Anomalous scaling of stock price dynamics within ARCH-models

We show that autoregressive-conditional-heteroskedasticity (ARCH) models can encompass the observed anomalous scaling properties of stock price dynamics remarkably well. We find that with a suitable choice of parameters, simple ARCH models can reproduce the non-standard scaling behavior of the central part of the probability distribution functions of stock prices at different time horizons, as empirically found for the Standard & Poors 500 (S&P 500) 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 ARCH processes, denoted here as LARCH models, still preserving the anomalies observed, permits to fit the 1 min S&P 500 distribution more accurately. Received 12 October 2000 and Received in final form 5 February 2001