Spectral calibration of exponential Lévy models

Abstract

Abstract. We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.

High-frequency Donsker theorems for Lévy measures

“…Jongbloed et al (2005) studied nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes. Belomestny and Reiß (2006) treat nonparametric estimation of Lévy processes in a financial model. Low-frequency observations were considered, e.g., by Neumann and Reiß (2009), Belomestny (2010), Gugushvili (2012) as well as Nickl and Reiß (2012), whereas Figueroa-López (2009, 2011 treats high-frequency observations.…”

Adaptive Estimation for Lévy Processes

“…Thus, the sample path can be expressed as: 8) where (X t ) is a centered square integrable pure-jump martingale:…”

“…[11], [52], [26], [2], [8] and [3], [15]). The nonparametric estimation of the Lévy density has been studied for a continuous time observation of the sample path on a time interval [0, T ] with T tending to infinity ( [33]) or for discrete time observations.…”

“…The chapter only deals with upper risk bounds, but to check the optimality of the estimators, lower bounds are needed: they are provided, in the high frequency setting by [31], [7], and in the low frequency setting by [8], [56], [46], [45]. Lower bound in the specific case of decompounding is obtained in [25].…”

A jump-diffusion Libor model and its robust calibration

“…Discussion The order of the error || ψ n −ψ n || L ∞ (U ) depends on N , the number of caplets available, as well as on the type of errors in caplet prices C n−1,j , −N ≤ j ≤ N. For example, assuming that the observed caplet prices are "exact", one can show (see [1]) that…”