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Multifractality in Finance: A deep understanding and review of Mandelbrot's MMAR

Author

Listed:
  • Federico Maglione

    (Department of Economics, University Of Venice C� Foscari)

Abstract

Beno�t Mandelbrot, the father of Fractal Geometry, developed a multifractal model for describing price changes. Despite the commonly used models, such as the Brownian motion, the Mutifractal Model of Asset Return (MMAR) takes into account scale-consistency, long-range dependence and heavy tails, thus having a great flexibility in depicting the real-market peculiarities. In section 2 a review of the mathematics involved into multifractals is presented; Section 3 is addresses to the extension of multifractality towards stochastic processes, introducing the crucial concept of local H\"older exponent of a function. Finally, Section 4 deeply analyze the mathematical properties of the scaling function which drives the ``wildeness'' of the process. The proof of Theorem 4.4 is unpublished and the generalization of a Mandelbrot's result, which highlights a possible alternative motivation for the presence of heavy tails and a connection with the Extreme Value Theory. Section 5 is devoted to the analysis of the connection between the scaling function, Multifractal Formalism and Large Deviation Theory, suggesting possible ways in order to estimate the quantities involved. Finally in Section 6 the MMAR is presented, listing all the theorems that make it a suitable model for financial modelling.

Suggested Citation

  • Federico Maglione, 2015. "Multifractality in Finance: A deep understanding and review of Mandelbrot's MMAR," Working Papers 2015:05, Department of Economics, University of Venice "Ca' Foscari".
    • Handle: RePEc:ven:wpaper:2015:05
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    References listed on IDEAS

    as
    1. Attilio L. Stella & Fulvio Baldovin, 2008. "Role of scaling in the statistical modeling of finance," Papers 0804.0331, arXiv.org.
    2. Benoit Mandelbrot, 1999. "Survey of Multifractality in Finance," Cowles Foundation Discussion Papers 1238, Cowles Foundation for Research in Economics, Yale University.
    3. Alessandro Andreoli & Francesco Caravenna & Paolo Dai Pra & Gustavo Posta, 2010. "Scaling and multiscaling in financial series: a simple model," Papers 1006.0155, arXiv.org, revised Apr 2012.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Schadner, Wolfgang, 2022. "U.S. Politics from a multifractal perspective," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).

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    More about this item

    Keywords

    Multifractal processes; scaling function; multifractal spectrum; long-range dependence; heavy tails; MMAR; Extreme Value Theory.;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

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