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Currency pairs: A look through the fractal dimension
The key to the existence of predictability in financial time series is the presence of characteristics that are both persistent and distinct from those of a random walk. It is not enough for a time series to be either different from a random walk or constant in character as neither would allow for the systematic extraction of profit from the market. If a time series is constant in character but equal to a random walk then no useful predictions can be made while if the character of the series is different from that of a random walk but oscillates widely the possibility to extract any benefit is also voided. Through this article we will use the fractal dimension to explore how the character of 16 Forex financial series changes through time as well as how we can infer from this information which series are the most suitable for the finding of historically profitable algorithmic trading strategies.
The first problem we face when attempting to evaluate deviations from a random walk is how to measure such deviations. The family of measurements related to Chaos theory, such as the Lyapunov exponent, Hurst exponent and fractal dimension are particularly useful for this purpose as not only do they provide a measurement of how a series deviates from a random walk but they also hint at the qualitative properties that change within the series when such deviations happen. On this article we will be using the fractal dimension for our study as its calculation is computationally cheap and the deviations are associated with the fractality within a series which is related with series complexity and the existence of patterns which are immediately relevant to potentially profitable trading.
Figure 1
To evaluate how the fractal dimension changes within a time series we can divide this data into slices of different sizes and then examine their distribution and its properties. For this article we take the one hour timeframe close data for each Forex pair from 19862016 and perform this procedure using different slice sizes.
Since efficient financial time series returns are not expected to follow normally distributed random walks we cannot use the traditionally expected value of 1.5 for the fractal dimension for this case as a way to compare against an efficient series. In order to derive adequate fractal dimension benchmarks for efficient series we calculate the average fractal dimension for slices of the chosen size across 100 random time series created using bootstraping with replacement. Since this procedure is computationally expensive it was performed only on a slice size of 2000 bars as shown in Figure 1. In this figure the average fractal dimension value for efficient series is showed in red while the mean value for the real series is shown in blue, along with the distribution of fractal dimension values created from all the slices within the data.