  We present a comparative analysis of multifractal properties of financial
time series built on stock indices from developing (WIG) and developed (S&P500)
financial markets. It is shown how the multifractal image of the market is
altered with the change of the length of time series and with the economic
situation on the market. We emphasize that the proper adjustment of scaling
range for multiscaling power laws is essential to obtain the multifractal image
of time series. We analyze in this paper multifractal properties of real
financial time series using H\"older $f(\alpha)$ representation and
multifractal-DFA method. It is also investigated how multifractal properties of
stocks change with variety of "surgeries" done on the initial real financial
time series. This way we reveal main phenomena on the market influencing its
multifractal dynamics. In particular, we focus on examining how multifractal
picture of real time series changes when one cuts off extreme events like
crashes or rupture points, and how fluctuations around the main trend in time
series influence the multifractal behavior of financial series in the long-time
horizon for both developed and developing markets.
