Estimation of the realized (co-)volatility vector: Large deviations approach

Realized statistics based on high frequency returns have become very popular in financial economics. In recent years, different non-parametric estimators of the variation of a log-price process have appeared. Among them are the realized quadratic (co-)variation which is perhaps the most well known example, providing a consistent estimator of the integrated (co-)volatility when the logarithmic price process is continuous. In this paper, we propose to study the large deviation properties of realized (co-)volatility. Our main motivation is to improve upon the existing limit theorems such as the weak law of large numbers or the central limit theorem which have been proved in different contexts. Our large deviations results can be used to evaluate and approximate tail probabilities of realized (co-)volatility. As an application we provide the large deviations for the standard dependence measures between the two assets returns such as the realized regression coefficients or the realized correlation. Our study should contribute to the recent trend of research on the (co-)variance estimation problems, which are quite often discussed in high-frequency financial data analysis.