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Scaling transformation and probability distributions for financial time series


  • Marc-Etienne BRACHET

    (Laboratoire de Physique Statistique, CNRS, ENS, France)

  • Erik TAFLIN

    (Direction Scientifique, AXA-UAP, France)

  • Jean Marcel TCHEOU

    (Laboratoire de Physique Statistique, ENS, France; Direction Scientifique, AXA-UAP, France)


The price of financial assets are, since Bachelier, considered to be described by a (discrete or continuous) time sequence of random variables, i.e a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works. In this letter we investigate the question of scaling transformation of price processes by establishing a new connexion between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments.

Suggested Citation

  • Marc-Etienne BRACHET & Erik TAFLIN & Jean Marcel TCHEOU, 1999. "Scaling transformation and probability distributions for financial time series," GE, Growth, Math methods 9901003, University Library of Munich, Germany.
  • Handle: RePEc:wpa:wuwpge:9901003
    Note: Type of Document - PostScript; prepared on TeX; to print on PostScript-color; pages: 12; figures: included-EPS

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    References listed on IDEAS

    1. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
    2. Adlai Fisher & Laurent Calvet & Benoit Mandelbrot, 1997. "Multifractality of Deutschemark/US Dollar Exchange Rates," Cowles Foundation Discussion Papers 1166, Cowles Foundation for Research in Economics, Yale University.
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    Cited by:

    1. Bacry, E. & Delour, J. & Muzy, J.F., 2001. "Modelling financial time series using multifractal random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 84-92.
    2. Struzik, Zbigniew R. & Siebes, Arno P.J.M., 2002. "Wavelet transform based multifractal formalism in outlier detection and localisation for financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 309(3), pages 388-402.
    3. Struzik, Zbigniew R., 2003. "Econonatology: the physics of the economy in labour," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 344-351.

    More about this item


    multifractal; scaling; exchange rate; stock index; non-linear group action;

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • D5 - Microeconomics - - General Equilibrium and Disequilibrium
    • D9 - Microeconomics - - Micro-Based Behavioral Economics

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