For equity options, does the implied vol change if the price of the underlying does?

Question

For example, consider S&P options.

My reasoning is rooted in the fact that VIX returns and S&P returns have a negative relationship, since VIX is a measure of S&P options' implied vol. Doesn't that imply that when the S&P goes down, the implied vol of S&P options is going up? So if you were to answer "what happens to the price of this option when the underlying increases 1%"... instead of just using the delta of the option, would it be intuitively more correct to then make an expected vega adjustment? If I a mistaken about something, please let me know.

Thanks!

Answer 1

Sometimes.

Two extreme models are assuming: a) ATM vol stays constant for a given moneyness (called sticky moneyness) or b) vol stays constant at fixed strikes (called sticky moneyness). A variation on a) is sticky delta. It's also possible to come up with models that are sort of a weighted average of these two extremes.

As Quantuple pointed out, work has been done by Derman to identify what the market is using; the answer is that it depends/varies over time. Googling on his name plus the terms above will get you a long way in finding more resources.