PhD Thesis Defense: "Truncated Realized Variations of Lévy Models: Optimality, Debiasing, and Implementation Approaches."

Abstract: Statistical inference for stochastic processes under high-frequency observations has been an active research area in econometrics and financial statistics for over twenty years. In this dissertation, we consider some aspects related to the estimation of the volatility of an Itô semimartingale in the presence of Lévy-type jumps based on truncated realized variations (TRV). Motivated by recent results that state that the MSE optimal threshold parameter is asymptotically proportional to the modulus of continuity of Brownian motion and the volatility itself, we first investigate the consistency and CLTs of estimation methods in which the volatility is iteratively estimated until the method stabilizes on a unique estimate. Secondly, we study the MSE optimal truncation level for a semiparametric tempered stable Lévy process of unbounded variation. We obtain an explicit second-order expansion of the optimal threshold in a high-frequency asymptotic regime, hence, generalizing earlier results that considered only stable Lévy processes and first-order asymptotics. As an application, a new estimation method is put forward. The method iteratively combines the generalized method of moment estimators and TRVs with the newly found small-time approximation for the optimal threshold. Finally, by developing new high-order expansions of the truncated moments of a Lévy process, we construct a new rate- and variance-efficient volatility estimator by applying a two-step debiasing procedure to TRV for a class of tempered stable Lévy processes of unbounded variation. Extensive Monte Carlo experiments indicate that our method outperforms other efficient alternatives in the literature in the setting covered by our theoretical framework.

Host: Jose Figueroa-Lopez