Investigación

 

Asymptotic behavior of the daily increment distribution of the IPC, the mexican stock market index

 

H.F. Coronel-Brizio and A.R. Hernandez-Montoya

 

Facultad de Física e Inteligencia Artificial, Universidad Veracruzana, Apartado Postal 475, Xalapa, Veracruz, Mexico, e-mail: hcoronel@uv.mx, alhernandez@uv.mx

 

Recibido el 9 de febrero de 2004;
aceptado el 21 de octubre de 2004

 

Abstract

In this work, a statistical analysis of the distribution of daily fluctuations of the IPC, the Mexican Stock Market Index is presented. A sample of the IPC covering the 13-year period 04/19/1990 - 08/21/2003 was analyzed and the cumulative probability distribution of its daily logarithmic variations studied. Results show that the cumulative distribution function for extreme variations, can be described by a Pareto-Levy model with shape parameters α= 3.634 ± 0.272 and α= 3.540 ± 0.278 for its positive and negative tails, respectively. This result is consistent with previous studies, where it has been found that 2.5 < α < 4 for other financial markets worldwide.

Keywords: Econophysics; stock market; Power-Law; stable distribution; Levý regime.

 

Resumen

Presentamos un análisis estadístico de la distribución de fluctuaciones diarias del índice de la Bolsa Mexicana de Valores, el llamado IPC (Índice de Precios y Cotizaciones). Estudiamos la función de distribución acumulativa de las diferencias logarítmicas diarias calculadas a partir de una muestra del IPC que cubre un periodo de 13 años, que empieza el 19/04/1990 y finaliza el 21/08/2003. Hallamos que esta función de distribución acumulativa puede describirse para los valores extremos de estas diferencias mediante una distribución de Pareto-Levy (ley potencia) con exponentes α= 3.634±0.272 y α= 3.540±0.278 en sus colas positiva y negativa respectivamente. Este resultado es consistente con estudios previos que muestran que 2.5 < α < 4 para los mercados financieros de diferentes partes del mundo.

Descriptores: Econofísica; bolsa de valores; ley potencia; distribución estable; régimen de Levý.

 

PACS: 01.75.+m; 02.50.-r; 89.65.Gh; 89.90.+n

 

DESCARGAR ARTÍCULO EN FORMATO PDF

 

References

1. B.M. Roehner, Patterns of Speculation. A Study in Observational Econophysics (Cambridge University Press, United Kingdom 2002) p. 25.         [ Links ]

2. Proceedings of the Workshop "Empirical Science of Financial Fluctuations. The Advent of Econophysics", edited by Hideki Takayasu (Workshop Organized by Nihon Keizai Shimbun, Tokyo 2000).         [ Links ]

3. J. Bouchaud, Physica A 313 (2002) 238.         [ Links ]

4. H. E. Stanley etal., Physica A 269 (1999) 156.         [ Links ]

5. Dietrich Stauffer, Int. J. Mod.Phys. C 11 (2000) 1081.         [ Links ]

6. R.N. Mantegna and H.E. Stanley, An Introduction to Econo-physics (Cambridge university Press, united Kingdom, 2000).         [ Links ]

7. L. Bachelier, Ph.D. Thesis, Théorie de la Spéculation, Annales Scientifiques de l'Ecole Normale Superieure III-17, (1900).         [ Links ]

8. M.F.M. Osborne, Brownian motion in the stock market, P.H. Cootner (Ed.), (The Random Character of Stock Market Prices, The MIT Press, Cambridge, MA 1964) p. 100.         [ Links ]

