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On a multi-timescale statistical feedback model for volatility fluctuations

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  • Lisa Borland

    (Evnine-Vaughan Associates, Inc.)

  • Jean-Philippe Bouchaud

    (Science & Finance, Capital Fund Management
    CEA Saclay;)

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    Abstract

    We study, both analytically and numerically, an ARCH-like, multiscale model of volatility, which assumes that the volatility is governed by the observed past price changes on different time scales. With a power-law distribution of time horizons, we obtain a model that captures most stylized facts of financial time series: Student-like distribution of returns with a power-law tail, long-memory of the volatility, slow convergence of the distribution of returns towards the Gaussian distribution, multifractality and anomalous volatility relaxation after shocks. At variance with recent multifractal models that are strictly time reversal invariant, the model also reproduces the time assymmetry of financial time series: past large scale volatility influence future small scale volatility. In order to quantitatively reproduce all empirical observations, the parameters must be chosen such that our model is close to an instability, meaning that (a) the feedback effect is important and substantially increases the volatility, and (b) that the model is intrinsically difficult to calibrate because of the very long range nature of the correlations. By imposing the consistency of the model predictions with a large set of different empirical observations, a reasonable range of the parameters value can be determined. The model can easily be generalized to account for jumps, skewness and multiasset correlations.

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    Bibliographic Info

    Paper provided by Science & Finance, Capital Fund Management in its series Science & Finance (CFM) working paper archive with number 500059.

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    Date of creation: Jul 2005
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    Handle: RePEc:sfi:sfiwpa:500059

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    7. Lisa Borland & Jean-Philippe Bouchaud, 2004. "A non-Gaussian option pricing model with skew," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 4(5), pages 499-514.
    8. Adrian Dragulescu & Victor Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(6), pages 443-453.
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    19. Longin, Francois M, 1996. "The Asymptotic Distribution of Extreme Stock Market Returns," The Journal of Business, University of Chicago Press, University of Chicago Press, vol. 69(3), pages 383-408, July.
    20. Lisa Borland, 2002. "A theory of non-Gaussian option pricing," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(6), pages 415-431.
    21. A. Dragulescu & V. M. Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Computing in Economics and Finance 2002, Society for Computational Economics 127, Society for Computational Economics.
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