Risk-Neutral CAPM Beta from Option-Implied Volatility and Correlation

Summary

The document compares the standard CAPM beta, built from physical-measure volatilities and correlations, with a risk-neutral beta that uses option-implied quantities. It describes a paper by Buss and Vilkov that reports the risk-neutral version better explains the cross-section of returns, then asks what model could produce that expression.

A response interprets the beta as the product of the risk-neutral correlation between an asset and the market and the ratio of their risk-neutral volatilities. It suggests implied variances may be obtained through a model, while implied correlations are more involved, and gives a proposed relationship between physical and risk-neutral correlations. The discussion is tentative: the respondent says they lack a clear intuition for implied correlation, and the excerpt supplies neither a derivation nor empirical details supporting the stated result. It therefore frames a modeling question rather than providing a complete risk-neutral CAPM construction.

Key ideas

• Risk-neutral beta replaces physical-measure volatility and correlation inputs with option-implied counterparts.
• The cited paper reports that risk-neutral beta better explains cross-sectional returns.
• Beta can be expressed using asset-market correlation and the ratio of their implied volatilities.
• The response suggests implied correlation is more difficult to model than implied variance.
• The excerpt leaves the interpretation and derivation of implied correlation unresolved.

Tags

# Risk-Neutral CAPM


# Risk-Neutral CAPM












In the paper Measuring Equity Risk with Option-implied Correlations, Buss and Vilkov replace the standard CAPM beta:

$$ \beta_{iM,t}^P=\frac{\sigma_{i,t}^P\sum_{j=1}^N w_j \sigma_{j,t}^P\rho_{ij,t}^P}{(\sigma_{M,t}^P)^2} $$

With a risk-neutral beta:

$$ \beta_{iM,t}^Q=\frac{\sigma_{i,t}^Q\sum_{j=1}^N w_j \sigma_{j,t}^Q\rho_{ij,t}^Q}{(\sigma_{M,t}^Q)^2} $$

And show that the later works better in explaining the cross-section of returns. My question is whether there is any simple model that would deliver the second expression. There are several models that deliver the formula for beta under $P$, but I am not sure if there is any that could deliver the one under $Q$.

## Answer by David Addison (score 1)

https://quant.stackexchange.com/a/32728

I do not have a good intuition about the meaning of implied correlation. Is this supposed to be correlation between some asset, $i$, and the index, $M$? Perhaps , maybe one way to think of risk-neutral beta is as the "correlated co-implied volatilities" of $M$ and $i$:

$Beta^Q_{M,i} = \rho^Q_{M,i} * \frac{\sigma^Q_{i}}{\sigma^Q_{M}}$

Getting the implied variances should be easy, a-la some model. Getting the implied correlations seems to a little more complex. The paper you reference defines implied correlation as:

$\rho_{ij_t}^Q = \rho_{ij,t}^P - a(1-rho_{i,j,t}^P)$

where $a$ appears to be some factor which is bounded between 0 and -1.

That's all I got. I wish I had a better intuition of what implied correlation was attempting to measure.

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.