Quantile estimation for Lévy measures

Abstract

Generalizing the concept of quantiles to the jump measure of a Lévy process, the generalized quantiles q ± τ > 0, for τ > 0, are given by the smallest values such that a jump larger than q + τ or a negative jump smaller than −q − τ , respectively, is expected only once in 1/τ time units. Nonparametric estimators of the generalized quantiles are constructed using either discrete observations of the process or using option prices in an exponential Lévy model of asset prices. In both models minimax convergence rates are shown. Applying Lepski's approach, we derive adaptive quantile estimators. The performance of the estimation method is illustrated in simulations and with real data.

Nonparametric implied Lévy densities

“…We finish this section with a comparison of our method with existing nonparametric estimators of the Lévy measure from option data. [11] uses penalized least squares while [4,5], [28], [29] and [30,31] use spectral-based techniques for recovering the Lévy measure of exponential Lévy models from options with fixed maturity T . We will restrict comparison to the existing spectral-based estimators as their rates of convergence are explicitly derived.…”

“…That is, they are based on option-based estimates of E Q t (e iux t+T +x t+T ) which is very similar to our use of the characteristic function E Q t (e iux t+T ). [4] and [28] work directly with E Q t (e iux t+T +x t+T ) in a setting of finite activity jumps while [29] and [30,31] use derivatives of it and allow for more general jump specifications similar to our use of derivatives of f T .…”

“…The major difference between the current paper and this strand of work, from a statistical point of view, is that we use option data for the inference which results in a very different statistical setup. Third, most closely related to the current paper is a body of work that considers nonparametric Lévy density estimation in the context of exponential Lévy models from options with fixed maturity, see e.g., [4,5], [11], [28], [29] and [30,31]. The major differences between the current work and these papers are two.…”

Bootstrap confidence bands for spectral estimation of Lévy densities under high-frequency observations

“…The literature on nonparametric estimation of Lévy measures or densities is broad. Recent contributions include Shimizu (2006), Figueroa-López (2009), Comte and Genon-Catalot (2009, 2015, Kappus and Reiß (2010), Duval (2013), and Bec and Lacour (2015) under the highfrequency setup (i.e., ∆ = ∆ n → 0 as n → ∞), and van Es et al (2007), Neumann and Reiß (2009), Gugushvili (2009), Chen et al (2010), Comte and Genon-Catalot (2010), Kappus and Reiß (2010), Belomestny (2011), Gugushvili (2012), Kappus (2014), Trabs (2015), and Belomestny and Reiß (2015) under the low-frequency setup (i.e., ∆ > 0 is fixed). Jongbload et al (2005) study nonparametric estimation of the Lévy measure for a Lévy driven Ornstein-Uhlenbeck process under high and low frequency observations.…”

Nonparametric inference on Lévy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observations

“…Further, Chen et al (2010) and Trabs (2015) investigate nonparametric estimation of a class of Lévy processes under the low-frequency set up. Belomestny et al (2019) studies nonparametric estimation of Lévy measures of the moving average Lévy processes under low-frequency observations.…”