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Scaling in financial prices: IV. Multifractal concentration

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  • B. B. Mandelbrot

Abstract

In the Brownian model, even the largest of N successive daily price increments contributes negligibly to the overall sample variance. The resulting 'absent' concentration justifies the role of variance in measuring Brownian volatility. Mandelbrot introduced in 1963 an alternative 'mesofractal model', in which the population variance is infinite. A significant proportion of the overall sample variance comes from an absolutely small number of large contributions, expressing a 'hard' form of concentration. To achieve a prescribed proportion of the overall measured variance, those 1900 and 1963 models require numbers of days of the order of N1 and N0, respectively. This paper shows that an intermediate possibility exists: a new and very flexible 'soft' form of concentration is provided by the 'multifractal' model Mandelbrot introduced in 1997. The standard 'extreme values' theory applies to mesofractals but multifractals behave very differently. The single largest contribution to sample variance is asymptotically negligible; however, an arbitrarily high proportion of the overall variance is contributed by a number of days of the order of ND, where 0

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  • B. B. Mandelbrot, 2001. "Scaling in financial prices: IV. Multifractal concentration," Quantitative Finance, Taylor & Francis Journals, vol. 1(6), pages 641-649.
    • Handle: RePEc:taf:quantf:v:1:y:2001:i:6:p:641-649
    DOI: 10.1088/1469-7688/1/6/306
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    Cited by:

    1. Wang, Yudong & Wu, Chongfeng & Yang, Li, 2016. "Forecasting crude oil market volatility: A Markov switching multifractal volatility approach," International Journal of Forecasting, Elsevier, vol. 32(1), pages 1-9.
    2. Wilhelm Berghorn & Sascha Otto, 2017. "Mandelbrot Market-Model and Momentum," International Journal of Financial Research, International Journal of Financial Research, Sciedu Press, vol. 8(3), pages 1-26, July.
    3. Gencay, Ramazan & Selcuk, Faruk, 2004. "Extreme value theory and Value-at-Risk: Relative performance in emerging markets," International Journal of Forecasting, Elsevier, vol. 20(2), pages 287-303.
    4. Fieberg, Christian & Günther, Steffen & Poddig, Thorsten & Zaremba, Adam, 2024. "Non-standard errors in the cryptocurrency world," International Review of Financial Analysis, Elsevier, vol. 92(C).
    5. Sutthisit Jamdee & Cornelis A. Los, 2005. "Multifractal Modeling of the US Treasury Term Structure and Fed Funds Rate," Finance 0502021, University Library of Munich, Germany.
    6. M. A. H. Dempster, 2011. "Benoit B. Mandelbrot (1924-2010): a father of Quantitative Finance," Quantitative Finance, Taylor & Francis Journals, vol. 11(2), pages 155-156.
    7. Wilhelm Berghorn & Martin T. Schulz & Markus Vogl & Sascha Otto, 2021. "Trend Momentum II: Driving Forces of Low Volatility and Momentum," International Journal of Financial Research, International Journal of Financial Research, Sciedu Press, vol. 12(3), pages 300-319, May.
    8. Mehmet Ali Balcı & Larissa M. Batrancea & Ömer Akgüller & Lucian Gaban & Mircea-Iosif Rus & Horia Tulai, 2022. "Fractality of Borsa Istanbul during the COVID-19 Pandemic," Mathematics, MDPI, vol. 10(14), pages 1-33, July.
    9. Chris Heyde, 2009. "Scaling issues for risky asset modelling," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 593-603, July.

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