  Many financial variables are found to exhibit multifractal nature, which is
usually attributed to the influence of temporal correlations and fat-tailedness
in the probability distribution (PDF). Based on the partition function approach
of multifractal analysis, we show that there is a marked finite-size effect in
the detection of multifractality, and the effective multifractality is the
apparent multifractality after removing the finite-size effect. We find that
the effective multifractality can be further decomposed into two components,
the PDF component and the nonlinearity component. Referring to the normal
distribution, we can determine the PDF component by comparing the effective
multifractality of the original time series and the surrogate data that have a
normal distribution and keep the same linear and nonlinear correlations as the
original data. We demonstrate our method by taking the daily volatility data of
Dow Jones Industrial Average from 26 May 1896 to 27 April 2007 as an example.
Extensive numerical experiments show that a time series exhibits effective
multifractality only if it possesses nonlinearity and the PDF has impact on the
effective multifractality only when the time series possesses nonlinearity. Our
method can also be applied to judge the presence of multifractality and
determine its components of multifractal time series in other complex systems.

Statistical Finance