Measuring skew

  1. Normalized or delta skew (ie dividing by ATM vol) Should be used when comparing skew by delta. For example 25d RR.
  1. Fixed skew
    1. If you use moneyness strikes (ie 90%-110% RR) then you should not divide by ATM vol and just use the diff. As vol increases, these strikes are closer together in standard dev space. If you divide by ATM vol (an increasing denominator) you will overstate the contraction in skew.
      The best measure of skew is one that is independent of the level of volatility. If this were not the case, then the measure would be partly based on volatility and partly on skew, which would make it more difficult to determine if skew was cheap or expensive.
      The most common method of measuring skew is to look at the difference in implied volatility between two strikes, for example 90%-110% skew or 90%-ATM skew. It is a common mistake to believe that strike skew should be divided by ATM volatility in order to take into account the fact that a 5pt difference is more significant for a stock with 20% volatility than 40% volatility. This ignores the fact that the strikes chosen (say 90%-110% for 20% volatility stocks) should also be wider for high volatility stocks (say 80%-120%, or two times wider, for 40% volatility stocks as the volatility is 2×20%). The difference in implied volatility should be taken between two strikes whose width between the strikes is proportional to the volatility (similar to taking the implied volatility of a fixed delta, eg, 25% delta). An approximation to this is to take the fixed strike skew, and multiply by volatility, as shown below. As the two effects cancel each other out, we can simply take a fixed strike skew without dividing by volatility.
      Difference in vol between 2 strikes = 90-110% Difference in vol between 2 strikes whose width increases with vol = 90-110% × ATM Skew = Difference in vol between 2 strikes whose width increases with vol/ATM vol
      • Skew = 90-110% × ATM/ATM
      • Skew = 90-110%
While delta skew is theoretically the best measure, in practice it is virtually identical to strike skew. As there is a R2 of 93% between delta skew and strike skew, we believe both are viable measures of skew (although strike skew is arguably more practical as it represents a more intuitive measure).