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The Dynamics of Financial Markets -- Mandelbrot's multifractal cascades, and beyond

  • Lisa Borland

    (Evnine-Vaughan Associates, Inc.)

  • Jean-Philippe Bouchaud

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

  • Jean-Francois Muzy

    (Centre de Recherche Paul Pascal, Pessac, FRANCE)

  • Gilles Zumbach

    (Consulting in Financial Research)

Registered author(s):

    This is a short review in honor of B. Mandelbrot's 80st birthday, to appear in W ilmott magazine. We discuss how multiplicative cascades and related multifractal ideas might be relevant to model the main statistical features of financial time series, in particular the intermittent, long-memory nature of the volatility. We describe in details the Bacry-Muzy-Delour multifractal random walk. We point out some inadequacies of the current models, in particular concerning time reversal symmetry, and propose an alternative family of multi-timescale models, intermediate between GARCH models and multifractal models, that seem quite promising.

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    Paper provided by Science & Finance, Capital Fund Management in its series Science & Finance (CFM) working paper archive with number 500061.

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    Date of creation: Jan 2005
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    Handle: RePEc:sfi:sfiwpa:500061
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    1. Liu, Yanhui & Cizeau, Pierre & Meyer, Martin & Peng, C.-K. & Eugene Stanley, H., 1997. "Correlations in economic time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 245(3), pages 437-440.
    2. Gilles Zumbach, 2004. "Volatility processes and volatility forecast with long memory," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 70-86.
    3. Andrew W. Lo, 1989. "Long-term Memory in Stock Market Prices," NBER Working Papers 2984, National Bureau of Economic Research, Inc.
    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. Marc Potters & Rama Cont & Jean-Philippe Bouchaud, 1996. "Financial markets as adaptative systems," Science & Finance (CFM) working paper archive 500037, Science & Finance, Capital Fund Management.
    6. Benoit Pochart & Jean-Philippe Bouchaud, 2002. "The skewed multifractal random walk with applications to option smiles," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 303-314.
    7. Benoit Pochard & Jean-Philippe Bouchaud, 2002. "The skewed multifractal random walk with applications to option smiles," Science & Finance (CFM) working paper archive 0204047, Science & Finance, Capital Fund Management.
    8. Laurent Calvet & Adlai Fisher, 2002. "Multifractality In Asset Returns: Theory And Evidence," The Review of Economics and Statistics, MIT Press, vol. 84(3), pages 381-406, August.
    9. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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    12. T. Lux, 2001. "Turbulence in financial markets: the surprising explanatory power of simple cascade models," Quantitative Finance, Taylor & Francis Journals, vol. 1(6), pages 632-640.
    13. Miccichè, Salvatore & Bonanno, Giovanni & Lillo, Fabrizio & Mantegna, Rosario N, 2002. "Volatility in financial markets: stochastic models and empirical results," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 756-761.
    14. Potters, Marc & Bouchaud, Jean-Philippe & Sestovic, Dragan, 2001. "Hedged Monte-Carlo: low variance derivative pricing with objective probabilities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 289(3), pages 517-525.
    15. Longin, Francois M, 1996. "The Asymptotic Distribution of Extreme Stock Market Returns," The Journal of Business, University of Chicago Press, vol. 69(3), pages 383-408, July.
    16. J-P. Bouchaud & M. Potters, 2001. "Welcome to a non-Black-Scholes world," Quantitative Finance, Taylor & Francis Journals, vol. 1(5), pages 482-483.
    17. Yanhui Liu & Pierre Cizeau & Martin Meyer & Chung-Kang Peng & H. Eugene Stanley, 1997. "Correlations in Economic Time Series," Papers cond-mat/9706021, arXiv.org.
    18. Lisa Borland, 2002. "A theory of non-Gaussian option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 415-431.
    19. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-47.
    20. Marc Potters & Jean-Philippe Bouchaud & Dragan Sestovic, 2000. "Hedged Monte-Carlo: low variance derivative pricing with objective probabilities," Science & Finance (CFM) working paper archive 500031, Science & Finance, Capital Fund Management.
    21. V. Plerou & P. Gopikrishnan & L. A. N. Amaral & M. Meyer & H. E. Stanley, 1999. "Scaling of the distribution of price fluctuations of individual companies," Papers cond-mat/9907161, arXiv.org.
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