A Multifractal Model of Asset Returns

Benoît Mandelbrot, Adlai Fisher and Laurent Calvet

No 1164, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

Abstract: This paper presents the multifractal model of asset returns ("MMAR"), based upon the pioneering research into multifractal measures by Mandelbrot (1972, 1974). The multifractal model incorporates two elements of Mandelbrot's past research that are now well-known in finance. First, the MMAR contains long-tails, as in Mandelbrot (1963), which focused on Levy-stable distributions. In contrast to Mandelbrot (1963), this model does not necessarily imply infinite variance. Second. the model contains long-dependence, the characteristic feature of fractional Brownian Motion (FBM), introduced by Mandelbrot and van Ness (1968). In contrast to FBM, the multifractal model displays long dependence in the absolute value of price increments, while price increments themselves can be uncorrelated. As such, the MMAR is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years. The distinguishing feature of the multifractal model is multi-scaling of the return distribution's moments under time-rescalings. We define multiscaling, show how to generate processes with this property, and discuss how these processes differ from the standard processes of continuous-time finance. The multifractal model implies certain empirical regularities, which are investigated in a companion paper.

Keywords: Multifractal model of asset returns; compund stochastic process; subordinated stochastic process; time deformation; trading time; scaling laws; multiscaling; self-similarity; self-affinity (search for similar items in EconPapers)
Pages: 33 pages
Date: 1997-09
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (203)

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Related works:
Working Paper: A Multifractal Model of Asset Returns (2011)
Working Paper: A Multifractal Model of Assets Returns (1999)
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