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# Estimating the Fractal Dimension of the S&P 500 Index using Wavelet Analysis

## Author

Listed:
• Erhan Bayraktar
• H. Vincent Poor
• Ronnie Sircar

## Abstract

S&P 500 index data sampled at one-minute intervals over the course of 11.5 years (January 1989- May 2000) is analyzed, and in particular the Hurst parameter over segments of stationarity (the time period over which the Hurst parameter is almost constant) is estimated. An asymptotically unbiased and efficient estimator using the log-scale spectrum is employed. The estimator is asymptotically Gaussian and the variance of the estimate that is obtained from a data segment of $N$ points is of order $\frac{1}{N}$. Wavelet analysis is tailor made for the high frequency data set, since it has low computational complexity due to the pyramidal algorithm for computing the detail coefficients. This estimator is robust to additive non-stationarities, and here it is shown to exhibit some degree of robustness to multiplicative non-stationarities, such as seasonalities and volatility persistence, as well. This analysis shows that the market became more efficient in the period 1997-2000.

## Suggested Citation

• Erhan Bayraktar & H. Vincent Poor & Ronnie Sircar, 2007. "Estimating the Fractal Dimension of the S&P 500 Index using Wavelet Analysis," Papers math/0703834, arXiv.org.
• Handle: RePEc:arx:papers:math/0703834
as

File URL: http://arxiv.org/pdf/math/0703834

## References listed on IDEAS

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8. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2006. "A Limit Theorem for Financial Markets with Inert Investors," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 789-810, November.
9. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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## Citations

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Cited by:

1. Alper Ozun & Atilla Cifter, 2008. "Modeling long-term memory effect in stock prices: A comparative analysis with GPH test and Daubechies wavelets," Studies in Economics and Finance, Emerald Group Publishing, vol. 25(1), pages 38-48, March.
2. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2006. "A Limit Theorem for Financial Markets with Inert Investors," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 789-810, November.
3. Stephanie Rendón de la Torre, 2012. "Estimación del coeficiente de Hurst con wavelets de índices accionarios de Turquía, Indonesia, México y Corea del Sur," Revista de Administración, Finanzas y Economía (Journal of Management, Finance and Economics), Tecnológico de Monterrey, Campus Ciudad de México, vol. 6(2), pages 27-50.
4. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "Queueing Theoretic Approaches to Financial Price Fluctuations," Papers math/0703832, arXiv.org.

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