Anomalous scaling of stock price dynamics within ARCH-models

We show that autoregressive-conditional-heteroskedasticity (ARCH) models ca n encompass the observed anomalous scaling properties of stock price dynami cs remarkably well. We find that with a suitable choice of parameters, simp le ARCH models can reproduce the non-standard scaling behavior of the centr al part of the probability distribution functions of stock prices at differ ent time horizons, as empirically found for the Standard & Poors 500 (S&P 5 00) index data, but fail to reproduce the shape of the S&P 500 distribution , in particular at the smallest time horizon (1 min). A linear version of A RCH processes, denoted here as LARCH models, still preserving the anomalies observed, permits to fit the 1 min S&P 500 distribution more accurately.