Calibration of self-decomposable L\'{e}vy models

TL;DR

This paper develops a nonparametric method for calibrating self-decomposable Lévy models with infinite jump activity, estimating key parameters and the jump density function from option prices, with proven convergence rates.

Contribution

It introduces an adaptive spectral estimation technique for the $k$-function and related parameters in self-decomposable Lévy models, addressing nonsmoothness at zero.

Findings

Achieves minimax convergence rates for parameter estimation.

Demonstrates the method's effectiveness through simulations.

Successfully applies the approach to real market data.

Abstract

We study the nonparametric calibration of exponential L\'{e}vy 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.