this paper is built. First, although raw returns are clearly leptokurtic, returns standardized by realized volatilities are approximately Gaussian. Second, although the distributions of realized volatilities are clearly right-skewed, the distributions of the logarithms of realized volatilities are approximately Gaussian. Third, the long-run dynamics of realized logarithmic volatilities are well approximated by a fractionally-integrated long-memory process. Motivated by the three ABDL empirical regularities, we proceed to estimate and evaluate a multivariate model for the logarithmic realized volatilities: a fractionally-integrated Gaussian vector autoregression (VAR) . Importantly, our approach explicitly permits measurement errors in the realized volatilities. Comparing the resulting volatility forecasts to those obtained from currently popular daily volatility models and more complicated high-frequency models, we find that our simple Gaussian VAR forecasts generally produce superior forecasts. Furthermore, we show that, given the theoretically motivated and empirically plausible assumption of normally distributed returns conditional on the realized volatilities, the resulting lognormal-normal mixture forecast distribution provides conditionally well-calibrated density forecasts of returns, from which we obtain accurate estimates of conditional return quantiles. In the remainder of this paper, we proceed as follows. We begin in section 2 by formally developing the relevant quadratic variation theory within a standard frictionless arbitrage-free multivariate pricing environment. In section 3 we discuss the practical construction of realized volatilities from high-frequency foreign exchange returns. Next, in section 4 we summarize the salient distributional features of r...

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 di#erence 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.

Abstract not found

A rapidly growing literature has documented important improvements in financial return volatility measurement and forecasting via use of realized variation measures constructed from high-frequency returns coupled with simple modeling procedures. Building on recent theoretical results in Barndorff-Nielsen and Shephard (2004a, 2005) for related bi-power variation measures, the present paper provides a practical and robust framework for non-parametrically measuring the jump component in asset return volatility. In an application to the DM/ $ exchange rate, the S&P500 market index, and the 30-year U.S. Treasury bond yield, we find that jumps are both highly prevalent and distinctly less persistent than the continuous sample path variation process. Moreover, many jumps appear directly associated with specific macroeconomic news announcements. Separating jump from non-jump movements in a simple but sophisticated volatility forecasting model, we find that almost all of the predictability in daily, weekly, and monthly return volatilities comes from the non-jump component. Our results thus set the stage for a number of interesting future econometric developments and important financial applications by separately modeling, forecasting, and pricing the continuous and jump components of the total return variation process.

This paper uses the Specific-to-General methodological approach that is widely used in science, in which problems with existing theories are resolved as the need arises, to illustrate a number of important developments in the modelling of univariate and multivariate financial volatility. Twenty frequently arising issues in analysing timevarying univariate and multivariate conditional volatility and stochastic volatility are discussed. In view of some of these difficulties, including the number of parameters to be estimated, and the computational complexities associated with multivariate conditional volatility models and both univariate and multivariate stochastic volatility models, automated inference is argued to be unhelpful to modelling in empirical financial econometrics. Some suggestions for future research are also presented. *The author wishes to acknowledge helpful discussions with Manabu Asai, Massimiliano

We compare the probabilistic properties of the non-Gaussian Ornstein-Uhlenbeck based stochastic volatility model of Barndorff-Nielsen and Shephard (2001) with those of the COGARCH process. The latter is a continuous time GARCH process introduced by the authors (2004). Many features are shown to be shared by both processes, but differences are pointed out as well. Furthermore, it is shown that the COGARCH process has Pareto like tails under weak regularity conditions.

We develop general model-free adjustment procedures for the calculation of unbiased volatility loss functions based on practically feasible realized volatility benchmarks. The procedures, which exploit the recent non-parametric asymptotic distributional results in Barndorff-Nielsen and Shephard (2002a) along with new results explicitly allowing for leverage effects, are both easy-to-implement and highly accurate in empirically realistic situations. On properly accounting for the measurement errors in the volatility forecast evaluations reported in Andersen, Bollerslev, Diebold and Labys (2003), the adjustments result in markedly higher estimates for the true degree of return volatility predictability.

Summary. Consider a semimartingale of the form Yt = Y0 + ∫ t 0 asds + ∫ t σs − dWs, 0 where a is a locally bounded predictable process and σ (the “volatility”) is an adapted right–continuous process with left limits and W is a Brownian motion. We consider the realised bipower variation process V (Y; r, s) n t = n r+s

A rapidly growing literature has documented important improvements in volatility measurement and forecasting performance through the use of realized volatilities constructed from high-frequency returns coupled with relatively simple reduced form time series modeling procedures. Building on recent theoretical results from Barndorff-Nielsen and Shephard (2003c) for related bi-power variation measures involving the sum of high-frequency absolute returns, the present paper provides a practical framework for non-parametrically measuring the jump component in the realized volatility measurements. Exploiting these ideas for a decade of high-frequency five-minute returns for the DM/ $ exchange rate, the S&P500 aggregate market index, and the 30-year U.S. Treasury Bond, we find the jump components to be distinctly less persistent than the contribution to the overall return variability originating from the continuous sample path component of the price process. Explicitly including the jump measure as an additional explanatory variable in an easy-to-implement reduced form model for the realized volatilities results in highly significant jump coefficient estimates at the daily, weekly and quarterly forecasts horizons. As such, our results hold promise for improved financial asset allocation, risk management, and derivatives pricing, by separate modeling, forecasting and pricing of the continuous and jump components of the total return variability.

Abstract not found