Higher Order Realized Power Variations of Semi-Martingales with Applications

Abstract

The realized power variations with even order of a discretely observed semi-martingale have been widely studied in literature, due to some important applications in finance, for example, estimating the integrated volatility and integrated quarticity. However, few works have paid attention to the realized power variations whose power indices are odd. In this paper, we derive some limit theorems for realized variations with odd functions of an Ito semi-martingale on the fixed time interval [0,T], observed discretely at a high frequency. In the continuous case, unlike the realized power variations of even order, for example the quadratic variation, they converge only in distribution (stably) after multiplied by some appropriate factors, which are related to the length of the sampling interval, and the limiting processes consist of centered Wiener integrals and Riemann integrals that play a role as asymptotic biases. The limit theorems for the general case containing jumps have also been derived. An important application of the result is to measure the realized skewness with high frequency data. Simulation studies for various models have been investigated. Finally, we provide some real applications.