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 ser ies - (i) The probability distribution of stock price fluctuations: Stock p rice fluctuations occur in all magnitudes, in analogy to earthquakes from t iny fluctuations to very drastic events, such as the crash of 19 October 19 87, sometimes referred to as "Black Monday". The distribution of price fluc tuations decays with a power-law tail well outside the Levy stable regime a nd describes fluctuations that differ by as much as 8 orders of magnitude. In addition, this distribution preserves its functional form for fluctuatio ns on time scales that differ by 3 orders of magnitude, from 1 min up to ap proximately 10 days. (ii) Correlations in financial time series: While pric e 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 inte rval [t, t + Deltat]. We find that N-Deltat displays long-range power-law c orrelations in time, which leads to the interpretation that the long-range correlations previously found for \G(Deltat)\ are connected to those of N-D eltat. (C) 2000 Elsevier Science B.V. All rights reserved.