Keywords
Pointwise Hölder Exponent
Wavelet Analysis
Stock Market Indices
Regularidad Local
Exponente Hölder Puntual
Análisis Wavelet
Wavelet Leaders
Indices Bursátiles
How to Cite
Abstract
There is evidence that signals from financial markets, such as stock indices, interest rates or commodities, have a multifractal nature. In recent years, many efforts have been made to relate the inefficiency of markets with the multifractal characteristics of this corresponding signals. These characteristics are summarized in the knowledge of the spectrum of singularities or multifractal spectrum that relates to the set of singular points of the signal with its corresponding Hausdorff dimension. The novel approach proposed in this paper, to study the dynamics of financial markets, is to analyze the evolution of the set of singular points or Hölder exponents of the series of exchanges, measured daily. We examined the “logarithmic returns” of stock indices from 9 countries in developed markets and 12 belonging to emerging markets from February 2006 to March 2009.The analysis reveals that the temporal variation of the local Hölder exponent point reflects the evolution of the crisis and identifies the historical events which have occurred during this phenomenon, from the minimum values of the Hölder exponent.
References
Bony, J.M. (1986) “Second microlocalization and propagation of singularities for semilinear hyperbolic equations”, in Hyperbolic Equations and Related Topics (Kata/Kyoto,1984); Academic Press: Boston: 11-49.
Jaffard, S.; Meyer, Y. (1996) “Wavelet methods for pointwise regularity and local oscillations of function”, Mem. Amer. Math. Soc. 123:587.
Jaffard, S. (1998) “Oscillation spaces: Properties and applications to fractal and multifractal functions”, J. Math. Phys. 39: 4129–4141.
Jaffard, S. (2004) “Wavelet techniques in multifractal analysis”, Proc. Sympos. Pure Math., AMS 72,(2): 91–151.
Kantelhardt, J.W.; Zschiegner,S. A.; Koscielny-Bunde, E.; Havlin, S.; Bunde, A.; Stanley, H. E. (2002) “Multifractal detrended fluctuation analysis of nonstationary time series”, Phys. A316: 87–114.
Legrand, P.; Lévy Vehel, J. (2003) “Local regularity-based image de-noising”, Image Processing, ICIP 2003. Proceedings 3: 377–380.
Lévy Vehel, J; Lutton, E. (2001) “Evolutionary signal enhancement based on Hölder regularity analysis”, in: Application in Evolutionary Computing, Lecture Notes in Computer Science; Springer, Berlin: 325–334.
Loutridis, S. (2007) “An algorithm for the characterization of time-series based on local regularity”, Phys. A381: 383–398.
Mallat, S. (1989) “Multiresolution approximations and wavelet orthonormal bases of L2”, Trans. Amer. Math. Soc., 315: 69–87.
Mallat, S. (2009)A Wavelet Tour of Signal Processing, The Sparse Way, 3rd Edition. Academic Press, Burlington.
Meyer, Y. (1997) Wavelets, Vibrations and Scalings, CRM Monograph Series, Vol. 9, AMS.
Seuret, S.; Lévy Vehel, J. (2002) “The local Hölder function of a continuous function”, Applied and Computational Harmonic Analysis, 13(3): 263–276.
Shang, P.; Lu, Y.; Kama, S. (2006) “The application of Hölder exponent to traffic congestion warning”, Phys. A370: 769–776.
Zunino, L.; Figliola, A.; Tabak, B.M.; Pérez, D.G.; Garavaglia, M.; Rosso, O. A. (2009) “Multifractal structure in Latin-American market indices”, Chaos, Solitons & Fractals 41,(5): 2330–2339.