From statistical point of view, many physical phenomena are characterized by non-gaussian p.d.f. with slowly declining tails. Those tails contain information about statistics of extreme events, i.e. fluctuations which greatly deviate from the distribution's mean.
In the case of financial markets the most extreme events are observed for the foreign exchange market - Forex. They are especially pronounced in a context of the so-called triangular relation, coupling the mutual exchange rates of each currency triple as required by the no-arbitrage condition. Here we consider high-frequency time series of deviations from the perfect triangle
relation expressed by a sum of the coupled exchange rate returns. The distribution of deviations in this case is distinguished by the tails which are exceptionally "fat" as compared with other financial data. This suggests that within a triangle the balance between currency quotations can be significantly violated, indicating an arbitrage opportunity.
The situation is even more intriguing when one considers the temporal organization of such time series. The Hurst-type analysis shows that the deviations possess strongly antipersistent behaviour and, moreover, both the Hoelder exponents and the fractal dimensions take negative values. Such a result may seem rather strange at first glance, but this kind of anomalies have already been discussed earlier, e.g. by B. Mandelbrot, in a context of some variant of the multifractal cascade: the two-valued canonical multifractal. According to our knowledge, in the case of financial data these
anomalous fractal characterictics are observed for the first time.
It is worth noting that such singular behaviour of both the distributions and the multifractal spectra indicates that dynamics of the considered processes goes beyond the standard multifractal formalism and should rather be approached within the framework of broad multifractals, proposed by Mandelbrot. |