Quantile estimation for Lévy measures

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.