A cautionary note on the detection of multifractal scaling in finance and economics

Abstract

The scaling properties of the multifractional Brownian motion (mBm), a generally not multifractal process is investigated, and it is argued that, when calibrated on actual financial time series, its partition function as well as its spectrum behave as those of genuine multifractal processes. The examples here provided, based on the analysis of two major stock indexes, are intended to solicit a prudent evaluation of the recent findings about the multifractal behaviour in finance and economics.

Notes

¹Given the metric spaces (X, d _X ) and (Y, d _Y ), the function f:X →Y is called a Hölder function with exponent β > 0 if, for each x,y∈X such that d _X (x, y)<1 there exists a constant k satisfying the condition

The hölderianity is sufficient (but not necessary) for the continuity for the process.

²See Ayache (2000) for a discussion on how to generalize the mBm in order to obtain possibly multifractal scaling.