We investigate the general problem of how to model the kinematics of stock prices without considering the dynamical causes of motion. We propose a stochastic process with long-range correlated absolute returns. We find that the model is able to reproduce the experimentally observed clustering, power law memory, fat tails and multifractality of real financial time series. We find that the distribution of stock returns is approximated by a Gaussian with log-normally distributed local variance and shows excellent agreement with the behavior of the NYSE index for a range of time scales.
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