We study nonparametric inference of stochastic models driven by stable Lévy processes. We introduce a nonparametric estimator of the stable index that achieves the parametric √ n rate of convergence. For the volatility function, due to the heavy-tailedness, the classical least-squares method is not applicable. We then propose a nonparametric least-absolute-deviation or median-quantile estimator and study its asymptotic behavior, including asymptotic normality and maximal deviations, by establishing a representation of Bahadur–Kiefer type. The result is applied to several major foreign exchange rates. © 2009 Elsevier B.V. All rights reserved.