Abstract
Multifractal detrended cross-correlation methodology is described and applied to Foreign exchange (Forex) market time series. Fluctuations of high-frequency exchange rates of eight major world currencies over 2010–2018 period are used to study cross-correlations. The study is motivated by fundamental questions in complex systems’ response to significant environmental changes and by potential applications in investment strategies, including detecting triangular arbitrage opportunities. Dominant multiscale cross-correlations between the exchange rates are found to typically occur at smaller fluctuation levels. However, hierarchical organization of ties expressed in terms of dendrograms, with a novel application of the multiscale cross-correlation coefficient, is more pronounced at large fluctuations. The cross-correlations are quantified to be stronger on average between those exchange rate pairs that are bound within triangular relations. Some pairs from outside triangular relations are, however, identified to be exceptionally strongly correlated as compared to the average strength of triangular correlations. This in particular applies to those exchange rates that involve Australian and New Zealand dollars and reflects their economic relations. Significant events with impact on the Forex are shown to induce triangular arbitrage opportunities which at the same time reduce cross-correlations on the smallest timescales and act destructively on the multiscale organization of correlations. In 2010–2018, such instances took place in connection with the Swiss National Bank intervention and the weakening of British pound sterling accompanying the initiation of Brexit procedure. The methodology could be applicable to temporal and multiscale pattern detection in any time series.
Highlights
In our study we focus on different signatures and statistical properties of multivariate time series with respect to triangular arbitrage, one may envisage a broader picture of such analysis, whereby one would like to uncover a specific kind of crosscorrelations in these time series which would help us to detect underlying interconnections useful for the system behavior prediction in future
The tails of the cumulative distributions of the highfrequency intra-day quotes exhibit non-Gaussian distribution of the rare events by means of the so-called fat tails. This clearly documents that large fluctuations in the logarithmic rate returns occur more frequently than one may expect from the Gaussian distribution
We have found that on average the cross-correlations of exchange rates for currencies in the triangular relationship are stronger than cross-correlations between exchange rates for currencies outside the triangular relationship
Summary
The data used in the present study have been obtained from the Dukascopy Swiss Banking Group [31]. The data set (see “Appendix”) comprises foreign exchange rates for the period 2010–2018 between pairs of 8 major currencies. For the purpose of this study, we consider the arithmetic average out of the bid and ask price for each exchange rate: R(t) = Rask(t) + Rbid(t). We consider the following time series of logarithmic returns of such exchange rates for each pair of currencies:
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