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

TL;DR

This paper introduces a novel estimation method for tempered stable Lévy models that improves accuracy in volatility and jump intensity estimation, especially for processes with higher jump activity or stochastic volatility.

Contribution

It combines an iterative semiparametric method of moments with a new small-time approximation for the optimal threshold in TRQV, enhancing estimation performance.

Findings

Outperforms existing methods in estimating volatility and Blumenthal-Getoor index.

Effective for models with stochastic volatility and high jump activity.

Validated through simulations on CGMY and Heston-type models.

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.