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Parallel cartoons of fractal models of finance

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  • Benoit B. Mandelbrot

Abstract

Having been crafted to welcome a new scientific journal, this paper looks forward but requires no special prerequisite. The argument builds on a technical wrinkle (used earlier but explained here fully for the first time), namely, the author’s grid-bound variant of Brownian motion B(t). While B(t) itself is additive, this variant is a multiplicative recursive process the author calls a ‘‘cartoon.’’ Reliance on this and related cartoons allows a new perspicuous exposition of the various fractal/multifractal models for the variation of financial prices. These illustrations do not claim to represent reality in its full detail, but suffice to imitate and bring out its principal features, namely, long tailedness, long dependence, and clustering. The goal is to convince the reader that the fractals/multifractals are not an exotic technical nightmare that could be avoided. In fact, the author’s models arose successively as proper, ‘‘natural,’’ and even ‘‘unavoidable’’ generalization of the Brownian motion model of price variation. Considered within the context of those generalizations, the original Brownian comes out as very special and narrowly constricted, while the fractal/multifractal models come out as nearly as simple and parsimonious as the Brownian. The cartoons are stylized recursive variants of the author’s fractal/multifractal models, which are even more versatile and realistic. Copyright Springer-Verlag Berlin Heidelberg 2005

Suggested Citation

  • Benoit B. Mandelbrot, 2005. "Parallel cartoons of fractal models of finance," Annals of Finance, Springer, vol. 1(2), pages 179-192, October.
    • Handle: RePEc:kap:annfin:v:1:y:2005:i:2:p:179-192
    DOI: 10.1007/s10436-004-0007-2
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    References listed on IDEAS

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    1. Benoit Mandelbrot & Howard M. Taylor, 1967. "On the Distribution of Stock Price Differences," Operations Research, INFORMS, vol. 15(6), pages 1057-1062, December.
    2. B. B. Mandelbrot, 2001. "Scaling in financial prices: III. Cartoon Brownian motions in multifractal time," Quantitative Finance, Taylor & Francis Journals, vol. 1(4), pages 427-440.
    3. B. B. Mandelbrot, 2001. "Stochastic volatility, power laws and long memory," Quantitative Finance, Taylor & Francis Journals, vol. 1(6), pages 558-559.
    4. B. B. Mandelbrot, 2001. "Scaling in financial prices: I. Tails and dependence," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 113-123.
    5. Laurent Calvet & Adlai Fisher & Benoit Mandelbrot, 1997. "Large Deviations and the Distribution of Price Changes," Cowles Foundation Discussion Papers 1165, Cowles Foundation for Research in Economics, Yale University.
    6. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-155, January.
    7. T. Lux, 2001. "Power laws and long memory," Quantitative Finance, Taylor & Francis Journals, vol. 1(6), pages 560-562.
    8. Mandelbrot, Benoit B, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices: Comment," Econometrica, Econometric Society, vol. 41(1), pages 157-159, January.
    9. Benoit Mandelbrot & Adlai Fisher & Laurent Calvet, 1997. "A Multifractal Model of Asset Returns," Cowles Foundation Discussion Papers 1164, Cowles Foundation for Research in Economics, Yale University.
    10. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    11. B. B. Mandelbrot, 2001. "Scaling in financial prices: II. Multifractals and the star equation," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 124-130.
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    Cited by:

    1. Estrada, Fernando, 2011. "Benoit Mandelbrot (1924 - 2011 ) : A Greek among Romans," MPRA Paper 30563, University Library of Munich, Germany.
    2. Yuli Radev, 2015. "New dynamic disequilibrium," Economic Thought journal, Bulgarian Academy of Sciences - Economic Research Institute, issue 6, pages 65-90.
    3. Fernando Estrada, 2011. "Benoit Mandelbrot (1924-2010): a Greek among Romans," History of Economic Ideas, Fabrizio Serra Editore, Pisa - Roma, vol. 19(1), pages 9-16.
    4. Ronny Mazzocchi, 2013. "Scope and Flaws of the New Neoclassical Synthesis," DEM Discussion Papers 2013/13, Department of Economics and Management.

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    More about this item

    Keywords

    Fractal; Multifractal; Cartoons; Roughness; Financial prices; G1;
    All these keywords.

    JEL classification:

    • G1 - Financial Economics - - General Financial Markets

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