  We study the nonparametric calibration of exponential L\'{e}vy models with
infinite jump activity. In particular our analysis applies to self-decomposable
processes 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 nonparametrically the $k$-function. 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.
