A generalised form of permutation entropy for observing stock market behaviour

We propose and investigate a generalised form of permutation entropy and its application to time-dependent financial data, in particular that of stock market behaviour. We survey the initial literature on the historical formulation of Shannon entropy and its relevance in financial markets, as well as on permutation entropy. Our proposal here considers a generalised permutation entropy to present a unified framework specifically within permutation entropy via the power sum of probabilities. We call this the power-sum permutation entropy (PSPE) measure. This PSPE measure is positioned within a temporal framework, and a rolling-window approach is outlined. We illustrate our results on key stock market indicators, namely the Johannesburg Stock Exchange’s (JSE’s) All Share and Top 40 indices, as well as the Standard and Poor’s (S&P) 500 index, for a period spanning 2006 through 2025, and reflect on how the PSPE measure could be used to observe the effect of key events within the financial market context.