9. B.B. Mandelbrot, J Business 36 (1963) 394.         [ Links ]

10. E.F. Fama, J Business 38 (1965) 34.         [ Links ]

11. R. N. Mantegna, Physica A 179 (1991) 232.         [ Links ]

12. R. N. Mantegna and H.E. Stanley, Nature 376 (1995) 46.         [ Links ]

13. R.N. Mantegna and H.E. Stanley, Phys.Rev. Lett. 73 (1994) 2946.         [ Links ]

14. R.N. Mantegna and H.E. Stanley, J. Stat. Phys. 89 (1997) 469.         [ Links ]

15. M.S. Baptista and I.L. Caldas, Physica A 284 (2000) 348.         [ Links ]

16. M.S. Baptista etal., Physica A 287 (2000) 91.         [ Links ]

17. P. Gopikrishnan et al., Eur. Phys. J.B 3 (1998) 139.         [ Links ]

18. The Mexican Stock Market index or Indice de Precios y Cotizaciones in spanish.         [ Links ]

19. In finance, referred to as "Stylized facts", these statistical properties of financial time series, in conjuntion with others such as the properties of intermittency, asymmetry in time scales, absence of autocorrelations, gain/loss assymetry, etc.

20. Rama, Cont. Quant. Finance 1 (2001) 223.         [ Links ]

21. P. Levy, Theorie de l'Addition des Variables Aléatories (Gauthier-Villars, Paris, 1937).         [ Links ]

22. B.V. Gnedenko and A. Kolmogorov, Limit Distribution for Sums of Independent Random Variable (Addison.Wesley, Cambridge MA, 1954).         [ Links ]

23. J. Voit, The Statistical Mechanics of Financial Markets, Second Edition (Springer-Verlag 2003) p. 85.         [ Links ]

24. V. Plerou, P. Gopikrishnan, L.A. Amaral, and H.E. Stanley, Quant. Finance 1 (2001) 262.         [ Links ]

25. V. Plerou, P. Gopikrishnan, L.A. Amaral, M. Meyer, and H.E. Stanley, Phys. Rev. E 60 (1999) 6519.         [ Links ]

26. P. Gopikrishnan et al., Phys. Rev. E 60 (1999) 5305.         [ Links ]

27. T. Lux, Applied Financial Economics 6 (1996) 463.         [ Links ]

28. J. Bouchaud, Quantitative Finance 1 (2001) 105.         [ Links ]

29. P. Cizeau, Y. Liu, M. Meyer, C.-K. Peng, and H.E. Stanley, Physica A 245 (1997)441.         [ Links ]

30. P. Cizeau, Y. Liu, M. Meyer, C.-K. Peng, and H. Eugene Stanley, Physica A 245 (1997) 441.         [ Links ]

31. Y. Liu et al., Phys. Rev. E 60 (1999) 1390.         [ Links ]

32. Some authors call returns to the difference of the natural logarithm of prices, and call normalized returns to our returns. For the case of high frequency data, each one approaches the other.

33. S. Salomon and P. Richmond, Physica A 299 (2001) 188.         [ Links ]

34. X. Gabaix et al., Physica A 324 (1996) 1.         [ Links ]

35. X. Gabaix et al., Nature 423 (2003).         [ Links ]

36. Bank of Mexico website: http://www.banxico.org.mx.         [ Links ]

37. In finance, volatility is a relative measure of price movement during a given time period. It can be modeled by the standard deviation of stock price changes. Econophysicists usually use absolute returns to model its dynamics.

38. The 2003 Nobel Price of Economy was awarded jointly to Robert F. Engle for a related topic:"... for methods ofanalyzing economic time series with time-varying volatility (ARCH)". Engle owns a M. S. in Physics by Cornell University (1966).

39. A. Bera and M. Higgins, Journal of Economic Survey 7 (1993) 305.         [ Links ]

40. S. Taylor, Mathematical Finance 4, 183-204.         [ Links ]

41. A.Z. Gorski, S. Drozdz, and J. Speth, Physica A 316 (2003) 496.         [ Links ]

42. R. Gencay, F. Selcuk, and B. Whitcher, An introduction to Wavelets and other filtering methods in Finance and Economics (Academic Press, San Diego 2001).         [ Links ]

43. T.D. Matteo T. Aste, and M.M. Dacorogna, Physica A 324 (2003) 183.         [ Links ]

44. R. Brun and Fons Rademakers, ROOT - An Object Oriented Data Analysis Framework, Proceedings AIHENP'96 Workshop, Lausanne, Sep. 1996, Nucl. Inst.Meth. in Phys. Res. A 389 (1997) 81. See also http://root.cern.ch/.         [ Links ]