Dominique, C-René and Rivera-Solis, Luis Eduardo (2011): Mixed fractional Brownian motion, short and long-term Dependence and economic conditions: the case of the S&P-500 Index. Forthcoming in: International Business and Management , Vol. Vol.3, No. No.2 (30. November 2011): pp. 1-13.
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The Kolmogorov-Mandelbrot-van Ness Process is a zero mean Gaussian process indexed by the Hurst Parameter (H). When it models financial data, a controversy arises as to whether or not financial data exhibit short or long-range dependence. This paper argues that the Mixed Fractional Brownian is a more suitable tool for the purpose as it leaves no room for controversy. It is used here to model the S&P-500 Index, sampled daily over the period 1950-2011. The main results are as follows: The S&P-500 Index is characterized by both short and long-term dependence. More explicitly, it is characterized by at least 12 distinct scaling pa-rameters that are, ex hypothesis, determined by investors’ approach to the market. When the market is dominated by “blue-chippers” or ‘long-termists’, or when bubbles are ongoing, the index is persistent; and when the market is dominated by “con-trarians”, the index jumps to anti-persistence that is a far-from-equilibrium state in which market crashes are likely to occur.
|Item Type:||MPRA Paper|
|Original Title:||Mixed fractional Brownian motion, short and long-term Dependence and economic conditions: the case of the S&P-500 Index|
|Keywords:||Gaussian Processes; Mixed Fractional Brownian Motion; Hurst Exponent; Local Self-similarity, Persistence; Anti-persistence; Definiteness of covariance Functions; Dissipative dynamic systems|
|Subjects:||C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models; Multiple Variables > C32 - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models
D - Microeconomics > D5 - General Equilibrium and Disequilibrium > D53 - Financial Markets
C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C16 - Specific Distributions
C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods
|Depositing User:||Luis Rivera|
|Date Deposited:||20. Nov 2011 14:20|
|Last Modified:||13. Feb 2013 15:24|
1] Alvarez-Ramirez, T. et al. (2008). “ Time-varying Hurst Exponent for US Stock Markets.” Physica A, 1959- 1969.  Aydogan, K. and Booth, G. (1988). “Are There Long Cycles in Common Stock Returns.” Southern Jour. of Econ.., 55, 141-149.  Baraktur, E. et al. (2003). “Estimating the Fractal Dimension of the S&P-500 Index Using Wavelet Analysis.” E-Quad Pa-per, Dept. of Electrical Eng., Princeton Univ., Princeton, N.J. 08544.  Cheung, Yin-Wang (1993). “Long-memory in Foreign Exchange Rates.” Jour. of Bus. And Econ. Stat., 11, 93-101.  Dominique, C-R., Rivera, S. L. and Desrosiers, F. (2011). “Determining the Value-at-risk in the Shadow of the Power Law.” The Jour. of Global Bus. and Tech., 7, 1-23.  Gençay, R., Selcuk, F. and Whither, B. (2001). “Scaling Properties of Foreign Exchange Volatility.” Physica A, 289, 249-266.  Gopikrishan, P. et al. (1999). “Scaling of the Distribution of Fluctuations in Financial Market Indices. “ Physica E, 60, 5305-5310.  Greene, M. T. and Fielitz, B. D. (1977). “Long-term Dependence in Common Stock Returns.” Jour. of Financial Econ., 4, 339-349.  Kaplan, L. M. and Jay Kuo, C. C. (1993). “Fractal Estimation from Noisy Data via Discrete Fractional Gaussian Noise and the Haar Basis.” IEEE Transactions, 41, 3554-3562.  Lo, A. (1991) “Long-term Memory in Stock Market Prices.” Econometrica, 59, 1279-1313.  Lobato, I. N. and Savin, N. E. (1998). “ Real and Spurious Long-memory Properties of Stock Market Data.” Jour. of Bus. and Econ. Stat., 16, 261-268.  Lux, T. (1996). “Long-term Stochastic Dependence in Financial Prices: Evidence from the German Stock Market.” Applied Econ. Letters, 3 701-706.  Mandelbrot, B. et al. (1997). “A Multi-fractal Model of Asset Returns.” Cowles Foundation for Econ. Res. Working Paper no 39.  Mandelbrot, B. and van Ness, J. (1968). “Fractional Brownian Motions, Fractional Noises and Applications.” SIAM Rev., 10, 422-437.  Miao, Y. et al. (2008). “On the Fractional Mixed Fractional Brownian Motion.” Applied Math. Science, 35, 1729-1738.  Peters, E. (1994). Fractal Analysis: Applying Chaos to Investment and Economics. New York: John Wiley and Sons.  Preciado, J. and Morris, H. (2008). “The Varying Behavior of U. S. Market Persistence.” Proceedings of the World Congress on Engineering and Computer Science 2008, held in San Francisco, USA on Oct. 22-24,2008.  Sottinen, T. (2003). “Fractional Brownian Motion in Finance and Queuing.” Ph.D Dissertation, Univ. of Helsinki, Finland.  Thale, C. (2009). “Further Remarks on Mixed Fractional Motion.” Applied Math. Science, 3, 1-17.  Yalamova, R. (2005).”Multifractal Spectrum of Financial Time Series.” MFC-180%20 www: pdftop.com.,1-21.  Zili, M. (2006). “On the Mixed Fractional Brownian Motion.” Jour. of Applied Math. And Stochastic Analyses, 2006, 1-9.