Calibration of self-decomposable Lévy models

Abstract

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.

Published in at http://dx.doi.org/10.3150/12-BEJ478 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (6)

• Spectral estimation of the fractional order of a Lévy process
• Statistical inference for time-changed Lévy processes via composite characteristic function estimation
• Confidence sets in nonparametric calibration of exponential Lévy models
• Calibration of self-decomposable Lévy models
• Option calibration of exponential Lévy models: Confidence intervals and empirical results
• Polar sets of anisotropic Gaussian random fields

Cited by in corpus (3)

• Confidence sets in nonparametric calibration of exponential Lévy models
• Calibration of self-decomposable Lévy models
• Adaptation to lowest density regions with application to support recovery