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We propose using the price range in the estimation of stochastic volatility models. We show theoretically, numerically, and empirically that range-based volatility proxies are not only highly efficient, but also approximately Gaussian and robust to microstructure noise. Hence range-based Gaussian quasi-maximum likelihood estimation produces highly efficient estimates of stochastic volatility models and extractions of latent volatility. We use our method to examine the dynamics of daily exchange rate volatility and find the evidence points strongly toward two-factor models with one highly persistent factor and one quickly mean-reverting factor. VOLATILITY IS A CENTRAL CONCEPT in finance, whether in asset pricing, portfolio choice, or risk management. Not long ago, theoretical models routinely assumed constant volatility ~e.g., Merton ~1969!, Black and Scholes ~1973!!. Today, however, we widely acknowledge that volatility is both time varying and predictable ~e.g., Andersen and Bollerslev ~1997!!, andstochastic volatility models are commonplace. Discrete- and continuous-time stochastic volatility models are extensively used in theoretical finance, empirical finance, and financial econometrics, both in academe and industry ~e.g., Hull and

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We consider various MIDAS (Mixed Data Sampling) regression models to predict volatility. The models differ in the specification of regressors (squared returns, absolute returns, realized volatility, realized power, and return ranges), in the use of daily or intra-daily (5-minute) data, and in the length of the past history included in the forecasts. The MIDAS framework allows us to compare models across all these dimensions in a very tightly parameterized fashion. Using equity return data, we find that daily realized power (involving 5-minute absolute returns) is the best predictor of future volatility (measured by increments in quadratic variation) and outperforms model based on realized volatility (i.e. past increments in quadratic variation). Surprisingly, the direct use of high-frequency (5-minute) data does not improve volatility predictions. Finally, daily lags of one to two months are sufficient to capture the persistence in volatility. These findings hold both in- and out-of-sample.

In this paper, we provide both qualitative and quantitative measures of the precision of measuring integrated volatility by realized volatility for a fixed frequency of observation. We start by characterizing for a general diffusion the dierence between realized and integrated volatility for a given frequency of observation. Then we compute the mean and variance of this noise and the correlation between the noise and the integrated volatility in the Eigenfunction Stochastic Volatility model of Meddahi (2001a). This model has as special cases log-normal, affine and GARCH diusion models. Using previous empirical results, we show that the noise is substantial compared with the unconditional mean and variance of integrated volatility, even if one employs five-minute returns. We also propose a simple approach to capture the information about integrated volatility contained in the returns through the leverage eect. We show that in practice, the leverage effect does not matter.

SUMMARY In the 20 years following the publication of the ARCH model, there has been a vast quantity of research uncovering the properties of competing volatility models. Wide-ranging applications to financial data have discovered important stylized facts and illustrated both the strengths and weaknesses of the models. There are now many surveys of this literature. This paper looks forward to identify promising areas of new research. The paper lists five new frontiers. It briefly discusses three-high-frequency volatility models, large-scale multivariate ARCH models, and derivatives pricing models. Two further frontiers are examined in more detail-application of ARCH models to the broad class of non-negative processes, and use of Least Squares Monte Carlo to examine non-linear properties of any model that can be simulated. Using this methodology, the paper analyses more general types of ARCH models, stochastic volatility models, long-memory models and breaking volatility models. The volatility of volatility is defined, estimated and compared with option-implied volatilities.

This paper is concerned with the Bayesian estimation and comparison of flexible, high dimensional multivariate time series models with time varying correlations. The model proposed and considered here combines features of the classical factor model with that of the heavy tailed univariate stochastic volatility model. A unified analysis of the model, and its special cases, is developed that encompasses estimation, filtering and model choice. The centerpieces of the estimation algorithm (which relies on MCMC methods) are (1) a reduced blocking scheme for sampling the free elements of the loading matrix and the factors and (2) a special method for sampling the parameters of the univariate SV process. The resulting algorithm is scalable in terms of series and factors and simulation-efficient. Methods for estimating the log-likelihood function and the filtered values of the time-varying volatilities and correlations are also provided. The performance and effectiveness of the inferential methods are extensively tested using simulated data where models up to 50 dimensions and 688 parameters are fitted and studied. The performance of our model, in relation to multivariate GARCH models, is also evaluated using a real data set of weekly returns on a set of 10 international stock indices. We consider the performance along two dimensions: the ability to correctly estimate the conditional covariance matrix of future returns and the unconditional and conditional coverage of the 5 % and 1% Value-at-Risk (VaR) measures of four pre-defined portfolios.

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This paper analyses multivariate high frequency financial data using realized covari-ation. 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.

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