Modelling stock price movements: multifractality or multifractionality?

Abstract

The scaling properties of two alternative fractal models recently proposed to characterize the dynamics of stock market prices are compared. The former is the Multifractal Model of Asset Return (MMAR) introduced in 1997 by Mandelbrot, Calvet and Fisher in three companion papers. The latter is the multifractional Brownian motion (mBm), defined in 1995 by Peltier and Levy Vehel as an extension of the very well-known fractional Brownian motion (fBm). We argue that, when fitted on financial time series, the partition function as well as the scaling function of the mBm, i.e. of a generally non-multifractal process, behave as those of a genuine multifractal process. The analysis, which concerns the daily closing prices of eight major stock indexes, suggests to evaluate prudently the recent findings about the multifractal behaviour in finance and economics.Multifractals, MMAR, Multifractionality, Stock prices,

Latitudinal variation of stochastic properties of the geomagnetic field

“…That is to say, there is no unique scaling exponent, but the presence of a full hierarchy of them (possibly with a stationary distribution, although not required in the generalized scale invariance framework). For instance, Bianchi and Planese (2007) found that in some cases the partition function as well as the scaling function of the mBm, i.e., of a generally non-multifractal process, behave as those of a genuine multifractal process. Indeed, previous work by Wanliss et al (2005) and Yu et al (2010) demonstrate the presence of multifractality in magnetospheric data at higher latitudes.…”

Efficient Markets and Behavioral Finance: A Comprehensive Multifractional Model

“…In its turn, the comprehension of the mBm requires to recall shortly the well-known fractional Brownian motion, from which we will start the discussion. Notice that the family of multifractional processes should not be confused with the more widely known multifractal processes (see [11,15]). …”

“…The model we are going to discuss with a specific concern to financial markets, as well as the estimation technique we will apply, will have a general validity and will be of interest to all those in research fields ((geo)physics (see [32,35,68,50]), data traffic and image analysis (see [39,67,46,54,40,48,47]), economics (see [20,15,16,66,10,56]) in which a parsimonious model is needed to describe complex dynamics. This capability of the model should not surprise: the MPRE represents an insightful development of the celebrated fractional Brownian motion (fBm), whose regularity (constant) parameter H is too restrictive to describe many reallife phenomena.…”

The structure of gold and silver spread returns

“…The investigation and subsequent identification of complex dynamics, including long-memory processes, in various financial time series has provoked debate in the empirical finance literature (e.g., Mandelbrot (2001) and Bianchi and Pianese (2007)), especially in terms of how the presence of these processes affect asset market efficiency (e.g., Eom et al (2008) and McCauley et al (2008)) and asset pricing (e.g., Ellis and Hudson (2007) and Takami et al (2008)). Furthermore, a key contribution of this paper is that we are able to assess the economic implications of the identified long-memory process by applying a series of trading rules to the gold-silver spread time-series.…”