Multifractal analysis of financial markets: a review

Abstract

Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

Signatures of the Crypto-Currency Market Decoupling from the Forex

“…where α is referred to as the singularity exponent or Hölder exponent, and f (α) is the corresponding singularity spectrum, often referred to as the multifractal spectrum. The particular case of individual time-series corresponds to the commonly used Multifractal Detrended Fluctuation Analysis (MFDFA) [41,5,42,43]. When h(q) is approximately constant, the signal is interpreted as monofractal and f (α) collapses down to a single point.…”

On the interplay between multiscaling and stock dependence

“…We carried out the analysis also using MFDFA(Jiang et al 2018), and the results are qualitatively equivalent, suggesting the analysis is not method dependent.…”

“…Financial time-series are characterized by the presence of so called stylized facts (Cont 2001, Chakraborti et al 2011. The most famous ones are: power law tails (Mantegna and Stanley 1995), volatility clustering (Ding et al 1993), multiscaling (Mantegna and Stanley 1995, 1999, Lux and Marchesi 1999, Dacorogna et al 2001, Calvet and Fisher 2002, Lux 2004, Di Matteo et al 2005, Di Matteo 2007, Jiang et al 2018 and the presence of a dependency structure between stocks (Mantegna 1999, Borghesi et al 2007, Aste and Di Matteo 2010, Tumminello et al 2010, Musmeci et al 2015a. Stylized facts deserve attention due to the hidden data information they provide to risk and asset managers.…”

Multifractal cross-correlations between the world oil and other financial markets in 2012–2017

“…The oil market thus constitutes an inseparable component of the global world financial system and, as such, it crucially influences its complexity characteristics (Kwapień & Drożdż, 2012). Description of the related dependences and correlations seems therefore to be the most natural by using methodology of time series analysis that allows to take care of the nonlinear effects and that has already proved fruitful in many domains of complexity, including the financial markets (Mandelbrot et al, 1997;Calvet & Fisher, 2002;Bouchaud et al, 2004;Grech & Mazur, 2004;Lux, 2008;Morales et al, 2012;Rak et al, 2015;Gubiec & Kutner, 2017;Jiang et al, 2018;Klamut et al, 2018). This methodology includes the Wavelet Transform Modulus Maxima -WTMM (Muzy et al, 1991), and a series of methods based on detrending with an increasing degree of generality.…”