Alternative measure of multifractal content and its application in finance

An alternative method for analysis of multifractal properties of time series is provided. We propose a new kind of measure of multifractality strength which takes into account the behavior of multifractal profile of the generalized Hurst exponent h(q) for all moment orders q and is not limited only to the edge values of moment orders describing the scaling properties of smallest and largest fluctuations of a given signal in multifractal detrended fluctuation analysis (MFDFA). The meaning of this new measure is clarified and its performance is investigated for synthetic multifractal data and also for examples of real signals originating from stock markets. We provide also the interpretation of the alternative method following the scaling law that links together the geometric mean value of properly normalized standard q-fluctuation function F2(q; τ) in MFDFA and the window length τ in which detrending of a signal is performed. We discuss in this context the influence of multifractal bias on the new measure, i.e., the influence of effects which give similar observed features as multiscaling properties however, are not generated by temporal multiscaling autocorrelation in data. It is shown that the proposed alternative measure is robust in some extend to nonstationarity in data. As a result one may avoid problems with interpretation of multifractal profile h(q) encountered in many real nonstationary signals investigated in the standard way.