# Can the concept of entropy be applied to financial time series?

I am not familiar with the concept of entropy for time series. I am looking for good reference papers and examples of use.

As a good starting point read this recent paper by Jing Chen:
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1734526

For a special use of the entropy concept for forecasting the '87-crash read this paper:
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=959547
(Although I tried to contact the authors to get the data to reproduce their findings, which they didn't send, it is still an enlightening read)

For a more popular exposition of the use of entropy in money management (key word 'Kelly formula') you should read this intelligent page turner by Poundstone: Fortunes Formula

EDIT: Quite an interesting paper is this one where Black-Scholes is derived through the use of concepts of relative entropy: http://www.mdpi.com/1099-4300/2/2/70/

• just went through your first paper. Very good, thank you for the link Apr 1, 2011 at 6:38

Google for granger causality and its general version, transfer entropy, for a measure of whether a time series has a causal relationship with another (measured by calculating how much the conditional entropy of a time series decreases if we know another one, conditioned on everything else we know).

I have applied the concept of entropy and more specifically conditional entropy to spreading (ie, as a pricing model to get a sense for value) & execution decisions. It's good for everytime you're facing a problem of the sort, given X what is the probability density function of Y.

Also, the concept of mutual information which evaluates mutual dependence between two (random) variables can be useful in many applications. Again spreading comes to mind, risk management, etc

No papers on hand, but as usual the wiki is pretty good one going

I tried to apply an entropy to time series of daily P/L on equity portfolios (developed markets).

I found out that there is a strong correlation among entropy and other risk measures such as standard deviation, VaR and CVaR. Hence entropy is a good risk measure.

Additionally, entropy calculation does not rely on any assumption about underlying data distribution thus it is suitable for distribution with fat-tails as these do not have second (standard deviation) and sometimes even first momentum (average).

Moreover, and I think this my crucial finding, entropy starts to grow earlier than other mentioned measures, soit can be employed as an early warning indicator.

For my calculation I used Kozachenko-Leonenko estimator (see link {10} on Wiki and here you can find its implementation in MatLab).

Regarding sources, I can recommend this one: Entropy: A new measure of stock market volatility?

• seeing that you calculated entropy on daily returns, does the Kozachenko-Leonenko estimator become inaccurate for small-sample (monthly) datasets? and are there any other reasons why someone would bother looking at entropy of stock returns when it is merely just correlated (a reflection of) any other risk measure? Oct 20, 2020 at 10:50
• @develarist: I would not say that KL estimator is not accurate, however, I think that daily returns can provide me with better insight, monthly returns can be too "averaged". On the bothering about entropy - the problem is that some currently used risk measures, for example standard deviation, are not well defined for all distribution (for example Cauchy distribution has infinite standard deviation but has finite entropy). Oct 21, 2020 at 7:33