Measuring Implied Volatility Skew by Its Effect on Option Prices

# When is vol smile skew considered high?


# When is vol smile skew considered high?












There are a lot of ways one could look at this question. I usually define skew to mean the slope of the vol smile in log moneyness space - ie the skew at strike $k_0=log(K_0/F)$ is:

$$\frac{d\sigma}{dk}|_{k=k_0}$$

However this slope may not be sufficient information on a standalone basis since that slope will be much less impressive if $\sigma(k_0)=50\%$ versus $\sigma(k_0)=10\%$. Therefore some people might normalize that slope by putting $\sigma(k_0)$ as a denominator to the slope in order to normalize the skew.

Other ideas might look at prices of put spreads or call spreads. For example, we could consider the 25 delta put strike and solve for the strike that make a costless 1xN put spread and consider how far the second leg is from the 25 delta put leg. Similar things can be done with a risk reversal. Take one strike and solve for the other leg that makes a costless structure in a risk reversal and compare how far each leg is from the forward - and similarly do that for 1XN costless risk reversal structures.

Another idea which is similar to the put/call spread approach is to compare the price of a digital option that uses the skew at $k_0$ to the Black-Scholes prices which does not use the skew. So for example if the Black-Scholes price of a digital put is 0.30, but the price with skew is 0.21, then the reason for that -0.09 difference is due to the skew correction factor which account for skew (as defined by the slope of the smile) times the vega. I actually like this measurement of skew - particularly if we compare the magnitude of the skew correction factor to the price of the Black-Scholes Digital price because it much more directly shows the effect of skew on prices which ultimately are what matter most!

So my untested proposal for when skew is large (on the put side) is to use Skew*Vega/N(-d2) as a metric - or in other words (where $C$ is a vanilla call - for vega the same as for a put):

$$\frac{\frac{d\sigma}{dK}\frac{\partial C}{\partial \sigma}}{N(-d2)}$$

The trouble with this methodology is that the vol smile that one fits can have numerical artifacts that can make the slope very different at the same strike given two different fitting algorithms that both fit the data more or less equally well - particularly near where the second derivative of the slope is highest. So perhaps some weighted average across different strikes with this kind of quantity is the right approach (i.e. some integral of this kind of quantity across all strikes).

I imagine the right answer is the one that can backtest the best for a skew strategy by comparing the historical values of these kinds of skew metrics - I built an engine that can backtest these kinds of ideas if I do a days worth of coding - was curious what others have found about what defines skew to be "high" or "low".

Apologies if this kind of question is a little too open ended for this forum. I am not sure there is truly a "right" answer here, but a discussion on skew is useful in my opinion!