This paper presents the ﬁrst empirical investigation of the Multifractal Model of Asset Returns (“MMAR”). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997), is an alternative to ARCH-type representations for modelling temporal heterogeneity in ﬁnancial returns. Typically, researchers introduce temporal heterogeneity through time-varying conditional second moments in a discrete time framework. Multifractality introduces a new source of heterogeneity through time-varying local regularity in the price path. The concept of local Hölder exponent describes local regularity. Multifractal processes bridge the gap between locally Gaussian (Itô) diﬀusions and jump-diﬀusions by allowing a multiplicity of Hölder exponents. This paper investigates multifractality in Deutschemark/US Dollar currency exchange rates. After ﬁnding evidence of multifractal scaling, we show how to estimate the multifractal spectrum via the Legendre transform. The scaling laws found in the data are replicated in simulations. Further simulation experiments test whether alternative representations, such as FIGARCH, are likely to replicate the multifractal signature of the Deutschemark/US Dollar data. On the basis of this evidence, the MMAR hypothesis appears more likely. Overall, the MMAR is quite successful in uncovering a previously unseen empirical regularity. Additionally, the model generates realistic sample paths, and opens the door to new theoretical and applied approaches to asset pricing and risk valuation. We conclude by advocating further empirical study of multifractality in ﬁnancial data, along with more intensive study of estimation techniques and inference procedures.
Fisher, Adlai; Calvet, Laurent; and Mandelbrot, Benoit, "Multifractality of Deutschemark/US Dollar Exchange Rates" (1997). Cowles Foundation Discussion Papers. 1414.