17 research outputs found
Dynamical variety of shapes in financial multifractality
The concept of multifractality offers a powerful formal tool to filter out
multitude of the most relevant characteristics of complex time series. The
related studies thus far presented in the scientific literature typically limit
themselves to evaluation of whether or not a time series is multifractal and
width of the resulting singularity spectrum is considered a measure of the
degree of complexity involved. However, the character of the complexity of time
series generated by the natural processes usually appears much more intricate
than such a bare statement can reflect. As an example, based on the long-term
records of S&P500 and NASDAQ - the two world leading stock market indices - the
present study shows that they indeed develop the multifractal features, but
these features evolve through a variety of shapes, most often strongly
asymmetric, whose changes typically are correlated with the historically most
significant events experienced by the world economy. Relating at the same time
the index multifractal singularity spectra to those of the component stocks
that form this index reflects the varying degree of correlations involved among
the stocks.Comment: 26 pages, 10 figure
Wavelet versus Detrended Fluctuation Analysis of multifractal structures
We perform a comparative study of applicability of the Multifractal Detrended
Fluctuation Analysis (MFDFA) and the Wavelet Transform Modulus Maxima (WTMM)
method in proper detecting of mono- and multifractal character of data. We
quantify the performance of both methods by using different sorts of artificial
signals generated according to a few well-known exactly soluble mathematical
models: monofractal fractional Brownian motion, bifractal Levy flights, and
different sorts of multifractal binomial cascades. Our results show that in
majority of situations in which one does not know a priori the fractal
properties of a process, choosing MFDFA should be recommended. In particular,
WTMM gives biased outcomes for the fractional Brownian motion with different
values of Hurst exponent, indicating spurious multifractality. In some cases
WTMM can also give different results if one applies different wavelets. We do
not exclude using WTMM in real data analysis, but it occurs that while one may
apply MFDFA in a more automatic fashion, WTMM has to be applied with care. In
the second part of our work, we perform an analogous analysis on empirical data
coming from the American and from the German stock market. For this data both
methods detect rich multifractality in terms of broad f(alpha), but MFDFA
suggests that this multifractality is poorer than in the case of WTMM.Comment: substantially extended version, to appear in Phys.Rev.
The foreign exchange market: return distributions, multifractality, anomalous multifractality and Epps effect
We present a systematic study of various statistical characteristics of
high-frequency returns from the foreign exchange market. This study is based on
six exchange rates forming two triangles: EUR-GBP-USD and GBP-CHF-JPY. It is
shown that the exchange rate return fluctuations for all the pairs considered
are well described by the nonextensive statistics in terms of q-Gaussians.
There exist some small quantitative variations in the nonextensivity
q-parameter values for different exchange rates and this can be related to the
importance of a given exchange rate in the world's currency trade. Temporal
correlations organize the series of returns such that they develop the
multifractal characteristics for all the exchange rates with a varying degree
of symmetry of the singularity spectrum f(alpha) however. The most symmetric
spectrum is identified for the GBP/USD. We also form time series of triangular
residual returns and find that the distributions of their fluctuations develop
disproportionately heavier tails as compared to small fluctuations which
excludes description in terms of q-Gaussians. The multifractal characteristics
for these residual returns reveal such anomalous properties like negative
singularity exponents and even negative singularity spectra. Such anomalous
multifractal measures have so far been considered in the literature in
connection with the diffusion limited aggregation and with turbulence. We find
that market inefficiency on short time scales leads to the occurrence of the
Epps effect on much longer time scales. Although the currency market is much
more liquid than the stock markets and it has much larger transaction
frequency, the building-up of correlations takes up to several hours - time
that does not differ much from what is observed in the stock markets. This may
suggest that non-synchronicity of transactions is not the unique source of the
observed effect
Wavelet-based discrimination of isolated singularities masquerading as multifractals in detrended fluctuation analyses
The robustness of two widespread multifractal analysis methods, one based on detrended fluctuation analysis and one on wavelet leaders, is discussed in the context of time-series containing non-uniform structures with only isolated singularities. Signals generated by simulated and experimentally-realized chaos generators, together with synthetic data addressing particular aspects, are taken into consideration. The results reveal essential limitations affecting the ability of both methods to correctly infer the non-multifractal nature of signals devoid of a cascade-like hierarchy of singularities. Namely, signals harboring only isolated singularities are found to artefactually give rise to broad multifractal spectra, resembling those expected in the presence of a well-developed underlying multifractal structure. Hence, there is a real risk of incorrectly inferring multifractality due to isolated singularities. The careful consideration of local scaling properties and the distribution of Holder exponent obtained, for example, through wavelet analysis, is indispensable for rigorously assessing the presence or absence of multifractality.Facultad de Ingenierí