Path properties of volatility: A statistical analysis

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We develop statistical theory to infer path properties of the volatility of a stochastic process. Volatility is the key quantity to describe uncertainty in the evolution of a stochastic process. Its path properties determine the smoothness and dependence structure of volatility and thus optimal estimation methods, forecasting techniques and the persistence of volatility. Even though path properties of volatility are of particular interest for applications and subject of current empirical work, there is so far no statistical groundwork. Recently, there have been important contributions to advance estimation and testing procedures on path properties of stochastic processes from direct observations. The key difference and difficulty is that volatility is latent and thus not directly observable. Based on statistics involving pre-estimated volatility, the two applicants started to work in this new direction in a first article focusing on change-point analysis. Exploiting the most innovative contributions regarding volatility estimation and inference on path properties from direct observations, we aim to establish a novel strand of research with optimal statistical approaches. Currently, conflicting models for volatility are put forward in the literature. In order to build adequate volatility models, more knowledge on the path properties of volatility is required. We consider a rich class of random volatility processes to reconcile different stylized facts. The theory developed in this project will provide evidence about which models are suitable and, moreover, if path properties are persistent or time-varying. Particular interest in this work is motivated by financial econometrics. Volatility is the prevailing concept to describe market risk in price evolutions. Reliable volatility estimates are thus key ingredients for risk analysis. Our main focus is on intra-day high-frequency data at highest available recording frequencies. Modeling and analysing such high-frequency data becomes more and more important as much volume, currently almost 70 %, is attributed to high-frequency trading. At the same time, specific market frictions need to be taken into account, inducing noisy price observations. We design approaches for two noise specifications: The classical centred market microstructure noise model and irregular noise suitable for recorded prices from limit order books. This is pursued in three complementary work packages. One work package, providing novel methodological groundwork, is addressed by the applicants. In an idealized framework, we determine the minimal amount of information necessary to identify certain path properties and construct efficient techniques for their recovery. The two doctoral projects focus on optimal methods for the two related yet different noisy observation models. The more involved structure of these realistic models ask for several innovations to address the identification of the path properties of volatility.