Is it possible to estimate the correlation between an equity and its IV, purely from its IV skew?

Question

If we know the options Implied Volatility (IV) skew for an equity, is it possible to calculate the probability of the equity moving, given a move in the IV?

We can define IV skew as the difference between IV at delta 0.25 compared to IV at delta 0.75.

Answer 1

The skew alone is not enough. You can see this by noting the one-to-one correspondence between volatility skew and terminal probability distributions, which is independent of price and volatility dynamics. (See my answer at QQplot of returns from implied vols (as opposed to historic vols) for a derivation of that dependence)

Now, if you choose a non-Black-Scholes model (such as the Heston model) and calibrate it, then you can use that model to compute, say, the maximum likelihood equity price given a certain level of implied volatility.