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 Lévy-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.
Mandelbrot, Benoit; Fisher, Adlai; and Calvet, Laurent, "A Multifractal Model of Asset Returns" (1997). Cowles Foundation Discussion Papers. 1412.