This paper describes capabilities of wavelets for financial time series analysis. In current research wavelet analysis is provided by using Continuous Wavelet Transform. For basic assumption of time series behavior is used so called Fractal Brownian Motion, which is general case of classical Brownian motion, than implies time series long-term memory behavior. In terms of wavelets this analysis is done by filtering financial time series that are playing the role of signal with filter, by using Gausian mother wavelet function. Saying absolutely precisely, Continuous Wavelet Transform is done in frequency domain by using Fourier images of both – Fractal Brownian motion process upper interval bound and Gausian mother wavelet functions. The stock index analysis is completed in terms of Fractal Brownian Motion, for specified parts of the process. The devision of process in smaller parts is done by using probability bands. Each part of the process was analyzed by using modified R/S analysis. R/S indicator fitting to FBM bound is made using least square error criteria in time domain. Similar fitting is made by using wavelet images which are the result of Direct Continuous Wavelet Transform.