Uses of Entropy in Financial Forecasting, Risk, and Execution
Summary
The document surveys ways entropy concepts can be applied to financial time series. Suggested uses include studying market behavior and crash forecasting, connecting entropy with the Kelly approach to money management, and deriving option-pricing results through relative entropy. It also describes transfer entropy as a way to assess directional dependence between series by measuring changes in conditional uncertainty, and mutual information as a measure of dependence that may inform spreads, risk management, and execution decisions.
One contributor reports using conditional entropy to model probability distributions relevant to pricing and execution. Another describes estimating entropy in daily equity-portfolio P&L with a nearest-neighbor estimator, reporting associations with standard deviation, VaR, and CVaR, and suggesting entropy might act as an earlier warning signal. These are personal observations rather than a systematic comparison: the discussion supplies no reproducible data or detailed validation, and one referenced crash study could not be replicated by its commenter. The examples therefore indicate possible applications, not established predictive performance.
Key ideas
- Entropy and conditional entropy can describe uncertainty in financial returns and decision problems.
- Transfer entropy measures how knowledge of one series changes uncertainty about another series.
- Mutual information can help characterize dependence relevant to spreads, execution, and risk management.
- A contributor reports that estimated portfolio P&L entropy tracked conventional risk measures and may rise sooner.
- The claims and applications are exploratory, with limited reproducibility and validation in the discussion.
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# Can the concept of entropy be applied to financial time series?
# 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.
## Answer by vonjd (score 17, accepted)
https://quant.stackexchange.com/a/880
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/
## Answer by Dan (score 9)
https://quant.stackexchange.com/a/2507
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).
## Answer by phil (score 7)
https://quant.stackexchange.com/a/882
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
## Answer by Martin Vesely (score 2)
https://quant.stackexchange.com/a/51817
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.