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Estimating The Fractal Dimension Of The S&P 500 Index Using Wavelet Analysis

Author

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  • ERHAN BAYRAKTAR

    (Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA)

  • H. VINCENT POOR

    (Department of Electrial Engineering, Princeton University, Princeton, NJ 08544, USA)

  • K. RONNIE SIRCAR

    (Department of Operations Research & Financial Engineering, Princeton University, E-Quad, Princeton, NJ 08544, USA)

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 ofNpoints 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 suggests that the market became more efficient in the period 1997–2000.

Suggested Citation

  • Erhan Bayraktar & H. Vincent Poor & K. Ronnie Sircar, 2004. "Estimating The Fractal Dimension Of The S&P 500 Index Using Wavelet Analysis," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(05), pages 615-643.
    • Handle: RePEc:wsi:ijtafx:v:07:y:2004:i:05:n:s021902490400258x
    DOI: 10.1142/S021902490400258X
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    Cited by:

    1. Vuorenmaa, Tommi A., 2005. "A wavelet analysis of scaling laws and long-memory in stock market volatility," Research Discussion Papers 27/2005, Bank of Finland.
    2. Lahiri, Ananya & Sen, Rituparna, 2020. "Fractional Brownian markets with time-varying volatility and high-frequency data," Econometrics and Statistics, Elsevier, vol. 16(C), pages 91-107.
    3. Deniz Kenan Kılıç & Ömür Uğur, 2018. "Multiresolution analysis of S&P500 time series," Annals of Operations Research, Springer, vol. 260(1), pages 197-216, January.
    4. Josselin Garnier & Knut Solna, 2018. "Emergence of Turbulent Epochs in Oil Prices," Papers 1808.09382, arXiv.org, revised Apr 2019.
    5. Josselin Garnier & Knut Solna, 2018. "Chaos and Order in the Bitcoin Market," Papers 1809.08403, arXiv.org, revised Apr 2019.
    6. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "A Limit Theorem for Financial Markets with Inert Investors," Papers math/0703831, arXiv.org.
    7. Kohei Hayashi & Kei Nakagawa, 2022. "Fractional SDE-Net: Generation of Time Series Data with Long-term Memory," Papers 2201.05974, arXiv.org, revised Aug 2022.
    8. Garnier, Josselin & Solna, Knut, 2019. "Emergence of turbulent epochs in oil prices," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 281-292.
    9. Xiao, Wei-Lin & Zhang, Wei-Guo & Zhang, Xi-Li & Wang, Ying-Luo, 2010. "Pricing currency options in a fractional Brownian motion with jumps," Economic Modelling, Elsevier, vol. 27(5), pages 935-942, September.
    10. Yipeng Yang & Allanus Tsoi, 2016. "A Level Set Analysis and A Nonparametric Regression on S&P 500 Daily Return," IJFS, MDPI, vol. 4(1), pages 1-24, February.
    11. 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.
    12. Liu, Xiaoquan & Cao, Yi & Ma, Chenghu & Shen, Liya, 2019. "Wavelet-based option pricing: An empirical study," European Journal of Operational Research, Elsevier, vol. 272(3), pages 1132-1142.
    13. Ozun, Alper & Cifter, Atilla, 2007. "Modeling Long-Term Memory Effect in Stock Prices: A Comparative Analysis with GPH Test and Daubechies Wavelets," MPRA Paper 2481, University Library of Munich, Germany.
    14. Kim, Kyong-Hui & Kim, Nam-Ung & Ju, Dong-Chol & Ri, Ju-Hyang, 2020. "Efficient hedging currency options in fractional Brownian motion model with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    15. 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.
    16. Raluca M. Balan & Ciprian A. Tudor, 2010. "Stochastic Heat Equation with Multiplicative Fractional-Colored Noise," Journal of Theoretical Probability, Springer, vol. 23(3), pages 834-870, September.
    17. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "Queueing Theoretic Approaches to Financial Price Fluctuations," Papers math/0703832, arXiv.org.
    18. repec:zbw:bofrdp:2005_027 is not listed on IDEAS
    19. Dominique, C-Rene, 2018. "Assessing the Entropies of the Feigenbaum Strange Attractor and the S&P-500 Index as Factors Driving the Production of Information in Market Economies," MPRA Paper 89873, University Library of Munich, Germany, revised 05 Nov 2018.
    20. Haven, Emmanuel & Liu, Xiaoquan & Shen, Liya, 2012. "De-noising option prices with the wavelet method," European Journal of Operational Research, Elsevier, vol. 222(1), pages 104-112.
    21. Mariusz Tarnopolski, 2017. "Modeling the price of Bitcoin with geometric fractional Brownian motion: a Monte Carlo approach," Papers 1707.03746, arXiv.org, revised Aug 2017.
    22. Frezza, Massimiliano, 2012. "Modeling the time-changing dependence in stock markets," Chaos, Solitons & Fractals, Elsevier, vol. 45(12), pages 1510-1520.
    23. Garnier, Josselin & Solna, Knut, 2019. "Chaos and order in the bitcoin market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 708-721.
    24. Jean de Carufel & Martin Brooks & Michael Stieber & Paul Britton, 2017. "A Topological Approach to Scaling in Financial Data," Papers 1710.08860, arXiv.org.

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