Advertisement

Fractality of profit landscapes and validation of time series models for stock prices

Regular Article

Abstract

We apply a simple trading strategy for various time series of real and artificial stock prices to understand the origin of fractality observed in the resulting profit landscapes. The strategy contains only two parameters p and q, and the sell (buy) decision is made when the log return is larger (smaller) than p (−q). We discretize the unit square (p,q) ∈ [0,1] × [0,1] into the N × N square grid and the profit Π(p,q) is calculated at the center of each cell. We confirm the previous finding that local maxima in profit landscapes are scattered in a fractal-like fashion: the number M of local maxima follows the power-law form MN a , but the scaling exponent a is found to differ for different time series. From comparisons of real and artificial stock prices, we find that the fat-tailed return distribution is closely related to the exponent a ≈ 1.6 observed for real stock markets. We suggest that the fractality of profit landscape characterized by a ≈ 1.6 can be a useful measure to validate time series model for stock prices.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics (Cambridge University Press, Cambridge, 2000)Google Scholar
  2. 2.
    J.P. Bouchard, M. Potters, Theory of Financial Risk and Derivative Pricing (Cambridge University Press, Cambridge, 2009)Google Scholar
  3. 3.
    S. Sinha, A. Chatterjee, A. Chakraborti, B.K. Chakrabarti, Econophysics: An Introduction (Wiley-VCH, Weinheim, 2010)Google Scholar
  4. 4.
    A. Grönlund, I.G. Yi, B.J. Kim, PLoS ONE 7, e33960 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    F. Black, The Journal of Political Economy 81, 637 (1973)CrossRefGoogle Scholar
  6. 6.
    J.C. Hull, Options, Futures, and Other Derivatives (Pearson Education Ltd., London, 2009)Google Scholar
  7. 7.
    B.B. Mandelbrot, J.W. Van Ness, SIAM Rev. 10, 422 (1968)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Z. Ding, C.W.J. Granger, R.F. Engle, Journal of Empirical Finance 1, 83 (1993)CrossRefGoogle Scholar
  9. 9.
    Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, H.E. Stanley, Phys. Rev. E 60, 1390 (1999)ADSCrossRefGoogle Scholar
  10. 10.
    R. Cont, Quantitative Finance 1, 223 (2001)CrossRefGoogle Scholar
  11. 11.
    G. Oh, S. Kim, J. Korean Phys. Soc. 48, S197 (2006)Google Scholar
  12. 12.
    R.N. Mantegna, Phys. Rev. E 49, 4677 (1994)ADSCrossRefGoogle Scholar
  13. 13.
    E. Pantaleo, P. Facchi, S. Pascazio, Phys. Scripta T135, 014036 (2009)ADSCrossRefGoogle Scholar
  14. 14.
    W. Schoutens, Lévy process in Finance (John Wiley & Sons Ltd., New York, 2003)Google Scholar
  15. 15.
    G. Fusai, A. Roncoroni, Implementing Models in Quantitative Finance: Methods and Cases (Springer-Verlag, Berlin, 2008)Google Scholar
  16. 16.
    D. Applebaum, Lévy Processes and Stochastic Calculus (Cambridge University Press, Cambridge, 2009)Google Scholar
  17. 17.
    L.E. Calvet, A.J. Fisher, Multifractal Volatility (Academic Press, Burlington, 2008)Google Scholar
  18. 18.
    L.E. Calvet, A.J. Fisher, Journal of Econometrics 105, 27 (2001).MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    L.E. Calvet, A.J. Fisher, Journal of Financial Econometrics 2, 49 (2004)CrossRefGoogle Scholar
  20. 20.
    P. Gopikrishnan, V. Plerou, L.A.N. Amaral, M. Meyer, H.E. Stanley, Phys. Rev. E 60, 5305 (1999)ADSCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.BK21 Physics Research Division and Department of PhysicsSungkyunkwan UniversitySuwonKorea
  2. 2.Division of Business AdministrationChosun UniversityGwangjuKorea

Personalised recommendations