Stanley, Physica A 428, 46 (2015).

    ADS  Google Scholar 

  82. L. Zunino et al., Chaos Solitons Fractals 41, 2331 (2009).

    ADS  Google Scholar 

  83. S. Arshad and S. A. R. Rizvi, Physica A 419, 158 (2015).

    ADS  Google Scholar 

  84. Q. S. Ruan, B. C. Yang and G. F. Ma, Physica A 468, 91 (2017).

    ADS  Google Scholar 

  85. S. Samadder, K. Ghosh and T. Basu, Fractals 21, 1350003 (2013).

    ADS  MathSciNet  Google Scholar 

  86. S. Dutta, D. Ghosh and S. Chatterjee, Physica A 463, 188 (2016).

    ADS  Google Scholar 

  87. Y. Yuan, X. T. Zhuang and X. Jin, Physica A 388, 2189 (2009).

    ADS  Google Scholar 

  88. Y. Yuan, X. T. Zhuang and Z. Y. Liu, Physica A 391, 3484 (2012).

    ADS  Google Scholar 

  89. M. Y. Bai and H. B. Zhu, Physica A 389, 1883 (2010).

    ADS  Google Scholar 

  90. I-C. Chen, H-J. Chen and H-C. Tseng, Int. J. Mod. Phys. B 23, 4713 (2009).

    ADS  Google Scholar 

  91. E. Green, W. Hanan and D. Heffernan, Eur. Phys. J. B 87, 129 (2014).

    ADS  Google Scholar 

  92. L. Czarnecki and D. Grech, Acta Phys. Pol. A 117, 623 (2010).

    Google Scholar 

  93. R. Rak, S. Drożdż and J. Kwapień, Physica A 374, 315 (2007).

    ADS  Google Scholar 

  94. P. Suárez-García and D. Gómez-Ullate, Physica A 394, 226 (2014).

    ADS  Google Scholar 

  95. G-X. Du and X-X. Ning, Physica A 387, 261 (2008).

    ADS  Google Scholar 

  96. S. J. H. Shahzad, S. M. Nor, W. Mensi and R. R. Kumar, Physica A 471, 351 (2017).

    ADS  Google Scholar 

  97. K. E. Lee and J. W. Lee, J. Korean Phys. Soc. 44, 668 (2004).

    ADS  Google Scholar 

  98. M-Y. Cha, J. W. Lee, D-S. Lee and D. H. Kim, Comput. Phys. Commun. 182, 216 (2011).

    ADS  Google Scholar 

  99. L. A. Quang et al., J. Korean Phys. Soc. 72, 1272 (2018).

    ADS  Google Scholar 

  100. A. Nobi and J. W. Lee, J. Korean Phys. Soc. 71, 733 (2017).

    ADS  Google Scholar 

  101. E. U. Patwary et al., J. Korean Phys. Soc. 71, 444 (2017).

    ADS  Google Scholar 

  102. A. Nobi and J. W. Lee, Chaos 27, 063107 (2017).

    ADS  Google Scholar 

  103. A. Nobi, S. Alam and J. W. Lee, Physica A 482, 337 (2017).

    ADS  Google Scholar 

  104. J. W. Lee and A. Nobi, Comput. Econ. 51, 195 (2018).

    Google Scholar 

  105. A. Nobi and J. W. Lee, Physica A 450, 85 (2016).

    ADS  Google Scholar 

  106. G-G. Ha, J. W. Lee and A. Nobi, J. Korean Phys. Soc. 66, 1802 (2015).

    ADS  Google Scholar 

  107. A. Nobi, S. E. Maeng, G-G. Ha and J. W. Lee, J. Korean Phys. Soc. 66, L1153 (2015).

    ADS  Google Scholar 

  108. A. Nobi, S. Lee, D. H. Kim and J. W. Lee, Phys. Lett. A 378, 2482 (2014).

    ADS  Google Scholar 

  109. A. Nobi, S. E. Maeng, G. G. Ha and J. W. Lee, Physica A 407, 135 (2014).

    ADS  Google Scholar 

  110. A. Nobi, S. E. Maeng, G. G. Ha and J. W. Lee, J. Korean Phys. Soc. 62, 569 (2013).

    ADS  Google Scholar 

  111. J. H. Lim et al., Phys. Lett. A 377, 2542 (2013).

    ADS  Google Scholar 

  112. S. E. Maeng, J. W. Lee and D-S. Lee, Phys. Rev. E 88, 022804 (2013).

    ADS  Google Scholar 

  113. S. E. Maeng, J. W. Lee and D-S. Lee, Phys. Rev. Lett. 108, 108701 (2012).

    ADS  Google Scholar 

  114. D-S. Lee, S. E. Maeng and J. W. Lee, J. Korean Phys. Soc. 60, 648 (2012).

    ADS  Google Scholar 

  115. H. W. Choi, S. E. Maeng and J. W. Lee, J. Korean Phys. Soc. 60, 657 (2012).

    ADS  Google Scholar 

  116. D. H. Kim et al., J. Korean Phys. Soc. 58, 396 (2011).

    Google Scholar 

  117. H. Wang, L. J. Xiang and R. B. Pandey, Physica A 391, 3496 (2012).

    ADS  Google Scholar 

  118. H. Kim, K. Yim, S. Kim and G. Oh, J. Korean Phys. Soc. 67, 2039 (2015).

    ADS  Google Scholar 

  119. X. Y. Zhuang, Y. Wei and F. Ma, Physica A 430, 101 (2015).

    ADS  Google Scholar 

  120. L. S. Yang, Y. M. Zhu and Y. D. Wang, Physica A 451, 357 (2016).

    ADS  Google Scholar 

  121. K. Matia, Y. Ashkenazy and H. E. Stanley, EPL 61, 422 (2003).

    ADS  Google Scholar 

  122. J. de Souza and S. M. D. Queiros, Chaos Solitons Fractals 42, 2512 (2009).

