Fourier Method for the Measurement of Univariate and Multivariate Volatility in the Presence of High Frequency Data

Abstract

The increasing availability of intraday data as well as the increase in computational power has created an interest in the calculation of volatility and correlation using high frequency data instead of the more commonly used daily/weekly/monthly data that is available. However, on a high frequency scale, financial time series are not evenly spaced or synchronous, and therefore standard methods of calculating volatility and correlation cannot be directly applied. This dissertation evaluates a method proposed by Malliavin and Mancino [22] that uses a method based on Fourier series analysis to calculate the univariate and multivariate volatility. This method is compared with the traditional methods of estimating volatility and cross-correlation from tick-by-tick (high frequency) data in the context of an emerging market. In the case of evenly spaced data, we found that the Fourier method compares very well with classical methods and provide smoother estimates. In the case of high frequency data, we confirmed and extended the results of Iori [34] and found that the Fourier method gives better results than the realised volatility estimator in terms of generating smooth estimates with a lower bias and root mean squared error, which are also less sensitive to the choice of returns time scale. It is conceptually superior to methods that use interpolation and is also model independent. In addition, the Fourier method guarantees a positive definite matrix, which is not the case with other classical methods. The dataset analysed in this paper is the two-and-a-half year tick-by-tick trades executed on the JSE Stock Exchange from May 2002 till October 2004. The dataset was provided by Deutsche Bank South Africa.

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Cite this paper

@inproceedings{Malherbe2007FourierMF, title={Fourier Method for the Measurement of Univariate and Multivariate Volatility in the Presence of High Frequency Data}, author={Chanel Malherbe}, year={2007} }