Return over volume statistics and the Moses effect in S&P 500 data

Abstract

Financial time series are usually analyzed by looking at the log-returns. This phenomenological approach is justified by the apparent multiplicativity of stochastic noise in the process. In fact, it hides the non-stationarity of the data. Here we shift the focus to the interplay of return and volume instead of log returns. We analyze statistics and causal relationships between the squared price and the volume increments of S&P 500 data. A new quantity, namely the ratio of price increments and the square root of the traded volume shows particular non-stationarity giving rise to a Hurst exponent of 0.65 by the so-called Moses effect.