# Covariance Matrix of Correlated Random Variable

Suppose I know or have estimated the covariance matrix for one random variable (for example an asset) and have: $$\begin{bmatrix} <\text{spot, spot}> & <\text{atmv, spot}> \\ <\text{spot, atmv}> & <\text{atmv, atmv}> \end{bmatrix}$$

where atmv is the at the money volatility (or can just be realized). Suppose then I know the beta or correlation of this asset A to asset B. How would you derive the covariance matrix B as a function of beta and covariance matrix for A?

• you say the covariance matrix for one random variable but you need to view atmv as another random variable. So, you actually have three random variables. asset A, asset B and atmv. So, when you say covariance matrix of B, what is the random variable that B is correlated with ? Commented Jul 14, 2023 at 18:41

$$return(b)=cor_{A,B}*return(A)+noise_1$$ so

$$return(b)=cor_{A,B}*(beta_{A,vol}*atmvol+noise_2)+noise_1$$

so the correlation between b and atm vol depends on the correlation assumption between atmvol and noise_1. So there's no right answer if you have to go through A. Why don't you just directly correlate atmvol and asset B?

Note $$cor_{A,B}$$ is the correlation multiplied by the ratio of standard deviations of A and B (usual relationship between regression beta and correlation coefficient)

• Arshdeep: I think you meant to write $\beta_{A,B}$ whereever you have written $cor_{A,B}$ above. Still, it's an insightful answer. Thanks. Commented Jul 15, 2023 at 1:33
• @markleeds yes I've clarified the notation in my comment, I was just being lazy and thought the point would be apparent either way. Commented Jul 15, 2023 at 1:43
• No problem. But, in the first two formulae, it should be changed also. Commented Jul 15, 2023 at 1:53
• Even if the notation has been clarified, denoting beta as cor is unnecessarily confusing. Commented Jul 15, 2023 at 6:11
• Whats the issue with simply assuming E(noise_1) = E(noise_2) = 0?
– roz
Commented Jul 17, 2023 at 15:32