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Renormalization and fixed points in finance, since 1962

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  • Mandelbrot, Benoit B.

Abstract

In diverse sciences that lack Hamiltonians, the analysis of complex systems is helped by the powerful tools provided by renormalization, fixed points and scaling. As one example, an intrinsic form of exact renormalizability was long used by the author in economics and related fields, most notably in finance. In 1962–3, its use led to a model of price variation founded on the (Cauchy-Polyà-Lévy) stable distribution, with striking data collapse that accounted for observed large deviations from Gaussianity. In 1965, a different form of exact renormalization led to fractional Brownian motion, which neglected large deviations but accounted for long dependence and the resulting non periodic cyclic behavior. Finally, from a seed planted in 1972, exact renormalizability and scaling led to a model of price variation of which the M1963 and M1965 models are special examples. This broader model, fractional Brownian motion in multifractal time, accounts simultaneously for both large deviations and long dependence. These three steps are in loose parallelism with space, time and joint renormalization in statistical physics. This presentation surveys the old works and many new developments described in the author's 1997 books on fractals and scaling in finance.

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  • Mandelbrot, Benoit B., 1999. "Renormalization and fixed points in finance, since 1962," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 263(1), pages 477-487.
    • Handle: RePEc:eee:phsmap:v:263:y:1999:i:1:p:477-487
    DOI: 10.1016/S0378-4371(98)00520-2
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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..
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    1. Gregory G. Brunk, 2003. "Swarming of innovations, fractal patterns, and the historical time series of US patents," Scientometrics, Springer;Akadémiai Kiadó, vol. 56(1), pages 61-80, January.
    2. Antoniou, Antonios & Vorlow, Constantinos E., 2005. "Price clustering and discreteness: is there chaos behind the noise?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 348(C), pages 389-403.
    3. Leontitsis, Alexandros & Vorlow, Constantinos E., 2006. "Accounting for outliers and calendar effects in surrogate simulations of stock return sequences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 368(2), pages 522-530.

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