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Calibration of self-decomposable Lévy models. (English) Zbl 1285.62101

Summary: We study the nonparametric calibration of exponential Lévy 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.

MSC:

62M09 Non-Markovian processes: estimation
62G05 Nonparametric estimation
62P05 Applications of statistics to actuarial sciences and financial mathematics
65C60 Computational problems in statistics (MSC2010)
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