Periodicities of Foreign Exchange Markets and the Directional Change Power Law

Abstract

This paper utilizes advanced methods from Fourier analysis in order to describe periodicities in financial ultrahigh frequency foreign exchange data. The Lomb-Scargle Fourier transform is used to take into account the irregularity in spacing in the time domain. It provides a natural framework for the power spectra of different inhomogeneous time-series processes to be easily and quickly estimated. Furthermore, an event-based approach in intrinsic time based on a power-law relationship is employed using different event thresholds to filter the foreign exchange tick-data. The calculated spectral density demonstrates that the price process in intrinsic time contains different periodic components for directional changes, especially in the medium-long term, implying the existence of stylized facts of ultrahigh frequency data in the frequency domain.In fact, Aldridge (2010) calculates the maximum gain as the sum of price ranges at each frequency, whereas the maximum potential gain (loss) at every frequency is determined by the (negative) sum of all per-period ranges at that frequency. 190 I. GIAMPAOLI ET AL.