Multifractality in Time Series

Abstract

We apply the concepts of multifractal physics to financial time series in order to characterize the onset of crash for the Standard & Poor's 500 stock index x(t). It is found that within the framework of multifractality, the "analogous" specific heat of the S&P500 discrete price index displays a shoulder to the right of the main peak for low values of time lags. On decreasing T, the presence of the shoulder is a consequence of the peaked, temporal x(t+T)-x(t) fluctuations in this regime. For large time lags (T>80), we have found that C_{q} displays typical features of a classical phase transition at a critical point. An example of such dynamic phase transition in a simple economic model system, based on a mapping with multifractality phenomena in random multiplicative processes, is also presented by applying former results obtained with a continuous probability theory for describing scaling measures.