A Theoretical Comparison Between Integrated andRealized Volatilities / A Theoretical Comparison Between Integrated and Realized Volatilities

In this paper, we provide both qualitative and quantitative measures of the cost of measuring the integrated volatility by the realized volatility when the frequency of observation is fixed. We start by characterizing for a general diffusion the difference between the realized and the integrated volatilities for a given frequency of observations. 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 examples, log-normal, affine and GARCH diffusion models. Using some previous empirical works, we show that the standard deviation of the noise is not negligible with respect to the mean and the standard deviation of the integrated volatility even if one considers returns at five minutes. We also propose a simple approach to capture the information about the integrated volatility contained in the returns through the leverage effect. Dans cet article, nous quantifions qualitativement et quantitativement la precision de la mesure de la volatilite integree par la volatilite realisee quand la frequence d'observations est fixee. Nous commencons par caracteriser pour une diffusion generale la difference entre les volatilites realisee et integree pour une frequence d'observations donnee. Ensuite, nous calculons l'esperance et la variance de ce bruit ainsi que sa correlation avec la volatilite integree en supposant que la diffusion est un modele a volatilite stochastique par fonctions propres de Meddahi (2001a). Ce modele contient, comme exemples particuliers, les modeles de diffusion log-normal, affine et GARCH. En utilisant certains resulats empiriques, nous montrons que l'ecart-type du bruit n'est pas negligeable par rapport a la moyenne et a l'ecart-type de la volatilite integree meme si on considere des rendements a cinq minutes. Nous proposons aussi une approche simple pour extraire l'information sur la volatilite integree contenue dans les rendements via l'effet de levier.