Quantile estimation for L\'evy measures

TL;DR

This paper introduces a new way to estimate quantiles of jump sizes in Lévy processes, providing nonparametric estimators with optimal convergence rates, applicable to both observed data and financial models, and demonstrating their effectiveness through simulations and real data.

Contribution

It generalizes the concept of quantiles to Lévy jump measures and develops adaptive, minimax-optimal nonparametric estimators for these quantiles.

Findings

Minimax convergence rates are established for the estimators.

Adaptive estimators are derived using Lepski's method.

Simulation and real data illustrate estimator performance.

Abstract

Generalizing the concept of quantiles to the jump measure of a L\'evy process, the generalized quantiles $q_{\tau}^{\pm}>0$, for $\tau>0$, are given by the smallest values such that a jump larger than $q_{\tau}^{+}$ or a negative jump smaller than $-q_{\tau}^{-}$, respectively, is expected only once in $1/\tau$ time units. Nonparametric estimators of the generalized quantiles are constructed using either discrete observations of the process or using option prices in an exponential L\'evy 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.