Calibration of self-decomposable Lévy models

We study the nonparametric calibration of exponential, self-decomposable L\'evy models, whose jump density can be characterized by the k-function, which is typically nonsmooth at zero. On the one hand the estimation of the drift, of the activity measure \alpha:=k(0+)+k(0-) and of analogous parameters for the derivatives of the k-function are considered and on the other hand we estimate the k-function outside of a neighborhood of zero. Minimax convergence rates are derived. Since the rates depend on \alpha, we construct estimators adapting to this unknown parameter. Our estimation method is based on spectral representations of the observed option prices and on a regularization by cutting off high frequencies. Finally, the procedure is applied to simulations and real data.