Research of alpha-stable distributions is especially important nowadays because they often occur in the analysis of financial data and information flows along computer networks. It has been found that financial data are often leptokurtic with heavy-tailed dsitributions; many authors, e.g., Rachev, Mittnik (2000), Kabašinskas et al. (2012), Sakalauskas et al. (2013) have proved that the most often used normal distribution is not the most suitable way to analyse economic indicators and suggested to replace it with more general, for example stable distributions. Since Rachev, Mittnik (2000), Kabašinskas et al. (2012), Sakalauskas et al. (2013) have estimated one-dimensional alpha-stable distributions a problem arises how to estimate multidimensional data. Although the problem has been investigated and analyzed for several decades it has not been solved yet (Press, 1972; Davydov, Paulauskas, 1999; Nolan, 1998; Kring et al. 2009; Ogata, 2013). Maximum likelihood method for the estimation of multivariate alpha-stable distributions by using the EM algorithm is presented in this work. Integrals included in the expressions of the estimates have been calculated using the Gaussian and Legendre-Gauss quadrature formulas. The constructed model can be used in stock market data analysis.