The work proposes a method for estimating the stability index of alpha-stable distributions by using moments of fractional order. Provided numerical modeling has fully justified all of the results. Comparative analysis of the efficiency among the proposed method of estimating the stability index and widely used methods was performed. Proposal method is much simpler, far more faster and substantially less memory required. Estimation of generalized Hurst exponent from time series of the ordinary Lévy process was performed. Multifractal fluctuation analysis method and evaluation based on stability index estimation were compared. The results of numerical modelling showed that proposed method for estimating the fractal properties of the ordinary Lévy process, based on stability index estimation via fractional order moments is a much more accurate.