Spectral Densities of Ultra - high Frequency Data

This paper suggests the application of advanced methods from Fourier Analysis in order to describe ultra-high frequent data in limit order books. Using Lomb’s normalized periodogram and Scargle’s Discrete Fourier Transforms to take account of the irregularity in spacing, the power spectra of different time series processes can be easily estimated, without immense computational effort caused by the large amount of observations. With empirical data extracted from the German XETRA system, the spectral analysis shows that the entire trading process contains various different periodic components. While duration and volume processes have a strong cyclical behavior in the low-frequency domain, seasonalities of price differences arise in the high-frequency domain. Contrarily, the time series of the spread reveals no periodicity, neither in the long term, nor in the middle or short term.