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When looking at an implied volatility surface, are there some intuitive conclusions that one can draw from the shape? E.g. the steepness of the wings, the skew etc?

If one for example compares two implied volatility surfaces, are there certain intuitions one could get about the market opinions on the two underlyings, based on the shape?

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In general, the implied surface exists, because the market does not believe in the Black-Scholes equation, i.e. they use different volatilies for different strikes and maturities. What does this tell you? Consider a certain maturity. Then, you only have a volatility smile/skew. If you have a particular steep curve on the left, OTM options are very expensive, i.e. the market is willing to pay rather high premia for options that insure losses. Why does the market do that? We do not know with certainty but this may indicate that market participants anticipate declining stock prices (this does not however mean that stocks prices really will decline).

Perhaps related to your questions is the estimation of RND (risk-neutral density). It is well-known that $$ q(x) = e^{rT} \frac{\partial^2 C(K)}{\partial K^2}\bigg|_{K=x} = e^{rT} \frac{\partial^2 P(K)}{\partial K^2}\bigg|_{K=x}$$ Thus, we can infer a risk-neutral density from observed European option prices. However, we need to be careful and first transform a risk-neutral density into a real-world density. Then indeed, you can see what market participants really expect for the market. Policy makers at central banks use this tool. See also Chapter 16 in Taylor (2005).

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