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_Δt - and the price change G_Δt for a given stock, over a time interval [t,t+Δt]. We find that N_Δt displays long-range power-law correlations in time, which leads to the interpretation that the long-range correlations previously found for |G_Δt| are connected to those of N_Δt.