Probability measures, L\'{e}vy measures and analyticity in time

We investigate the relation of the semigroup probability density of an infinite activity L\'{e}vy process to the corresponding L\'{e}vy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the L\'{e}vy measure and the third method uses the analytic continuation of the L\'{e}vy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6114 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Econometrics of testing for jumps in financial economics using bipower variation

In this paper we provide an asymptotic distribution theory for some non-parametric tests of the hypothesis that asset prices have continuous sample paths. We study the behaviour of the tests using simulated data and see that certain versions of the tests have good finite sample behaviour. We also apply the tests to exchange rate data and show that the null of a continuous sample path is frequently rejected. Most of the jumps the statistics identify are associated with governmental macroeconomic announcements.Bipower variation; Jump process; Quadratic variation; Realised variance; emimartingales; Stochastic volatility.

Econometric Analysis of Realised Covariation: High Frequency Covariance, Regression and Correlation in Financial Economics

This paper analyses multivariate high frequency financial data using realised covariation. We provide a new asymptotic distribution theory for standard methods such as regression, correlation analysis and covariance. It will be based on a fixed interval of time (e.g. a day or week), allowing the number of high frequency returns during this period to go to infinity. Our analysis allows us to study how high frequency correlations, regressions and covariances change through time. In particular we provide confidence intervals for each of these quantities.Power variation; Realised correlation; Realised covolatility; Realised regression; Realised variance; Semimartingales; Covolatility

Integrated OU Processes

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 economics, for OU processes are used as models of instantaneous volatility in stochastic volatility (SV) models. In this case an intOU process can be regarded as a model of integrated volatility. Hence the tail behaviour of the intOU process will determine the tail behaviour of returns generated by SV models.Background driving Levy process; Chronometer; Co-break; Econometrics; Integrated volatility; Kumulant function; Levy density; Option pricing; OU processes; Stochastic volatility

Power and bipower variation with stochastic volatility and jumps

This paper shows that realised power variation and its extension we introduce here called realised bipower variation is somewhat robust to rare jumps. We show realised bipower variation estimates integrated variance in SV models --- thus providing a model free and consistent alternative to realised variance. Its robustness property means that if we have an SV plus infrequent jumps process then the difference between realised variance and realised bipower variation estimates the quadratic variation of the jump component. This seems to be the first method which can divide up quadratic variation into its continuous and jump components. Various extensions are given. Proofs of special cases of these results are given. Detailed mathematical results will be reported elsewhere.

How accurate is the asymptotic approximation to the distribution of realised volatility?

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 the realised volatility.Levy process; Mixed Gaussian limit; OU process; Quadratic variation; Realised power variation; Realised volatility; Square root process; Stochastic volatility; Superposition.

Estimating quadratic variation using realised volatility

This paper looks at some recent work on estimating quadratic variation using realised volatility (RV) - that is sums of M squared returns. When the underlying process is a semimartingale we recall the fundamental result that RV is a consistent estimator of quadratic variation (QV). We express concern that without additonal 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 volatility 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 or volatility functions, although we have to impose some weak regularity assumptions. We illustrate the use of the limit theory on some exchange rate data. We show that even with the large values of M and RV is sometimes a quite noisy estimator of integrated volatilityPower variation; Quadratic variation; Realised volatility; Semimartingale; Volatility.

A Feasible Central Limit Theory for Realised Volatility Under Leverage

In this note we show that the feasible central limit theory for realised volatility and realised covariation recently developed by Barndorff-Nielsen and Shephard applies under arbitrary diffusion based leverage effects. Results from a simulation experiment suggest that the feasible version of the limit theory performs well in practice.Euler approximation; Functional central limit theory; Quadratic variation; Realised volatility; Stochastic volatility.

Econometric analysis of realised volatility and its use in estimating stochastic volatility models

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 the realised volatility error - the difference between realised volatility and the actual volatility. These properties can be used to allow us to estimate the parameters of stochastic volatility models.Econometrics; Higher order variation; Kalman filter; Leverage; Levy process; OU process; Quarticity; Quadratic variation; Realised volatility; Square root process; Stochastic volatility; Subordination; Superposition.