A theoretical comparison between integrated and realized volatilities

Texte intégral

(1)CAHIER 30-2001. A THEORETICAL COMPARISON BETWEEN INTEGRATED AND REALIZED VOLATILITIES. Nour MEDDAHI.

(2) CAHIER 30-2001. A THEORETICAL COMPARISON BETWEEN INTEGRATED AND REALIZED VOLATILITIES Nour MEDDAHI1. 1. Centre de recherche et développement en économique (C.R.D.E.), CIRANO and Département de sciences économiques (Université de Montréal), and CEPR. December 2001. _______________________ The author would like to thank Bryan Campbell, Jean-Marie Dufour, René Garcia, Benoit Perron, Éric Renault, and Neil Shephard for helpful comments on an earlier draft. The author acknowledges financial support from Fonds FCAR, IFM2, and MITACS, and is solely responsible for any remaining errors..

(3) RÉSUMÉ Dans cet article, nous quantifions qualitativement et quantitativement la précision de la mesure de la volatilité intégrée par la volatilité réalisée quand la fréquence d’observations est fixée. Nous commençons par caractériser pour une diffusion générale la différence entre les volatilités réalisée et intégrée pour une fréquence d’observations donnée. Ensuite, nous calculons l’espérance et la variance de ce bruit ainsi que sa corrélation avec la volatilité intégrée en supposant que la diffusion est un modèle à volatilité stochastique par fonctions propres de Meddahi (2001a). Ce modèle contient, comme exemples particuliers, les modèles de diffusion log-normale, affine et GARCH. En utilisant certains résultats empiriques, nous montrons que l’écart-type du bruit n’est pas négligeable par rapport à la moyenne et à l’écart-type de la volatilité intégrée même si on considère des rendements à cinq minutes. Nous proposons aussi une approche simple pour extraire l’information sur la volatilité intégrée contenue dans les rendements via l’effet de levier. Mots clés : volatilité intégrée, volatilité réalisée, générateur infinitésimal, modèles à volatilité stochastique par fonctions propres, effet de levier, moments exacts. ABSTRACT. 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. Key words : integrated volatility, realized volatility, infinitesimal generator, eigenfunction stochastic volatility models, leverage effect, exact moments.

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