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.
Existe evidencia de que señales provenientes de los mercados financieros, tales como índices bursátiles, tasas de interés, variaciones de precios de productos básicos, tienen naturaleza multifractal. En los últimos años se han hecho esfuerzos para relacionar la ineficiencia de los mercados con las características multifractales de sus correspondientes señales. Estas características se resumen en el conocimiento del espectro de singularidades o espectro multifractal que relaciona al conjunto de puntos singulares de la señal con su correspondiente dimensión de Hausdorff. La novedosa aproximación que se propone en este trabajo, para el estudio de la dinámica de los mercados financieros, es el estudio de la evolución de los puntos singulares o exponentes Hölder locales de las series de índices bursátiles, medidos diariamente. Se analizaron los “retornos logarítmicos” de los índices bursátiles de 9 países pertenecientes a mercados desarrollados y 12 pertenecientes a mercados emergentes, desde febrero de 2006 hasta marzo de 2009. El análisis revela que la variación temporal del exponente Hölder puntual refleja la evolución de la crisis y detecta los eventos históricos que se desarrollaron durante este fenómeno, a partir de los valores mínimos del exponente Hölder puntual.
]]>
The authors wish to thank the support of the Agencia para la Promoción Científica y Técnica, CONICET, the Universidad Nacional de General Sarmiento and the Universidad de San Martín, Argentina.
References
1 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. [ Links ]
2 Jaffard, S.; Meyer, Y. (1996) “Wavelet methods for pointwise regularity and local oscillations of function”, Mem. Amer. Math. Soc. 123: 587. [ Links ] 3 Jaffard, S. (1998) “Oscillation spaces: Properties and applications to fractal and multifractal functions”, J. Math. Phys. 39: 4129–4141. [ Links ] 4 Jaffard, S. (2004) “Wavelet techniques in multifractal analysis”, Proc. Sympos. Pure Math., AMS 72,(2): 91–151. [ Links ] 5 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. [ Links ] 6 Legrand, P.; L´evy Vehel, J. (2003) “Local regularity-based image denoising”, Image Processing, ICIP 2003. Proceedings 3: 377–380. [ Links ] 7 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. [ Links ] 8 Loutridis, S. (2007) “An algorithm for the characterization of timeseries based on local regularity”, Phys. A381: 383–398. [ Links ] 9 Mallat, S. (1989) “Multiresolution approximations and wavelet orthonormal bases of L2”, Trans. Amer. Math. Soc., 315: 69–87. [ Links ] 10 Mallat, S. (2009) A Wavelet Tour of Signal Processing, The Sparse Way, 3rd Edition. Academic Press, Burlington. [ Links ] 11 Meyer, Y. (1997) Wavelets, Vibrations and Scalings, CRMMonograph Series, Vol. 9, AMS. [ Links ] 12 Seuret, S.; Lévy Vehel, J. (2002) “The local Hölder function of a continuous function”, Applied and Computational Harmonic Analysis, 13(3): 263–276. [ Links ] 13 Shang, P.; Lu, Y.; Kama, S. (2006) “The application of Hölder exponent to traffic congestion warning”, Phys. A370: 769–776. [ Links ] 14 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. [ Links ]
*Correspondencia a: Alejandra Figliola. Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J. M. Gutiérrez 1150, C.P. 1613, Los Polvorines, Provincia de Buenos Aires, Argentina, E-Mail: afigliol@ungs.edu.ar Mariel Rosenblatt. Misma dirección que / Same addressas A. Figliola, E-Mail: mrosen@ungs.edu.ar Eduardo P. Serrano. Centro de Matemática Aplicada, Universidad Nacional de San Martín, Campus Miguelete, 25 de Mayo y Francia. C.P. 1650, San Martín, Provincia de Buenos Aires, Argentina. E-Mail: eserrano@unsam.edu.ar
*Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J. M. Gutiérrez 1150, C.P. 1613, Los Polvorines, Provincia de Buenos Aires, Argentina, E-Mail: afigliol@ungs.edu.ar †Misma dirección que / Same address as A. Figliola, E-Mail: mrosen@ungs.edu.ar ‡Centro de Matemática Aplicada, Universidad Nacional de San Martín, Campus Miguelete, 25 de Mayo y Francia. C.P. 1650, San Martín, Provincia de Buenos Aires, Argentina. E-Mail: eserrano@unsam.edu.ar