Multi-Fractal Processes as Models for Financial Returns: A First Assessment

Abstract

Multi-fractal processes have been proposed as a new formalism for modeling the time series of returns in finance. The major attraction of these processes is their capability of generating various degrees of long-memory in different powers of returns - a feature that has been found to characterize virtually all financial prices. Furthermore, elementary variants of multi-fractal models are very parsimonious formalizations as they are essentially one-parameter families of stochastic processes. The aim of this paper is to provide a first assessment of the goodness-of-fit of this new class of models by applying them to four long time series from different financial markets (one exchange rate, two stock market indices and the price of gold). Our results are very encouraging in that the estimated models provide an astonishingly good fit to the unconditional distribution of the data and do even outperform estimates from a GARCH(1,1) specification. However, we also remark that a trade-off exists between goodness-of-fit for the unconditional distribution and the capability of the estimated processes to match the autocorrelation patterns of various moments.