Multi-scale description and prediction of financial time series

Abstract

A new method is proposed that allows a reconstruction of time series based on higher order multi-scale statistics given by a hierarchical process. This method is able to model financial time series not only on a specific scale but for a range of scales. The method itself is based on the general n-scale joint probability density, which can be extracted directly from given data. It is shown how based on this n-scale statistics, general n-point probabilities can be estimated from which predictions can be achieved. Exemplary results are shown for the German DAX index. The ability to model correctly the behaviour of the original process for different scales simultaneously and in time is demonstrated. As a main result it is shown that this method is able to reproduce the known volatility cluster, although the model contains no explicit time dependence. Thus a new mechanism is shown how, in a stationary multi-scale process, volatility clustering can emerge.