It is argued that the study of the correct specification of returns distributions has attractive implications in financial economics. This study estimates Levy-stable (fractal) distributions that can accurately account for skewness, kurtosis, and fat tails. The Levy-stable family distributions are parametrized by the Levy index (α), 0 < (α), ≤ 2, and include the normal distribution as a special case (α = 2). The Levy index, α, is the fractal dimension of the probability space. The unique feature of Levy-stable family distributions is the existence of a relationship between the fractal dimension of the probability space andthe fractal dimensionof the time series. This relationshipis simply expressed in terms of Hurst exponent (H), i.e. α = 1/ H. In addition, Hurst exponent is related to long-memory effects. Thus, estimating the Levy index allows us to determine long-memory effects. The immediate practical implication of the present work is that on the one hand we estimate the shape of returns distributions...