Turbulence in financial markets: the surprising explanatory power of simple cascade models

Price changes in financial markets have been found to share many of the features characterizing turbulent flows. In particular, a number of recent contributions have highlighted that time series from both stock and foreign exchange markets possess multifractal statistics, i.e. the scaling behaviour of absolute moments is described by a convex function. These findings have stimulated the application of certain cascade models from statistical physics to financial data. Extant work in this area has so far been confined to parameter estimation and visual comparison of empirical and theoretical scaling properties. The lack of rigorous statistical measures of goodness of fit in the literature on turbulence has, however, impeded a comparison of these new models with standard approaches in empirical finance. Here we try to fill this gap and provide a first assessment of two elementary cascade models based on elementary goodness-of-fit criteria. As it turns out, these relatively simple one-parameter models are not only capable of accommodating the multiscaling behaviour of price changes, but also provide a perplexingly good fit of the unconditional distribution of the data. In a double-blind test, we would, in fact, be unable to reject identity of the data-generating processes underlying empirical records and simulated data from stochastic cascades.