Weekly fractal dimensions of yen–dollar tick-by-tick exchange rates

In this study, we measure the fractal dimension of dollar/yen exchange rates for each week. The tick-by-tick data set of the bid/ask prices submitted by the brokers is used. For analyzing the data, we use two time scales: the physical time and the number of the submitted prices. The data are resampled to one-minute interval when the physical time is used. Using the number of the submitted prices, we can analyze all the data in the order of the records. We calculate the fractal dimensions on these two time scales by Higuchi's method. The main results are as follows. The weekly fractal dimensions vary with time. The fluctuations of the prices tend to have rather strong self-similarity when the market is volatile. The self-similarity is stronger on the physical time than on the number of prices. By using either time scale, the measured fractal dimensions exceed 1.5, which suggest anti-persistence. For any week, this anti-persistence is stronger on the number of the prices than on the physical time. However, the differences between these two dimensions of the same week little vary with time, since the variations of the two synchronize.