Estimation of volatility in a high-frequency setting: a short review

Our aim is to give an overview of the topic of estimation of volatility, in a high-frequency setting. We emphasize the various possible situations, relative to the underlying process (continuous, or with jumps having finite, or infinite, activity) and to the observation scheme (with microstructure noise or not, under a regular sampling scheme or not). We try to explain a variety of methods, including the most recent ones. Each method is quickly sketched, with comments on its range of applicability. Most results are given in the form of a theorem, with a precise description of the assumptions needed, but of course without proof, and some results are simply mentioned in a somewhat loose way. We consider only the one-dimensional case, although occasional comments are made about the multivariate case. We totally omit the nowadays hot topic when the number of assets is very large, meaning that this number increases as the frequency increases: this is unfortunately not compatible with a "short" review as this one.