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Reading Implied Volatility Skew and Risk-Neutral Densities

Article Quant Q&A · Author: Michael
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Summary

An implied volatility surface reflects option prices that vary by strike and maturity, unlike the constant volatility assumption in the basic Black–Scholes model. Looking at one maturity at a time, a steep downside wing means out-of-the-money puts are relatively expensive. This can indicate demand for downside insurance or concern about falling prices, but the shape alone cannot establish what market participants expect the underlying to do.

The discussion also explains that the second strike derivative of European call or put prices can be used to infer a risk-neutral density. That density is derived from option prices under risk-neutral pricing and is not automatically a real-world forecast. Converting it into a real-world distribution requires additional assumptions or adjustments. The answer mentions use of this technique by central banks but provides no empirical comparison of surfaces or procedure for making that conversion.

Key ideas

  • Implied volatility varies across option strikes and maturities because market prices depart from the constant-volatility model assumption.
  • A steep downside skew is consistent with expensive demand for protection against losses.
  • Option prices can be differentiated twice with respect to strike to infer a risk-neutral density.
  • A risk-neutral density should not be read directly as a real-world probability forecast.

Tags

OptionsVolatilityDerivatives pricingStatistics
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# Intuition behind Implied Volatility Surface


# Intuition behind Implied Volatility Surface












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?

## Answer by Kevin (score 1)

https://quant.stackexchange.com/a/46623

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).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.