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A Theoretical Comparison Between Integrated and Realized Volatilities

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  • Nour Meddahi

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. 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-normal, affine et GARCH. En utilisant certains résulats 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 considéré 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.

Suggested Citation

  • Nour Meddahi, 2001. "A Theoretical Comparison Between Integrated and Realized Volatilities," CIRANO Working Papers 2001s-71, CIRANO.
  • Handle: RePEc:cir:cirwor:2001s-71
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    More about this item

    Keywords

    integrated volatility; realized volatility; infinitesimal generator; eigenfunction stochastic volatility models; leverage effect; exact moments; volatilité intégrée; volatilité réalisée; générateur infinitésimal; modèles à volatilité stochastique par fonctions propres; effet de levier; moments exacts;
    All these keywords.

    JEL classification:

    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General

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