Multifractal analysis and extensive statistical tests are performed upon intraday minutely data within individual trading days for four stock market indexes (including HSI, SZSC, S&P500, and NASDAQ) to check whether the indexes (instead of the returns) possess multifractality. We find that the mass exponent $\tau(q)$ is linear and the singularity $\alpha(q)$ is close to 1 for all trading days and all indexes. Furthermore, we find strong evidence showing that the scaling behaviors of the original data sets cannot be distinguished from those of the shuffled time series. Hence, the so-called multifractality in the intraday stock market indexes is merely an illusion.
Deep Dive into Multifractality in stock indexes: Fact or fiction?.
Multifractal analysis and extensive statistical tests are performed upon intraday minutely data within individual trading days for four stock market indexes (including HSI, SZSC, S&P500, and NASDAQ) to check whether the indexes (instead of the returns) possess multifractality. We find that the mass exponent $\tau(q)$ is linear and the singularity $\alpha(q)$ is close to 1 for all trading days and all indexes. Furthermore, we find strong evidence showing that the scaling behaviors of the original data sets cannot be distinguished from those of the shuffled time series. Hence, the so-called multifractality in the intraday stock market indexes is merely an illusion.
Econophysics is an emerging interdisciplinary field applying concepts, theories, and tools borrowed from statistical physics, nonlinear sciences, applied mathematics, and complexity sciences to understand the complex self-organizing behaviors of financial markets [1,2,3,4]. This field has become to flourish since the pioneering work of Mantegna and Stanley on the scaling behavior in the dynamics of the Standard & Poor's 500 index [5], which is closely related to the Pareto-Lévy law proposed by Mandelbrot in the description of cotton price fluctuations [6]. Econophysicists have uncovered remarkable similarities between financial markets and turbulent flows [1,4]. Such analogues include (but not limited to) the evolution of probability densities of financial returns [7] based on the variational theory in turbulence [8,9,10,11], inverse statistics in stock markets [12,13,14] motivated by the inverse structure function analysis of velocity [15,16,17,18,19], scale-invariant distribution of multipliers defined from volatility of equities [20] and from dissipating energy [21,22,23,24], and intermittency and multifractality of asset returns [7,25].
Indeed, the multifractal nature of equity returns is one of the most important stylized facts. A small part of this literature contains the studies on the foreign exchange rate [7,25,26,27,28,29,30,31], gold price [28], commodity price [32], returns of stock price or indexes [32,33,34,35,36,37,38,39,40,41], and so on. We note that the quantity price (or its logarithm) in financial markets is the analogue of velocity in turbulence. Similarly, the counterpart of velocity difference in fluid mechanics is the asset return. In this framework, it is natural that numerous multifractal analyses have been carried out on the returns for financial equities similar to the velocity differences for turbulent flows.
However, there are exceptions, where analysis is performed on several indexes directly rather than their variations (the returns) and the presence of multifractality in the several indexes is claimed [42,43,44]. Specifically, they performed multifractal analysis on the intraday high-frequency data of Hang Seng Index (HSI), Shanghai Stock Exchange Composite Index (SSEC), and Shenzhen Stock Exchange Composite Index (SZEC) within individual trading days. The extracted “multifractal” spectra f (α) were then utilized to predict abnormal price movements and serve as a risk measure in risk management. It seems to us that a careful scrutiny on the obtained multifractality should be undertaken based on the extremely narrow spectra of the singularity α. Two problems arise, casting doubts on the aforementioned analysis [45].
Firstly, based on the multifractal theory, there exists a constant α(t) for each moment t such that the investigated measure µ on the neighbor B(t, l) of x scale with l when the scale l → 0, µ (B(t, l)) ∼ l α(t) .
(1) The measure µ is singular at arbitrary moment t with the singularity strength being α(t). When µ is defined as the sum of index prices within a given time interval, µ (B(t, l)) is approximately proportional to l, that is, α(t) ≈ 1 for all t. This suggests that the measure µ does not possess multifractal nature. This inference is further supported by the fact that the span of singularity strength ∆α = α max -α min ≈ 0 in the real data [42,43,44].
Secondly, in the analysis of multifractality in turbulence or high-frequency financial data, the moment order q should not be greater than 8 in order to make the partition function converge. Specifically, it is shown that the size of a time series should be no less than one million to ensure the estimate of its eighth order partition function statistically significant [19,46]. The situation is similar for high-frequency financial data [20]. Hence, it is of little significance to compute partition function for higher orders. In the analysis of minutely (or five-minute) data within a time period of one day [42,43,44], the size of the intraday high-frequency data is no more than 240 while the moment order is taken to be -120 q 120. This usually broad interval of q casts further doubts on the reported multifractality in the indexes.
Despite of the specific considerations discussed above, it is worthwhile to put further comments in general on the investigation of multifractality in financial data. The multifractal features in financial series have attracted great interests, however, the origin and significance of the extracted “multifractality” is less concerned. On one hand, it has been shown that an exact monofractal financial model can lead to an artificial multifractal behavior [47]. On the other hand, a time series of the price fluctuations possessing multifractal nature usually has either fat tails in the distribution or long-range temporal correlation or both [48]. However, possessing long memory is not sufficient for the presence of multifractality and one has to have a nonlinear process with long-me
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