# What is the implied volatility skew?

I often hear people talking about the skew of the volatility surface, model, etc... but it appears to me that a clear standard definition is not unanimously in place among practitioners.

So here is my question: Does anyone has a clear and unifying definition that can be stated in mathematical terms of what skew is in the context of risk neutral pricing? If not, do you have a set of definitions of the skew with respect to a specific context?

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Given that I found an entire paper in the JOD devoted to this question, I think it is arguably not "soft," so I removed that tag. Skew is also a technical term, not "jargon", so I removed that tag, too. –  Tal Fishman Aug 25 '11 at 16:54

Scott Mixon argues in What Does Implied Volatility Skew Measure that among all measures of implied volatility skew, the (25 delta put volatility - 25 delta call volatility)/50 delta volatility is the most descriptive and least redundant (volatility is Black-Scholes implied volatility). His paper, recently published in the Journal of Derivatives, gives a number of both theoretical and empirical arguments in favor of this measure. He distinguishes between "skew", which is a measure of the slope of the implied volatility curve for a given expiration date, and "skewness", which is the skewness of an option implied, risk neutral probability distribution. To calculate the latter, one needs to have a theoretical framework or model, whereas the former is easily observable from options prices.

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Skew is indeed a widely used word and can represent one of the following:

• Skew(ness) - 3rd standardized moment that represents assymetry of the distribution (olaker metioned it his answer).

• (Volatility) skew - is observable property of implied volatility surface that can be seen on the market after the 1987 crash. It shows that OTM puts (high demand) are usually have higher price (for the same expiry) than OTM calls (high supply to buy protective puts).

In the first assumption you can think about IV surface = term structure (IV changes over time) + volatility skew (IV changes with strikes)

• Skew can also represent term in volatility model that adds adjustment to represent volatility skew which itself is a subject of proper calibration. For example see If Skew Fits article where local volatility model has the following form:

$${\sigma_t} = \sigma_{atm}+\sigma_{skew}+\sigma_{kurt}.$$

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"It shows that OTM puts (high demand) are usually have higher price (for the same expiry) than OTM calls (high supply to buy protective puts)." -- this is incorrect both factually and grammatically. –  quant_dev Feb 9 '11 at 16:07

First we must define what we mean by implied volatility. Let $c_{BS}(t,S(t),K,T;\sigma)$ denote the price of the call option with strike price $K$ and maturity $T$ in the Black-Scholes model with the volatility $\sigma$ (emphasized in the argument). Furthermore, let $c_{MA}(t,S(t),K,T;\sigma)$ denote the corresponding price on the market.

The volatility $\sigma_{imp}$ is defined by the specific volatility for which $$c_{BS}(t,S(t),K,T;\sigma) = c_{MA}(t,S(t),K,T)$$ for some fixed $t$.

This implied volatility is very much dependent on the strike price $K$ (which is quite intuitive). It is a well known phenomenon on the market that the maps $K \longmapsto \sigma_{imp}(K)$ have a so called smiley or skewed shape.

By smiley shape we mean that $\sigma_{imp}(K)$ has a convex shape with high values for both small and large values of $K$ and a minimum around the forward price $Se^{rt}$. This smiley phenomenon is very common in connection with options on currencies, but not so common for options on stocks.

In connection to stocks we often observe a so called skewed shape of the maps $K \longmapsto \sigma_{imp}(K)$. Meaning that the implied volatility is high for low strikes and not as high for high strikes. It doesn't even have to have a minimum around the forward price.

This is an example of a so called smiley shape:

This is an example of a so called skewed shape:

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The skew of a distribution is a measure of its assymetry. Let $X_n$ be a discrete process (say, of daily returns) with mean 0 and noncentered volatility $\sigma$. Then the noncentered skew is defined as $$\frac{1}{n}\sum_{k=1}^{n}\frac{X_k^3}{\sigma}.$$ It will be positive is $X_n$ and $X_n^2$ are positively correlated and negative if they are negatively correlated. Roughly speaking, the skew expresses the correlation between the move of a random process and its volatility.

The volatility skew is the slope of the graph of implied volatility versus strike. A negative skew corresponds to a downward slope which is observed in equity options.

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I'm curious - is it possible to estimate the correlation between the equity and its IV, if all you know is the IV skew? –  Contango Jun 6 '11 at 0:43

Implied volatility skew is simply collection of implied volatilities on the same underlying instrument for a given expiration. Term "implied volatility skew" is only loosely connected to statistical definition of skewness. Implied volatility surface is the collection of implied volatilities on the same underlying for several expirations.

If BS formula were to be true, you would expect implied volatility to be a constant across strikes and expirations. This is not true, and deviation (from constant) is referred to as skew.

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