Maximum likelihood estimation of stable Paretian models

Stable Paretian distributions have attractive properties for empirical modeling in finance, because they include the normal distribution as a special case but can also allow for heavier tails and skewness. A major reason for the limited use of stable distributions in applied work is due to the facts that there are, in general, no closed-form expressions for its probability density function and that numerical approximations are nontrivial and computationally demanding. Therefore, Maximum Likelihood (ML) estimation of stable Paretian models is rather difficult and time consuming. Here, we study the problem of ML estimation using fast Fourier transforms to approximate the stable density functions. The performance of the ML estimation approach is investigated in a Monte Carlo study and compared to that of a widely used quantile estimator. Extensions to more general distributional models characterized by time-varying location and scale are discussed.