文档介绍：A Multifractal Model of Asset Returns
Benoit Mandelbrot∗
Department of Mathematics, Yale University and
IBM T. J. Watson Research Center
Adlai Fisher and Laurent Calvet†
Department of Economics, Yale University
Cowles Foundation Discussion Paper #1164
This Draft: September 15, 1997
First Draft: October 1996
THIS VERSION TEXT ONLY
Download figures at /∼fisher/
∗10 Hillhouse Avenue, New Haven, CT 06520-8283. e-mail: ******@watson.
†28 Hillhouse Avenue, New Haven, CT 06520-1972. e-mail: fi******@, ******@
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 L´evy-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), in-
troduced 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 distri-
bution 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.
Keywords: Multifractal Model