文档介绍：Q UANTITATIVE F INANCE V OLUME 1 (2001) 427–440 RESEARCH PAPER
I NSTITUTE OF P HYSICS P UBLISHING quant.
Scaling in financial prices:
III. Cartoon Brownian motions in
multifractal time
Benoit B Mandelbrot
Sterling Professor of Mathematical Sciences, Yale University, New Haven,
CT 06520-8283, USA
Received 22 March 2001
Abstract
This article describes a versatile family of functions that are increasingly
roughened by essive interpolations. They reproduce, in the simplest way
possible, the main features of financial prices: continually varying volatility,
discontinuity or concentration, and the fact that many changes fall far outside
the mildly behaving Brownian ‘norm’. Being illuminating but distorted and
plete, these constructions deserve to be called ‘cartoons’. They address
both the observed variation of financial prices and the generalized model the
author introduced in 1997, namely, Brownian motion in multifractal time.
Special cases of the same construction provide cartoons of the Bachelier
model—the Wiener Brownian motion—or the two models the author
proposed in the 1960s, namely, Levy´ stable and fractional Brownian motions.
The cartoons are the embodiment of the author’s ‘principle of scaling in
economics’. While rich in structure, they are unexpectedly parsimonious,
puted, and pared to one another by being associated
with points in a square ‘phase diagram’.
1. Introduction to as BMMT. When followed in the ordinary clock time, the
Brownian motions in question will be either the original one
Financial prices, such as those of securities, commodities, due to Wiener, to be referred to as WBM, or the fractional one,
foreign exchange or interest rates, are largely unpredictable to be referred to as FBM. Other authors, such as Calvet and
but one must evaluate the odds for or against some desired Fisher (2001), prefer to refer to BMMT as MMAR.
or feared es, the most extreme being ‘ruin’. Those The key terms, ‘fractional’ and ‘multifractal