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The European Physical Journal B

, Volume 76, Issue 4, pp 513–527 | Cite as

Higher-order phase transitions on financial markets

  • A. Kasprzak
  • R. Kutner
  • J. Perelló
  • J. Masoliver
Focus Section on Applications of Physics in Financial Analysis

Abstract

Statistical and thermodynamic properties of the anomalous multifractal structure of random interevent (or intertransaction) times were thoroughly studied by using the extended continuous-time random walk (CTRW) formalism of Montroll, Weiss, Scher, and Lax. Although this formalism is quite general (and can be applied to any interhuman communication with nontrivial priority), we consider it in the context of a financial market where heterogeneous agent activities can occur within a wide spectrum of time scales. As the main general consequence, we found (by additionally using the Saddle-Point Approximation) the scaling or power-dependent form of the partition function, Z(q’). It diverges for any negative scaling powers q’ (which justifies the name anomalous) while for positive ones it shows the scaling with the general exponent τ(q’). This exponent is the nonanalytic (singular) or noninteger power of q’, which is one of the pilar of higher-order phase transitions. In definition of the partition function we used the pausing-time distribution (PTD) as the central one, which takes the form of convolution (or superstatistics used, e.g. for describing turbulence as well as the financial market). Its integral kernel is given by the stretched exponential distribution (often used in disordered systems). This kernel extends both the exponential distribution assumed in the original version of the CTRW formalism (for description of the transient photocurrent measured in amorphous glassy material) as well as the Gaussian one sometimes used in this context (e.g. for diffusion of hydrogen in amorphous metals or for aging effects in glasses). Our most important finding is the third- and higher-order phase transitions, which can be roughly interpreted as transitions between the phase where high frequency trading is most visible and the phase defined by low frequency trading. The specific order of the phase transition directly depends upon the shape exponent \(\alpha \) defining the stretched exponential integral kernel. On this basis a simple practical hint for investors was formulated.

