Why Implied Volatility Skew Cannot Identify Equity–Volatility Correlation

Summary

The document asks whether an equity’s implied volatility skew can reveal the probability of an equity move conditional on a change in implied volatility. It defines skew as the difference between implied volatilities at two option deltas and explains why that cross-sectional option measure does not, by itself, identify the joint dynamics of price and volatility.

The answer notes that volatility skew corresponds to a terminal probability distribution, which does not specify the path dynamics linking equity prices and implied volatility. To estimate a conditional relationship, an analyst must add assumptions through a model. A calibrated framework such as Heston can then be used to calculate a model-dependent quantity, such as the most likely equity price at a given implied volatility. Such an estimate depends on the chosen model and calibration; skew alone does not determine it.

Key ideas

• An implied volatility skew alone cannot determine equity–volatility correlation.
• Skew informs the risk-neutral terminal price distribution but does not specify price and volatility dynamics.
• Conditional price estimates require additional modeling assumptions.
• A calibrated stochastic volatility model can produce model-dependent conditional estimates.

Tags

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


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












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 by Brian B (score 3, accepted)

https://quant.stackexchange.com/a/1670

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 How to derive the implied probability distribution from B-S volatilities? 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.

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