Truncated Realized Variations of Lévy Models: Optimality, Debiasing, and Implementation Approaches

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 thesis, we consider some aspects related to the estimation of the volatility of an Itô semimartingale in the presence of Lévy-type jumps, which is of fundamental importance in derivatives pricing evaluation, risk management and portfolio allocation. The main technique we use is the Truncated Realized Variation (TRV) that is both rate- and variance-efficient, in the Cramer-Rao lower bound sense, when jumps are of bounded variation. Motivated by recent results that state that the optimal threshold parameter, in the mean squared error (MSE) sense, is asymptotically proportional to the modulus of continuity of Brownian motion and the volatility itself, we first investigate the consistency and central limit theorems (CLTs) of estimation methods in which the volatility is iteratively estimated until the method stabilizes on a unique estimate. To this end, we establish some new consistency and CLT results for TRVs with random data-driven thresholds. 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. Our method shows superior performance, especially when the jump activity index is larger than 3/2. 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