Financial markets can serve as an example of complex systems with extremely complicated structure. It is not a surprise, thus, that analysis of these systems is a difficult issue, which requires involving of the concepts borrowed both from physics and from mathematics.

Due to the fact that self-similarity is one of the best documented properties of financial data, an approach that seems to be most promising in describing the financial markets is the fractal analysis. Moreover, besides the standard fractal characteristics of data like the mono- and multifractality, the self similar character of financial signals manifests itself also in the existence
of a hierarchy of the log-periodic oscillation patterns preceding and following the financial crashes and the trend reversals.

In our present analysis we consider a few major world stock indices from period 2004-2009. We show that the reversal date of the upward trend in Nov 2007 can be estimated much earlier from the log-periodic oscillations starting in March 2006. We identify the self-similar nature of the accelerating and decelerating log-periodic trends, showing that fully-developed log-periodic
structures at lower levels constitutes in fact the individual oscillations at higher levels. Furthermore, a frequency analysis of the oscillations confirms the hypothesis of the existence of a universal scaling factor λ, responsible for condensing of the oscillations, equal to 2.

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