I've been thinking about this question for some time. In addition to @Brian B's answer, giving here another route to constructing the skew for asset $Y$ given the skew for another asset $X$, where $X_t$ is a positive price process.
I'll state the assumptions first:
- $d\ln (Y_t/Y_0) = \beta d\ln (X_t/X_0) + d\ln (Z_t/Z_0)$, and $d \ln X_t\, d\ln Z_t = 0$
- $\beta$ is constant or at most deterministic and can be regarded as the regression coefficient of logreturns
- $Z_t$ is also a positive process that drives the error term $d\ln Z_t$ of the regression and has a known distribution $q(z)$
From assumptions (1) and (2) it follows that $$ \frac{Y_T}{Y_t} = \left(\frac{X_T}{X_t}\right)^\beta \frac{Z_T}{Z_t} $$
The price of a vanilla option on $Y$ is then $$ E_t \left[ \left(Y_T - K\right)_+ \right] = E_t \left[ \left(\frac{Y_t}{X_t^\beta Z_t}X_T^\beta Z_T - K\right)_+ \right] $$ Since by assumption (1) $Z$ is independent of $X$, and by assumption (3) the distribution of $Z$ is known, we can write $$ E_t \left[ \left(\frac{Y_t}{X_t^\beta Z_t}X_T^\beta Z_T - K\right)_+ \right] = \int_0^\infty E_t \left[ \left(\frac{Y_t}{X_t^\beta Z_t}z X_T^\beta - K\right)_+ \right] q(z) dz $$ The expectation in the integrand is a claim on $X^\beta_T$ and can be synthesised using plain vanilla options on $X_T$ by making use of the Carr and Madan formula. Hence, since $q(z)$ is assumed to be known (for example the ubiquitous lognormal distribution), you can calculate options on $Y_T$ and infer the corresponding implied volatilities.
Remarks:
- Typically $\beta$ and $q(z)$ are inferred from historical data since the regression and error is based on historical data. For pricing purposesyou'd therefore have to make an educated guess about the risk-neutral values
- Other than the assumptions, no approximations are used, i.e. the computation of the skew of $Y$ is `exact'.
To the best of my knowledge the approach outlined above I have not seen in prior literature about this topic and hence original (whatever that means). I may at some point post a note about this on ssrn.com with some numerical examples. However, I am curious, if you decide to use this, what results you obtain.
Hope this helps.