The components of empirical multifractality in financial returns

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 reserved, (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 reserved. 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. 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 Hurst index and 0.5. Our method also applies to other financial or physical variables and other multifractal formalism. Introduction. – There are a wealth of studies showing that financial markets exhibit multifractal nature [1–3]. Many different methods have been applied to characterize the hidden multifractal behavior in finance, for instance, the fluctuation scaling analysis [4–6], the structure function (or heightheight correlation function) method [1, 7–17], the multiplier method [18], the multifractal detrended fluctuation analysis (MF-DFA) [19–27], the partition function method [28–38], the wavelet transform approaches [39–42], 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] and can serve as a measure of market risk by introducing a new concept termed multifractal volatility [35]. An important and subtle issue of multifractality is about its origin. An even critical question is to ask whether the extracted multifractality is intrinsic or apparent. Indeed, it has been shown that an exact monofractal financial model can lead (a)e-mail: [email protected] to an artificial multifractal behavior [43]. It is usually argued in the Econophysics community that the sources of multifractal nature in financial time series are the fat tails and/or the longrange temporal correlation [19]. However, possessing long memory is not sufficient for the presence of multifractality and one has to have a nonlinear process with long-memory in order to have multifractality [44]. In many cases, the null hypothesis that the reported multifractal nature is stemmed from the large fluctuations of prices cannot be rejected [45]. In this Letter, we focus on the multifractal detrended fluctuation analysis of financial logarithmic returns defined as r(t) = ln[P (t)/P (t− 1)] (1) where P (t) is the price at time t. Specifically, we use the daily data of the Dow Jones Industrial Average (DJIA) from 26 May 1896 to 27 April 2007 (totally 30147 trading days) to illustrate the method and results. The reason is simply that most studies in this direction use MF-DFA on stock returns. Nevertheless, the methodology is quite general and also applies in the study of other financial variables and other multifractal analysis.