Scaling and correlation in financial time series

We discuss the results of three recent phenomenological studies focussed on understanding the distinctive statistical properties of financial time series - (i) The probability distribution of stock price fluctuations: Stock price fluctuations occur in all magnitudes, in analogy to earthquakes from tiny fluctuations to very drastic events, such as the crash of 19 October 1987, sometimes referred to as "Black Monday". The distribution of price fluctuations decays with a power-law tail well outside the Levy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. In addition, this distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. (ii) Correlations in financial time series: While price fluctuations themselves have rapidly decaying correlations, the magnitude of fluctuations measured by either the absolute value or the square of the price fluctuations has correlations that decay as a power-law, persisting for several months. (iii) Volatility and trading activity: We quantify the relation between trading activity measured by the number of transactions N-Deltat - and the price change G(Deltat) for a given stock, over a time interval [t, t + Deltat]. We find that N-Deltat displays long-range power-law correlations in time, which leads to the interpretation that the long-range correlations previously found for \G(Deltat)\ are connected to those of N-Deltat. (C) 2000 Elsevier Science B.V. All rights reserved.