The components of empirical multifractality in financial returns

Abstract

Abstract. -We perform a systematic investigation on the components of the empirical multifractality of financial returns using the daily data of Dow Jones Industrial Average from 26 May 1896 to 27 April 2007 as an example. The temporal structure and fat-tailed distribution of the returns are considered as possible influence factors. The multifractal spectrum of the original return series is compared with those of four kinds of surrogate data: (1) shuffled data that contain no temporal correlation but have the same distribution, (2) surrogate data in which any nonlinear correlation is removed but the distribution and linear correlation are preserved, (3) surrogate data in which large positive and negative returns are replaced with small values, and (4) surrogate data generated from alternative fat-tailed distributions with the temporal correlation preserved. We find that all these factors have influence on the multifractal spectrum. We also find that the temporal structure (linear or nonlinear) has minor impact on the singularity width ∆α of the multifractal spectrum while the fat tails have major impact on ∆α, which confirms the earlier results. In addition, the linear correlation is found to have only a horizontal translation effect on the multifractal spectrum in which the distance is approximately equal to the difference between its DFA scaling exponent and 0.5. Our method can also be applied to other financial or physical variables and other multifractal formalisms.Introduction. -There are a wealth of studies showing that financial markets exhibit multifractal nature [1][2][3]. Many different methods have been applied to characterize the hidden multifractal behavior in finance, for instance, the fluctuation scaling analysis [4][5][6], the structure function (or height-height correlation function) method [1,[7][8][9][10][11][12][13][14][15][16][17], the multiplier method [18], the multifractal detrended fluctuation analysis (MF-DFA) [19][20][21][22][23][24][25][26][27], the partition function method [28][29][30][31][32][33][34][35][36][37][38], the wavelet transform approaches [39][40][41][42], to list a few. There are also efforts seeking for applications of the extracted multifractal spectra. Some researchers reported that the observed multifractal singularity spectrum has predictive power for price fluctuations [29,31,38], can serve as a measure of market risk by introducing a new concept termed multifractal volatility [35], and can be used to quantify the inefficiency of markets [43].

The origins of multifractality in financial time series and the effect of extreme events

“…A wide smoot h spect rum does not imply t hat t he dat a act ually has mult ifract al scaling. T he spect ra in t his paper seem shift ed t o t he right in comparison t o t hose in t he lit erat ure [20,23,38]. T his can be accounted for by the updated definit ion of τ (q) in St ep 7 in Sect ion 2.…”

“…T his is a very wide range in comparison wit h ot her st udies [12,20,26,38] where it is st andard t o use q ∈ [− 5, 5]. However, for smaller ranges of q, less of t he mult ifract al spect rum is revealed.…”

“…Ot her st udies [16,20,22,42,43] have found mult ifract al scaling in shuffl ed financial dat a. However, as no explicit invest igat ion of t he logarit hmic plot s and t heir local slopes was conduct ed, t he conclusion t hat mult ifract al scaling is present is not just ified.…”

“…P erforming mult ifract al analysis helps t o increase our knowledge about t he financial syst em and furt her charact erise it . Many st udies have found mult ifract al scaling in financial dat a [20][21][22][23]. An underst anding of t his mult ifract al st ruct ure can enable deeper underst anding of t he dynamics of financial market s. If it is found t o be a universal feat ure of financial dat a, it provides an addit ional benchmark by which t o measure t he fitness of financial models.…”

Quantitative Characteristics of Correlations of Meteorological Data

“…Multifractal structures were identified in systems from various areas such as physics [2][3][4][5], biology [6][7][8], chemistry [9,10], economics [11][12][13] and even music [14][15][16][17][18][19]. One of the most popular methods of the multifractal analysis is multifractal detrended correlation (MFDFA) [20][21][22] and cross-correlation analysis (MFCCA) [23,24].…”