Towards A Multifractal Paradigm Of Stochastic Volatility?

This paper examines the behavior of volatility and trading volume in an extended Mixture-of-Distribution Hypothesis framework. According to this Hypothesis, both volatility and volume are subordinated to the same latent, stochastic variable: the information flow. One way to enlarge this modeling in order to capture long range dependecies observed in these two variables is to use multi-component volatility models. However, traditional multi-component volatility models seem no longer enough to describe the more complex behavior of the volatility. This is why we introduce in this class of models, a stochastic cascade model of volatility, inherited from the multifractal paradigm. In the empirical section we provide some evidence in favor of this approach. Both volatility and volume are fractionally integrated processes with similar hyperbolic decay rates, but the long range dependence of their power transforms exhibits significant differences. Moreover, we emphasize the existence of lagged cross-correlations between the volatilities at different sampling frequencies, first pointed out by Mülleret al.[59], which historically opened the door to a (stochastic) cascade scheme of the volatility in the literature. A multiscaling analysis of the structure functions (generalized volatilities) confirms this result. The empirical results are provided with data on the French stock Alcatel.