Gaussian Entropy and the Limits of Entropy-Implied Volatility

Summary

The document examines whether volatility derived from the entropy of Gaussian returns provides an independent measure. For a Gaussian distribution, differential entropy is an analytical function of its standard deviation, so solving that relationship for volatility simply recovers the volatility implied by the entropy value. The post questions whether this is circular when the entropy itself is calculated from sample volatility.

Its key lesson is that algebraic inversion does not create new information: if entropy was computed from volatility, the resulting entropy-implied volatility is not an independent estimate. The discussion does not provide an alternative entropy estimator, empirical comparison, or trading use case, so it leaves open whether entropy estimated through another method could be useful. It also presents an inversion formula that appears inconsistent with the stated Gaussian entropy equation, so the formula should be checked before use.

Key ideas

• For Gaussian returns, differential entropy is determined by the distribution’s volatility.
• Inverting that relationship recovers volatility when entropy was calculated from volatility.
• An algebraic transformation alone does not provide an independent volatility estimate.
• The document does not assess alternative entropy estimators or practical trading applications.

Tags

# Entropy-implied volatility requires itself to be calculated?


# Entropy-implied volatility requires itself to be calculated?












\begin{align} H &= \frac{1}{2} \ln (2\pi\sigma^2) + \frac{1}{2}\\ &= \frac{1}{2} \ln (2\pi e \sigma^2) \end{align} is the analytical solution for the entropy of a Gaussian random variable, such as a returns series, in the textbook Financial Machine Learning.

The author then re-works one of the above into a formula for entropy-implied volatility by isolating $\sigma$ by itself, and renaming it $\sigma_H$:

$$\sigma_{H} = \frac{e^{H} - \frac{1}{2}}{\sqrt{2\pi}}$$

but isn't this useless because, in order to calculate $H$, you need sample volatility $\sigma$ anyway? In other words, to calculate $\sigma_H$, you need $H$, which itself in turn requires $\sigma$, meaning $\sigma = \sigma_H$. It's fair enough that $H$ is a function of $\sigma$, but $\sigma_H$ is a function of $H$ as well as its own self, $\sigma$

Is the $\sigma_H$ formula just a small exercise in algebra that is circularly redundant given that it itself ($\sigma$) is required as an input?

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.