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I often hear people talking about the skew of the volatility surface, model, etc... but it appears to me that there isn't a clear standard definition unanimously used by practitioners.
So here is my question: Does anyone have 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?
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.
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)
See also Volatility Skew FAQ for brief explanation.
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:
$${\sigma_t} = \sigma_{atm}+\sigma_{skew}+\sigma_{kurt}.$$
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:
The skew of a distribution is a measure of its asymmetry. Let $X_n$ be a discrete process (say, of daily returns) with mean 0 and non-centered volatility $\sigma$. Then the non-centered skew is defined as $$\frac{1}{n}\sum_{k=1}^{n}\frac{X_k^3}{\sigma}.$$ It will be positive if $X_n$ 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.
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.
Old and golden question, and maybe a new perspective:
As the previous answers have pointed out, distinction needs to made between "skewness" and "skew". The former is the third moment of returns, and the latter is what volatility traders/portfolio managers usually associate with the difference between two implied volatilities straddling the ATM.
From a risk-neutral perspective, skewness can be traded by trading the difference between the log contract and the entropy contract.
Pricing and replicating skew, in a self-financing way is more difficult. The paper below proposes first of all to define (forward start) skew as the difference between the (forward start) volswap and its "dual", which is defined in the paper and can be regarded as the counterpart for volatility what the entropy contract is for variance. Then it is shown that this difference can indeed be identified with the difference between two implied volatilities, and a dynamic replication strategy is given.
