On multi-scaling in volatility processes

Among the statistical features of financial volatility processes, the most challenging for statistical inference purposes are non-gaussianity and non-stationarity. In this study I present results from experiments aimed to empirically modeling return generating processes in which the underlying volatility dynamics are subject to changes in scaling regimes. This means that the usually observed power law behavior of self-similar and long range dependent processes might be subject to dynamics which vary with the resolution levels. The observed information are encompassed in a model based on scale-dependent information packets that reflect heterogeneous activity in the financial market. I show how the volatility structure of a stock return index can be cast in a latent variable form and suggest how this model can account for regime switching when risk and returns are explicitly linked. To this end, I find that wavelets and multi-resolution analysis are very useful in modeling volatility and in interpreting its structure, since they deliver both sparse representations and coherent decompositions, and show the multi-scaling features inherent to the data set