Inferring an Asset’s Volatility Covariance Structure from Beta
Summary
The document considers how to infer a second asset’s covariance structure from that of a first asset and a relationship between their returns. One answer models the second asset’s return as a constant beta times the first asset’s return plus a small independent error. Assuming the error is negligible, the second asset’s instantaneous variance scales with the square of beta, the return–variance covariance with its cube, and the variance-of-variance term with its fourth power.
The answer connects return–variance covariance to the short-dated at-the-money implied volatility slope, suggesting that slope can be inferred under the same assumptions. These results rely on a specific return model, independent noise, and a short-dated implied-volatility approximation; they are not a general transformation for any estimated covariance matrix. Another answer cautions that the result depends on how the noise relates to volatility and recommends directly estimating the relationship between the second asset and its volatility when possible.
Key ideas
- The proposed mapping assumes the second asset’s return is beta times the first asset’s return plus independent noise.
- Neglecting the noise makes instantaneous variance scale with beta squared.
- Under the same assumptions, return–variance covariance scales with beta cubed and variance-of-variance with beta to the fourth power.
- The return–variance covariance is related to the short-dated at-the-money implied volatility slope.
- The inferred structure is assumption-dependent, and direct estimation may be preferable.
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Full text
# Covariance Matrix of Correlated Random Variable
# 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?
## Answer by Frido (score 1)
https://quant.stackexchange.com/a/81338
Interesting question, a pity I didn't see it earlier. A bit late but maybe still useful for the OP and/or others:
First of all, when speaking about covariance it is more usual to speak of covariance of returns or price changes instead of covariance of prices.
With that in mind, let $X$ and $Y$ be the assets and assume that $$ \frac{dY}{Y} = \beta \frac{dX}{X} + \varepsilon $$ where $\varepsilon$ is a small error term, independent of $dX/X$, and $\beta$ is some constant.
It is a well-known result that the short-dated limit of the ATM IV is the spot / instantaneous volatility. So let's assume that the following covariance matrix is known: $$ E \left[ \left( \frac{dX}{X} \right)^2 \right] = v_X dt, \quad E \left[ \left( \frac{dX}{X} dv_X \right) \right] = \rho f(v_X) dt, \quad E \left[ \left(dv_X \right)^2 \right] = g(v_X) dt $$ where $\rho$ is the correlation between $X$ and it's instantaneous variance $v_X$, and $f(v_X), g(v_X)$ are deterministic functions of $v_X$.
It follows that $$ E \left[ \left( \frac{dY}{Y} \right)^2 \right] = v_Y dt = \beta^2 v_X dt + \varepsilon^2 \implies v_Y \approx \beta^2 v_X $$ We can then write $$ dv_Y \approx \beta^2 dv_X $$ Hence, $$ E \left[ \left( \frac{dY}{Y} dv_Y \right) \right] = \beta^3 E \left[ \left( \frac{dX}{X} dv_X \right) \right] = \beta^3 \rho f(v_X) dt $$ and $$ E \left[ \left(dv_Y \right)^2 \right] = \beta^4 E \left[ \left( dv_X \right)^2 \right] = \beta^4 g(v_X) dt $$ which is your desired covariance matrix for $Y$.
Remark: The quantities $E \left[ \left( \frac{dY}{Y} dv_Y \right) \right]$ and $E \left[ \left( \frac{dX}{X} dv_X \right) \right]$ are actually proportional to the respective short-dated ATM slope. Hence the ATM slope of $Y$ can be inferred from the ATM slope of $X$ under the assumption that $\varepsilon$ is indeed small.
## Answer by Arshdeep (score 0)
https://quant.stackexchange.com/a/76128
$return(b)=cor_{A,B}*return(A)+noise_1$ 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)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.