Keywords

Partition Function Interevent Time High Frequency Trading Frequency Trading Multifractal Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    J. Perelló, J. Masoliver, A. Kasprzak, R. Kutner, Phys. Rev. E 78, 036108 (2008) CrossRefADSGoogle Scholar
  2. 2.
    W.A. Fuller, Introduction to Statistical Time Series (J. Wiley, Ames Iowa, 1976) Google Scholar
  3. 3.
    M.B. Pristley, Non-linear and Non-Stationary Time Series Analysis (Acad. Press, London, 1988) Google Scholar
  4. 4.
    B. Torrésani, Special Issue on Wavelet and Time-Frequency Analysis, J. Math. Phys. 39 (1998) Google Scholar
  5. 5.
    J.C. Sprott, Chaos and Time-Series Analysis (Acad. Press, London, 2003) Google Scholar
  6. 6.
    Handbook of Time Series Analysis: Recent Theoretical Developments and Applications, edited by B. Schelter, M. Winterhalder, J. Timmer (Wiley-VCH, Weinheim, 2006) Google Scholar
  7. 7.
    C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49, (1994) 1685 Google Scholar
  8. 8.
    J.W. Kantelhardt, E. Koscielny-Bunde, H.H.A Rego, S. Havlin, A. Bunde, Physica A 295, 441 (2001) MATHCrossRefADSGoogle Scholar
  9. 9.
    J.W. Kantelhardt, S.A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, H.E. Stanley, Physica A 316, 82 (2002) CrossRefADSGoogle Scholar
  10. 10.
    A. Carbone, G. Castelli, H.E. Stanley, Phys. Rev. E 69, 026105 (2004) CrossRefADSGoogle Scholar
  11. 11.
    E. Alessio, A. Carbone, G. Cstelli, V. Frappietro, Eur. Phys. J. B 27, 197 (2002) ADSGoogle Scholar
  12. 12.
    D. Grech, Z. Mazur, Acta Phys. Pol. B 36, 2403 (2005) ADSGoogle Scholar
  13. 13.
    A.R. Bishop, G. Grüna, B. Nicolaenko, Physica D 23, 1 (1987) Google Scholar
  14. 14.
    A. Aharony, J. Feder, Physica D 38, 1 (1989) CrossRefMathSciNetADSGoogle Scholar
  15. 15.
    F. Schmitt, D. Schertzer, S. Lovejoy, Appl. Stochastic Models Data Anal. 15, 29 (1999) MATHCrossRefGoogle Scholar
  16. 16.
    O. Pont, J.M.D. Delgado, A. Turiel, C.J. Pérez-Vincente, New J. Phys. (2008), in print Google Scholar
  17. 17.
    J.F. Muzy, J. Delour, E. Bacry, Eur. Phys. J. B 17, 537 (2000) CrossRefADSGoogle Scholar
  18. 18.
    R.N. Mantegna, Physica A 179, 232 (1991) CrossRefADSGoogle Scholar
  19. 19.
    D. Sornette, Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools (Springer-Verlag, Berlin, 2000) Google Scholar
  20. 20.
    F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Physica A 287, 468 (2000) CrossRefADSGoogle Scholar
  21. 21.
    E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965) CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    H. Scher, M. Lax, Phys. Rev. B 7, 4491 (1973) CrossRefMathSciNetADSGoogle Scholar
  23. 23.
    H. Scher, E.W. Montroll, Phys. Rev. B 12, 2455 (1975) CrossRefADSGoogle Scholar
  24. 24.
    G. Pfister, H. Scher, Phys. Rev. B 15, 2062 (1977) CrossRefADSGoogle Scholar
  25. 25.
    G. Pfister, H. Scher, Adv. Phys. 27, 747 (1978) CrossRefADSGoogle Scholar
  26. 26.
    E.M. Montroll, B.J. West, in Fluctuation Phenomena, SSM, Vol. VII, edited by E.W. Montroll, J.L. Lebowitz (North-Holland, Amsterdam, 1979), p. 61 Google Scholar
  27. 27.
    E.M. Montroll, M.F. Shlesinger, in Nonequilibrium Phenomena II, From Stochastics to Hydrodynamics, SSM, Vol. XI, edited by J.L. Lebowitz, E.M. Montroll (North-Holland, Amsterdam, 1984), p. 1 Google Scholar
  28. 28.
    G.W. Weiss, A Primer of Random Walkology, in Fractals in Science, edited by A. Bunde, S. Havlin, (Springer-Verlag, Berlin, 1995), Chap. 5, p. 119 Google Scholar
  29. 29.
    E.J. Moore, J. Phys. C: Proc. Phys. Soc. London 7, 339 (1974) ADSGoogle Scholar
  30. 30.
    E. Scalas, R. Gorenflo, F. Mainardi, Phys. Rev. E 69, 011107 (2004) CrossRefMathSciNetADSGoogle Scholar
  31. 31.
    J. Masoliver, M. Montero, J. Perelló, G.H. Weiss, J. Economic Behavior Org. 61, (2006) 577 Google Scholar
  32. 32.
    R. Kutner, Chem. Phys. 284, 481 (2002) CrossRefADSGoogle Scholar
  33. 33.
    R. Kutner, F. Switała, Eur. Phys. J. B 33, 495 (2003) CrossRefADSGoogle Scholar
  34. 34.
    R. Kutner, F. Switała, Quant. Finance 3, 201 (2003) CrossRefMathSciNetGoogle Scholar
  35. 35.
    K.W. Kehr, R. Kutner, K. Binder, Phys. Rev. B 23, 4931 (1981) CrossRefADSGoogle Scholar
  36. 36.
    J.W. Haus, K.W. Kehr, Phys. Rep. 150, 263 (1987) CrossRefADSGoogle Scholar
  37. 37.
    G. Zumofen, J. Klafter, A. Blumen, Models for Anomalous Diffusion, in Disorder Effects on Relaxational processes. Glasses, Polymers, Proteins, edited by R. Richert, A. Blumen (Springer, Berlin, 1994), Chap. 8, p. 251 Google Scholar
