This is a draft Chapter from a book by the authors on “L´evy Driven Volatility Models”.

In this paper we study the detailed distributional properties of integrated non-Gaussian OU (intOU) processes. Both exact results and approximate results are given. We emphasise the study of the tail behaviour of the intOU process. Our results have many potential applications in financial...

We consider kernel-based estimators of integrated variances in the presence of independent market microstructure effects. We derive the bias and variance properties for all regular kernel-based estimators and derive a lower bound for their asymptotic variance. Further we show that the...

The availability of intra-data on the prices of speculative assets means that we can use quadratic variation like measures of activity in financial markets, called realised volatility, to study the stochastic properties of returns. Here we derive the moments and the asymptotic distribution of...

We propose a new measure of risk, based entirely on downwards moves measured using high frequency data. Realised semivariances are shown to have important predictive qualities for future market volatility. The theory of these new measures is spelt out, drawing on some new results from...

In a recent paper we have introduced the class of realised kernel estimators of the increments of quadratic variation in the presence of noise. We showed that this estimator is consistent and derived its limit distribution under various assumptions on the kernel weights. In this paper we extend...

In this paper we provide a systematic study of the robustness of probability limits and central limit theory for realised multipower variation when we add finite activity and infinite activity jump processes to an underlying Brownian semimartingale.

Limit distribution results on realised power variation, that is sums of absolute powers of increments of a process, are derived for certain types of semimartingale with continuous local martingale component, in particular for a class of flexible stochastic volatility models. The theory cover,...

In this paper we study the reliability of the mixed normal asymptotic distribution of realised volatility error, which we have previously derived using the theory of realised power variation. Our experiments suggests that the asymptotics is reliable when we work with the logarithmic transform of...

This paper discusses two classes of distributions, and stochastic processes derived from them: modified stable (MS) laws and normal modified stable (NMS) laws. This extends corresponding results for the generalised inverse Gaussian (GIG) and generalised hypberbolic (GH) or normal generalised...