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1. INTRODUCTION
In this paper we discuss the limiting theory for a novel, unifying class of nonparametric measures of the variation of financial prices. The theory covers commonly used estimators of variation such as realized volatility, but it also encompasses more recently suggested quantities such as realized power variation and realized bipower variation. We considerably strengthen existing results on the latter two quantities, deepening our understanding and unifying their treatment. We will outline the proofs of these theorems, referring for the very technical, detailed formal proofs of the general results to a companion probability theory paper, Barndorff-Nielsen, Graversen, Jacod, Podolskij, and Shephard (2006). Our emphasis is on exposition, explaining where the results come from and how they sit within the econometrics literature.
Our theoretical development is motivated by the advent of complete records of quotes or transaction prices for many financial assets. Although market microstructure effects (e.g., discreteness of prices, bid/ask bounce, irregular trading, etc.) mean that there is a mismatch between asset pricing theory based on semimartingales and the data at very fine time intervals it does suggest the desirability of establishing an asymptotic distribution theory for estimators as we use more and more highly frequent observations. Papers that directly model the impact of market frictions on realized volatility include Zhou (1996), Bandi and Russell (2003), Hansen and Lunde (2006), Zhang, Mykland, and Aït-Sahalia (2005), Barndorff-Nielsen, Hansen, Lunde, and Shephard (2004), and Zhang (2004). Related work in the probability literature on the impact of noise on discretely observed diffusions can be found in Gloter and Jacod (2001a, 2001b), whereas Delattre and Jacod (1997) report results on the impact of rounding on sums of functions of discretely observed diffusions. In this paper we ignore these effects.
Let the d -dimensional vector of the log prices of a set of assets follow the process
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At time t ≥ 0 we denote the log prices as Yt . Our aim is to calculate measures of the variation of the price process (e.g., realized volatility) over discrete time intervals (e.g., a day or a month). Without loss of generality we can study the mathematics of this by simply looking at what happens when we have n high-frequency observations on the time interval