Fourier series method for measurement of multivariate volatilities

Realized Volatility: A Review

“…Nielsen and Frederiksen (2006) also evaluated the finite sample accuracy of different estimators of the integrated variance under the presence of microstructure noise and possible jumps. The authors considered three estimators: the realized variance, the estimator based on Fourier series (Barucci and Reno, 2002a,b;Malliavin and Mancino, 2002), and finally, the wavelet estimator of Høg and Lunde (2003). The main conclusion of the article is that the Fourier estimator is preferable when compared to the other two and, most surprisingly, it has a slightly better finite sample performance (in terms of MSE) than the bias-corrected kernel-based estimators as in Hansen and Lunde (2006b).…”

“…The authors also provided recommendations for practical implementations of such estimators. Hoshikawa et al (2008) compared the multivariate version of the Fourier estimator of Malliavin and Mancino (2002), the HY estimator, and the classical realized covariance estimator. The authors found that the HY estimator performs the best among the alternatives in view of the bias and the MSE, while the other estimators were shown to have possibly heavy bias, mostly toward the origin.…”

Theories on the Relationship between Price Process and Stochastic Volatility Matrix with Compensated Poisson Jump Using Fourier Transforms

“…Malliavin and Mancino [3] presented the computation of a time series volatility using Fourier series analysis method from observations of a semimartingale data. The method was nonparametric and model free.…”

“…Kanatani [4] derived a linear interpolation bias of realized volatility. He used Fourier series estimator proposed by Malliavin and Mancino [3] to avoid the biasness. He examined the theoretical relationship between realized volatility and Fourier estimator and showed that the Fourier estimator was most efficient than the realized volatility.…”

Cross-Correlation Measures in the High-Frequency Domain

“…This method also requires synchronous and homogeneous time series. Barucci and Renò(2002), Renò(2003) have adapted a Fourier method developed by Malliavin and Mancino(2002) to the computation of FX rates correlations. The Fourier method can be directly applied to the actual time series to obtain correlation statistics.…”