Dynamical Variety of Shapes in Financial Multifractality

The concept of multifractality offers a powerful formal tool to filter out a 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 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 the 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.