Estimation of Tempered Stable L\'{e}vy Models of Infinite Variation

Abstract

We propose a new method for the estimation of a semiparametric tempered stable L\'{e}vy model. The estimation procedure combines iteratively an approximate semiparametric method of moment estimator, Truncated Realized Quadratic Variations (TRQV), and a newly found small-time high-order approximation for the optimal threshold of the TRQV of tempered stable processes. The method is tested via simulations to estimate the volatility and the Blumenthal-Getoor index of the generalized CGMY model as well as the integrated volatility of a Heston-type model with CGMY jumps. The method outperforms other efficient alternatives proposed in the literature when working with a L\'evy process (i.e., the volatility is constant), or when the index of jump intensity $Y$ is larger than $3/2$ in the presence of stochastic volatility.