Information measure for financial time series: Quantifying short-term market heterogeneity

A well-interpretable measure of information has been recently proposed based on a partition obtained by intersecting a random sequence with its moving average. The partition yields disjoint sets of the sequence, which are then ranked according to their size to form a probability distribution function and finally fed in the expression of the Shannon entropy. In this work, such entropy measure is implemented on the time series of prices and volatilities of six financial markets. The analysis has been performed, on tick-by-tick data sampled every minute for six years of data from 1999 to 2004, for a broad range of moving average windows and volatility horizons. The study shows that the entropy of the volatility series depends on the individual market, while the entropy of the price series is practically invariant for the six markets. Finally, a cumulative information measure – the Market Heterogeneity Index – derived from the integral of the entropy measure, is introduced for obtaining the weights of an Efficient Portfolio. A comparison with the weights obtained by using the Sharpe ratio – a traditional risk diversity measure – is also reported.