Estimating quadratic variation using realised variance

This paper looks at some recent work on estimating quadratic variation using realised variance (RV) - that is sums of M squared returns. This econometrics has been motivated by the advent of the common availability of high frequency financial return data. When the underlying process is a semimartingale we recall the fundamental results that RV is a consistent estimator of quadratic variation (QV). We express concern that without additional assumptions it seems difficult to given any measure of uncertainty of the RV in this context. The position dramatically changes when we work with a rather general SV model - which is a special case of the semimartingale model. Then QV is integrated variance and we can derive the asymptotic distribution of the RV and its rate of convergence. These results do not require us to specify a model for either the drift of volatility functions, although we have to impose some weak regulatrity assumptions. We illustrate the use of the limit theory on some exchange rate data and some stock data. We show that even with large values of M the RV is sometimes a quite noisy estimator of integrated variance.