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Fractal Market Time

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  • James McCulloch
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    Abstract

    Ane and Geman (2000) observed that market returns appear to follow a conditional Gaussian distribution where the conditioning is a stochastic clock based on cumulative transaction count. The existence of long range dependence in the squared and absolute value of market returns is a 'stylized fact' and researchers have interpreted this to imply that the stochastic clock is self-similar, multi-fractal (Mandelbrot, Fisher and Calvet; 1997) or mono-fractal (Heyde; 1999). We model the market stochastic clock as the stochastic integrated intensity of a doubly stochastic Poisson (Cox) point process of the cumulative transaction count of stocks traded on the New York Stock Exchange (NYSE). A comparative empirical analysis of a self-normalized version of the stochastic integrated intensity is consistent with a mono-fractal market clock with a Hurst exponent of 0.75.

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    File URL: http://www.qfrc.uts.edu.au/research/research_papers/rp311.pdf
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    Bibliographic Info

    Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number 311.

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    Length: 42
    Date of creation: 01 Aug 2012
    Date of revision:
    Handle: RePEc:uts:rpaper:311

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    Keywords: market time deformation; long range dependent; stochastic clock; fractal activity time; doubly stochastic binomial point process;

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    1. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
    2. James McCulloch, 2005. "Relative Volume as a Doubly Stochastic Binomial Point Process," Research Paper Series 146, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. 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 637-54, May-June.
    4. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
    5. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-24, October.
    6. Brock, William A. & Kleidon, Allan W., 1992. "Periodic market closure and trading volume : A model of intraday bids and asks," Journal of Economic Dynamics and Control, Elsevier, vol. 16(3-4), pages 451-489.
    7. Eugene F. Fama, 1963. "Mandelbrot and the Stable Paretian Hypothesis," The Journal of Business, University of Chicago Press, vol. 36, pages 420.
    8. Le Fol, Gaëlle & Mercier, Ludovic, 1998. "Time Deformation: Definition and Comparisons," Economics Papers from University Paris Dauphine 123456789/12729, Paris Dauphine University.
    9. Anat R. Admati, Paul Pfleiderer, 1988. "A Theory of Intraday Patterns: Volume and Price Variability," Review of Financial Studies, Society for Financial Studies, vol. 1(1), pages 3-40.
    10. Lancelot F. James & Antonio Lijoi & Igor Prünster, 2006. "Conjugacy as a Distinctive Feature of the Dirichlet Process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 33(1), pages 105-120.
    11. O. E. Barndorff-Nielsen & S. Z. Levendorskii, 2001. "Feller processes of normal inverse Gaussian type," Quantitative Finance, Taylor & Francis Journals, vol. 1(3), pages 318-331.
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    Cited by:
    1. Batten, Jonathan A. & Kinateder, Harald & Wagner, Niklas, 2014. "Multifractality and value-at-risk forecasting of exchange rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 401(C), pages 71-81.

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