# Does the Shannon entropy of stock returns change over time?

Shannon entropy, $$H(X) = -\sum_{i=1}^n p(x) \ln p(x)$$ is a probabilistic measure of randomness or disorder within a random variable's probability distribution or histogram.

If we take rolling window segments, or snapshots, of a full-sample time series of stock returns $$X$$, can we expect the Shannon entropy of these windows to consistently increase or decrease as $$t\rightarrow \infty$$? i.e. each window has its own pdf.

or would their entropy merely be a function of the volatility (clustering) in each window, given that the spread of a distribution (volatility) and its randomness (entropy) are inextricably linked?

In thermodynamics, time entropy naturally grows with time, so mustn't there be a connection between the Shannon entropy and time entropy of a stock's returns?

How about the entropy of stock returns using expanding windows instead?

• Just note, in thermodynamics, the $\Delta S \ge 0$, i.e. it grows or stay same. The entropy is same in case of reversible processes. I would expect something similar for stock. For example, for well established companies, $\Delta S \approx 0$. However, as uncertainty on market increases or decreases, also entropy would increases or decreases. Very nice question! Commented Oct 19, 2020 at 4:54
• This can be of interest for you: quant.stackexchange.com/questions/879/… Commented Oct 19, 2020 at 4:57
• Last comment to entropy in thermodynamics, it can also decrease but you have to do some work. The decrease in entropy has to be compensanted by its increase somewhere else. Commented Oct 19, 2020 at 5:01
• This is closely related to whether stock returns are stationary. If they are, the unconditional distribution is not a function of time. Then the true entropy, defined using unconditional distribution should be constant.
– fes
Commented Oct 19, 2020 at 5:42
• @MartinVesely what is $\Delta S$ in thermodynamics? Commented Oct 19, 2020 at 14:49