A theoretical comparison between integrated and realized volatility

Abstract

SUMMARYIn 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 difference 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 diffusion 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 effect. We show that in practice, the leverage effect does not matter.

Volatility forecast comparison using imperfect volatility proxies

“…A loss function, L, is ''robust'' if the ranking of any two (possibly imperfect) volatility forecasts, h 1t and h 2t , by expected loss is the same whether the ranking is done using the true conditional variance, σ 2 t , or some conditionally unbiased volatility proxy,σ 2 t . That is, Meddahi (2001) showed that the ranking of forecasts on the basis of the R 2 from the Mincer-Zarnowitz regression:…”

Realized Volatility: A Review

“…and Barndorff-Nielsen and Shephard (2005b), Meddahi (2002), Gonçalves and Meddahi (2005), and Nielsen and Frederiksen (2006) studied the finite sample behavior of the limit theory given in (3). The main conclusion is that (3) is poorly sized, but…”

“…Based on the theoretical results of Barndorff-Nielsen and Shephard (2002), , and Meddahi (2002), several recent studies have documented the properties of realized volatilities constructed from high frequency data. However, as will be discussed later, microstructure effects introduce a severe bias on the daily volatility estimation.…”

Indirect inference with time series observed with error

“…Meddahi (2002) proves that η t (∆) has a nonzero mean, when the drift in prices is non-zero, and is heteroskedastic. The correlation between IV and η t (∆) is zero when there is no leverage effect Shephard, 2002b andMeddahi, 2002).…”