Multi-Fractality in Foreign Currency Markets

Abstract

The standard hypothesis concerning the behavior of asset returns states that they follow a random walk in discrete time or a Brownian motion in continuous time. The Brownian motion process is characterized by a quantity, called the Hurst exponent, which is related to some fractal aspects of the process itself. For a standard Brownian motion (sBm) this exponent is equal to 0.5. Several empirical studies have shown the inadequacy of the sBm. To correct for this evidence some authors have conjectured that asset returns may be independently and identically Pareto-Levy stable (PLs) distributed, whereas others have asserted that asset returns may be identically - but not independently - fractional Brownian motion (fBm) distributed with Hurst exponents, in both cases, that differ from 0.5. In this paper we empirically explore such non-standard assumptions for both spot and (nearby) futures returns for five foreign currencies: the British Pound, the Canadian Dollar, the German Mark, the Swiss Franc, and the Japanese Yen. We assume that the Hurst exponent belongs to a suitable neighborhood of 0.5 that allows us to verify if the so-called Fractal Market Hypothesis (FMH) can be a reasonable generalization of the Efficient Market hypothesis. Furthermore, we also allow the Hurst exponent to vary over time which permits the generalization of the FMH into the MultiFractal Market Hypothesis (MFMH).