On the chaos in the foreign exchange rates and cryptocurrencies

Abstract

The behavior of the foreign exchange and cryptocurrency markets was studied from the perspective of the theory of dynamical systems. Using the phase space reconstruction procedure under the validity of Takens' theorem (1981). The presence of serial dependence was investigated through the BDS test, the property of sensitivity to initial conditions through the Lyapunov maximum exponent and the distinction between deterministic and stochastic signals observing the behavior of the E2(d) function in Cao's method (1997). Evaluating 17 exchange rate log-return series, evidence of serial dependence, possibly non-linear, was found in 11 of them. As for sensitivity to initial conditions, no series has shown conclusive results on such a property. All series presented evidence that they follow processes of a random nature and non-Gaussian increments, in the same way as the cryptocurrency log-return series. Of the 10 series of cryptocurrencies, the IID hypothesis was rejected for 8 of them, and none presented a conclusive result regarding a positive Lyapunov exponent. As a conclusion, no consistent characteristics of chaotic dynamics were found for the foreign exchange and digital currency markets in the analyzed period.