  38. 38.
    T. Wichmann, K.W. Kehr, J. Phys.: Condens. Matter 7, (1995) 717 Google Scholar
  39. 39.
    J. Klafter, G. Zumofen, M.F. Shlesinger, in Lévy Flights and Related Topics in Physics, LNP, 450, edited by M.F. Shlesinger, G.H. Zaslavsky, U. Frisch (Springer, Berlin, 1995), p. 196 Google Scholar
  40. 40.
    C. Monthus, J.-P. Bouchaud, J. Phys. A: Math. Gen. 29, 3847 (1996) MATHCrossRefADSGoogle Scholar
  41. 41.
    G. Zumofen, J. Klafter, M.F. Shlesinger, Lévy Flights and Lévy Walks Revisited, in Anomalous Diffusion. From Basics to Applications, LNP, 519, edited by R. Kutner, A. Pękalski, K. Sznajd-Weron (Springer, Berlin, 1999) p. 15 Google Scholar
  42. 42.
    R. Kutner, M. Regulski, Physica A 264, 107 (1999) CrossRefGoogle Scholar
  43. 43.
    R. Hilfer, Fractional Time Evolution, in Applications of Fractional Calculus in Physics, edited by R. Hilfer (World Scient., Singapore, 2000), p. 87 Google Scholar
  44. 44.
    D. ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, Cambridge, 2000) Google Scholar
  45. 45.
    M. Kozłowska, R. Kutner, Physica A 357, 282 (2005) CrossRefADSGoogle Scholar
  46. 46.
    B. Mandelbrot, R.L. Hudson, The (Mis)Behavior of Markets. A Fractal View of Risk, Ruin, and Reward (Basic Books, New York, 2004) Google Scholar
  47. 47.
    L. Li, Y. Meurice, Phys. Rev. D 73, 036006-1 (2006) CrossRefADSGoogle Scholar
  48. 48.
    Th.H. Halsey, M.H. Jensen, L.P. Kadanoff. I. Procaccia, B.I. Shraiman, Phys. Rev. A 33, 1141 (1986) MATHCrossRefMathSciNetADSGoogle Scholar
  49. 49.
    C. Beck, F. Schlögl, Thermodynamics of chaotic systems. An introduction (Cambridge Univ. Press, Cambridge, 1995) Google Scholar
  50. 50.
    L. Harris, J. Financ. Quant. Anal. 22, 127 (1987) CrossRefGoogle Scholar
  51. 51.
    W.M. Fong, W.F. Lab-sane, Quant. Finance 3, 184 (2003) CrossRefMathSciNetGoogle Scholar
  52. 52.
    B. Castaing, Y. Gagne, E. Hopfinger, Physica D 46, 177 (1990) MATHCrossRefADSGoogle Scholar
  53. 53.
    B. Chabaud, A. Naert, J. Peinke, F. Chillà, B. Castaing, B. Hebral, Phys. Rev. Lett. 73, 3227 (1994) CrossRefADSGoogle Scholar
  54. 54.
    C. Tsallis, Braz. J. Phys. 39, 337 (2009) CrossRefGoogle Scholar
  55. 55.
    C. Beck, E.G.D. Cohen, Physica A 322, 267 (2003) MATHCrossRefMathSciNetADSGoogle Scholar
  56. 56.
    C. Beck, E.G.D. Cohen, S. Rizzo, Europhysics News 36/6, 189 (2005) Google Scholar
  57. 57.
    J.-P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990) CrossRefMathSciNetADSGoogle Scholar
  58. 58.
    J.P. Bouchaud, in Lévy Flights and Related Topics in Physics, LNP, 450, edited by M.F. Shlesinger, G.H. Zaslavsky, U. Frisch (Springer, Berlin, 1995), p. 239 Google Scholar
  59. 59.
    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000) MATHCrossRefMathSciNetADSGoogle Scholar
  60. 60.
    R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics, Correlations and Complexity in Finance (Cambridge Univ. Press, Cambridge, UK, 2000) Google Scholar
  61. 61.
    W. Paul, J. Baschnagel, Stochastic Processes, From Physics to Finance (Springer, Berlin 1999) Google Scholar
  62. 62.
    Th. Niemeijer, J.M.J. van Leeuwen, in Phase Transitions and Critical Phenomena, edited by C. Domb, M.S. Green (Acad. Press, London, 1976), Vol. 6 Google Scholar
  63. 63.
    B.B. Mandelbrot, C.J.G. Evertsz, in Fractals and Disordered Systems, 2nd revised and enlarged edn., edited by A. Bunde, Sh. Havlin (Springer, Berlin, 1996) Google Scholar
  64. 64.
    H.E. Stanley, Fractals and Multifractals: The Interplay of Physics and Geometry in Fractals and Disordered Systems, Second Revised and Enlarged Edition, edited by A. Bunde, Sh. Havlin (Springer, Berlin 1996). Google Scholar
  65. 65.
    R. Badii, A. Politi, Complexity. Hierarchical structures and scaling in physics (Cambridge Univ. Press, Cambridge 1997) Google Scholar
  66. 66.
    W.G. Hanan, J. Gough, D.M. Heffernan, Phys. Rev. E 63, 011109 (2000) CrossRefADSGoogle Scholar
  67. 67.
    N.F. Johnson, P. Jefferies, P.M. Hui, Financial Market Complexity (Oxford Univ. Press, Oxford, 2003) Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • A. Kasprzak
    • 1
  • R. Kutner
    • 1
  • J. Perelló
    • 2
  • J. Masoliver
    • 2
  1. 1.Faculty of Physics, University of WarsawWarsawPoland
  2. 2.Departament de Física FonamentalUniversitat de BarcelonaBarcelonaSpain

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