A Multifractal Model of Asset Returns

33 Pages Posted: 21 Apr 1998

See all articles by Benoit B. Mandelbrot

Benoit B. Mandelbrot

Yale University - International Center for Finance; IBM Corporation - Thomas J. Watson Research Center

Adlai J. Fisher

University of British Columbia (UBC) - Sauder School of Business

Laurent E. Calvet

EDHEC Business School - Department of Economics & Finance; CEPR

Date Written: September 15, 1997

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 multiscaling 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.

JEL Classification: C22, G12, F31

Suggested Citation

Mandelbrot, Benoit B. and Fisher, Adlai J. and Calvet, Laurent E., A Multifractal Model of Asset Returns (September 15, 1997). Cowles Foundation Discussion Paper No. 1164; Sauder School of Business Working Paper. Available at SSRN: https://ssrn.com/abstract=78588

Benoit B. Mandelbrot

Yale University - International Center for Finance ( email )

Box 208200
10 Hillhouse Avenue
New Haven, CT 06520-8200
United States
914-945 1712 (Phone)
914-945 4149 (Fax)

IBM Corporation - Thomas J. Watson Research Center ( email )

Route 134
Kitchawan Road
Yorktown Heights, NY 10598
United States
914-945 1712 (Phone)
914-945 4149 (Fax)

Adlai J. Fisher (Contact Author)

University of British Columbia (UBC) - Sauder School of Business ( email )

2053 Main Mall
Vancouver, BC V6T 1Z2
Canada
604-822-8331 (Phone)
604-822-4695 (Fax)

HOME PAGE: http://finance.sauder.ubc.ca/~fisher

Laurent E. Calvet

EDHEC Business School - Department of Economics & Finance ( email )

France

CEPR ( email )

33 Great Sutton Street
London, EC1V 0DX
United Kingdom

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