    ADS  Google Scholar 

  123. R. Rak, S. Drożdż and J. Kwapień and P. Oşwięcimka, EPL 112, 48001 (2015).

    ADS  Google Scholar 

  124. D. O. Cajueiro and B. M. Tabak, Chaos Solitons Fractals 40, 497 (2009).

    ADS  Google Scholar 

  125. Z-Y. Su, Y-T. Wang and H-Y. Huang, J. Korean Phys. Soc. 54, 1395 (2009).

    ADS  Google Scholar 

  126. G. Oh, S. Kim and C. Eom, J. Korean Phys. Soc. 56, 982 (2010).

    Google Scholar 

  127. S. Lahmiri, Physica A 486, 183 (2017).

    ADS  Google Scholar 

  128. E. Alessio, A. Carbone, G. Castelli and V. Frappietro, Eur. Phys. J. B 27, 197 (2002).

    ADS  Google Scholar 

  129. A. Carbone, G. Castelli and H. E. Stanley, Physica A 344, 267 (2004).

    ADS  MathSciNet  Google Scholar 

  130. A. Carbone, G. Castelli and H. E. Stanley, Phys. Rev. E 69, 026105 (2004).

    ADS  Google Scholar 

  131. L. M. Xu et al., Phys. Rev. E 71, 051101 (2005).

    ADS  Google Scholar 

  132. Y-H. Shao et al., Sci. Rep. 2, 835 (2012).

    Google Scholar 

  133. G-F. Gu and W-X. Zhou, Phys. Rev. E 82, 011136 (2010).

    ADS  Google Scholar 

  134. A. Y. Schumann and J. W. Kantelhardt, Physica A 390, 2637 (2011).

    ADS  Google Scholar 

  135. P. Horta, S. Lagoa and L. Martins, Int. Rev. Financ. Anal. 35, 140 (2014).

    Google Scholar 

  136. S. S. He and Y. D. Wang, Physica A 467, 11 (2017).

    ADS  MathSciNet  Google Scholar 

  137. D. Xiao and J. Wang, Int. J. Nonlinear Sci. Numer. Simul. 15, 265 (2014).

    MathSciNet  Google Scholar 

  138. W. Y. Cheng and J. Wang, EPL 102, 18004 (2013).

    ADS  Google Scholar 

  139. W. J. Zhou, Y. G. Dang and R. B. Gu, Physica A 392, 1429 (2013).

    ADS  Google Scholar 

  140. M. Holschneider, J. Stat. Phys. 50, 963 (1988).

    ADS  Google Scholar 

  141. A. Arneodo, G. Grasseau and M. Holschneider, Phys. Rev. Lett. 61, 2281 (1988).

    ADS  MathSciNet  Google Scholar 

  142. J. F. Muzy, E. Bacry and A. Arneodo, Phys. Rev. E 47, 875 (1993).

    ADS  Google Scholar 

  143. B. Lashermes, S. G. Roux, P. Abry and S. Jaffard, Eur. Phys. J. B 61, 201 (2008).

    ADS  Google Scholar 

  144. E. Serrano and A. Figliola, Physica A 388, 2793 (2009).

    ADS  MathSciNet  Google Scholar 

  145. W-X. Zhou, Phys. Rev. E 77, 066211 (2008).

    ADS  Google Scholar 

  146. W-B. Shi, P-J. Shang, J. Wang and A-J. Lin, Physica A 403, 35 (2014).

    ADS  Google Scholar 

  147. A. J. Lin, P. J. Shang and H. C. Zhou, Nonlinear Dyn. 78, 485 (2014).

    Google Scholar 

  148. Y. J. Yang, J. P. Li and Y. M. Yang, Physica A 481, 23 (2017).

    ADS  Google Scholar 

  149. G. X. Cao and W. Xu, Physica A 444, 505 (2016).

    ADS  Google Scholar 

  150. J-M. Ghez and S. Vaienti, J. Stat. Phys. 57, 415 (1989).

    ADS  Google Scholar 

  151. J. F. Muzy, E. Bacry and A. Arneodo, Phys. Rev. Lett. 67, 3515 (1991).

    ADS  Google Scholar 

  152. J. F. Muzy, E. Bacry and A. Arneodo, Phys. Rev. E 47, 875 (1993).

    ADS  Google Scholar 

  153. A. N. Pavlov et al., Physica A 300, 310 (2001).

    ADS  Google Scholar 

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Acknowledgments

This work was supported by an Inha University Research Grant.

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Authors and Affiliations

  1. Department of Physics, Inha University, Incheon, 22212, Korea

    Nam Jung, Quang Anh Le, Biseko J. Mafwele, Hyun Min Lee, Seo Yoon Chae & Jae Woo Lee

  2. Institue of Natural Basic Sciences, Inha University, Incheon, 22212, Korea

    Jae Woo Lee

  3. Institue of Advanced Computational Sciences, Inha University, Incheon, 22212, Korea

    Jae Woo Lee

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  1. Nam Jung
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  2. Quang Anh Le
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Jung, N., Le, Q.A., Mafwele, B.J. et al. Fractality and Multifractality in a Stock Market’s Nonstationary Financial Time Series. J. Korean Phys. Soc. 77, 186–196 (2020). https://doi.org/10.3938/jkps.77.186

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