# 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 $Q$.

• A good question for the authors... – Alex C Feb 25 '17 at 14:42
• What is $\tau$ in the second expression? – Kiwiakos Feb 25 '17 at 15:36
• Typo. :) Thanks. I have corrected it. – phdstudent Feb 25 '17 at 15:36
• Is tu in you risk-neutral beta also typo? – Gordon Feb 27 '17 at 14:12
• Yes. It ended up there after correcting the first typo. Thanks again. It is corrected. – phdstudent Feb 27 '17 at 14:24

## 1 Answer

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.

• That I already know. It is just a parametric modelling under mild assumptions. Not at all what the question asks. – phdstudent Feb 28 '17 at 8:41