Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, generic interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in reference [23], naturally emerge. We then propose a simple solvable stochastic volatility model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover. its extension to a multivariate context. in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided.