Variance and Covariance Risk Premia Across Physical and Risk-Neutral Measures

Summary

The document considers whether a return variance-covariance matrix should be identical under the physical probability measure and the risk-neutral measure. Its answer emphasizes that equality is not generally supported by empirical evidence. In particular, it points to a variance risk premium, meaning that variance inferred under the risk-neutral measure can differ from variance observed under the physical measure.

For covariances, the response says the evidence is less extensive because options are not available for every stock pair. It nevertheless cites research documenting an aggregate market correlation risk premium. These observations suggest that assuming identical matrices should be treated as a modeling choice rather than a universal empirical fact. The excerpt offers references to prior studies but does not explain their methods, report numerical estimates, or derive conditions under which the matrices might coincide. Its discussion is therefore a concise empirical caution, not a complete theoretical treatment of measure changes or covariance estimation.

Key ideas

• The response says variance is not empirically identical under physical and risk-neutral measures.
• A variance risk premium is cited as evidence of a difference between the measures.
• Evidence on individual asset-pair covariance differences is described as limited by option availability.
• An aggregate market correlation risk premium is cited as evidence related to covariance risk.
• The excerpt provides references but no derivation or numerical results.

Tags

# Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$


# Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$












I'd like to understand why $\Sigma$ is the same under both measures $\mathbb{P}$ and $\mathbb{Q}$. Is it an assumption or a general fact based on theoretical concepts?

## Answer by phdstudent (score 4, accepted)

https://quant.stackexchange.com/a/55972

Just to expand on Alex answer.

Empirically it is simply not true. Focusing on the diagonal of the variance-covariance matrix, we know that there is a large variance risk premium. Take a look at table 3 from Carr and Wu (2009).

Regarding covariances we do not have much evidence, because there are no options on every single pair of stocks. However, we do know that there is an overall market correlation risk premium (Driessen, Maenhout and Vilkov (2008))

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.