Appendix - Measuring Skew & Smile Derivatives-based Portfolio Solutions
MOMENTS DESCRIBE THE PROBABILITY DISTRIBUTION In order to explain skew and smile, we shall break down the probability distribution of log returns into moments. Moments can describe the probability distribution (The combination of all moments can perfectly explain any distribution as long as the distribution has a positive radius of convergence or is bounded). From the formula below we can see that the zero-th moment is 1 (as the sum of a probability distribution is 1, as the probability of all outcomes is 100%). The first moment is the expected value (ie, mean or forward) of the variable. The second, third and fourth moments are variance, skew and kurtosis, respectively. For moments of two or greater it is usual to look at central moments, or moments about the mean (we cannot for the first moment as the first central moment would be 0). We shall normalize the central moment by dividing it by σ ^{n} in order to get a dimensionless measure. The higher the moment, the greater the number of data points that are needed in order to get a reasonable estimate.Raw moment = E(X ^{k}) = -∞∫∞ x^{k} f(x)Normalised central moment = E( (X - μ) ^{k} ) /σ^{k} = ∫ (x - μ)^{k} f (x) /σ^{k}where f (x) is the probability distribution function^{}
Options with high volga benefit from volatility of volatilityJust as an option with high gamma benefits from high stock price volatility, an option with high volga benefits from volatility of volatility. The level of volatility of volatility can be calculated in a similar way to how volatility is calculated from stock prices (taking log returns is recommended for volatility as well). The more OTM an option is, the greater the volatility of volatility exposure. This is because the more implied volatility can change, the greater the chance of it rising and allowing an OTM option to become ITM. This gives the appearance of a ‘smile’, as the OTM option’s implied volatility is lifted while the ATM implied volatility remains constant. Stock returns have positive excess kurtosis and are leptokurtic.Kurtosis is always positive as this moment is zero for a point distribution. Hence, excess kurtosis (kurtosis -3) is usually used. The kurtosis (or normalised fourth moment) of the normal distribution is three; hence, normal distributions have zero excess kurtosis (and are known as mesokurtic). High kurtosis distributions (eg, stock price log returns) are known as leptokurtic, whereas low kurtosis distributions (pegged currencies that change infrequently by medium-sized adjustments) are known as platykurtic. IMPLIED VOLATILITY SMIRK IS A COMBINATION OF SKEW AND SMILEThe final ‘smirk’ for options of the same maturity is the combination of skew (3rd moment) and smile (4th moment). The exact smirk depends on maturity. Kurtosis (or smile) can be assumed to decay with maturity by dividing by time (assuming stock price is led by Lévy processes) and, hence, is most important for near-dated expiries. For medium- (and long-) dated expiries, the skew effect will dominate kurtosis, as skew usually decays by the reciprocal of the square root of time (for more details, see the section Modelling Volatility Surfaces). Skew for equities is normally negative and therefore have mean < median < mode (max) and a greater probability of large negative returns (the reverse is true for positively skewed distributions). For far-dated maturities, the effect of both skew and kurtosis fades; hence, implieds converge to a flat line for all strikes. Skew can be thought of as the effect of changing volatility as spot moves, while smile can be thought of as the effect of jumps (up or down). THERE ARE THREE WAYS TO MEASURE SKEW There are three main ways skew can be measured. While the first is the most mathematical, in practice the other two are more popular with market participants. **Third moment****Strike skew**(i.e. 90%-110%)**Delta skew**(i.e. [25 delta put – 25 delta call] / 50 delta)
1. THIRD MOMENT IS DEFINITION OF CBOE SKEW INDEXCBOE have created a skew index on the S&P500. This index is based on the normalised third central moment; hence, it is strike independent. The formula for the index is given below. For normal negative skew, if the size of skew increases, so does the index (as negative skew is multiplied by -10). Skew = 100 – 10 × 3 ^{rd} moment2. STRIKE SKEW SHOULD NOT BE DIVIDED BY VOLATILITYThe 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, i.e. 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 Skew = (90-110% × ATM) / ATM Skew = 90-110% Empirically, 90%-100% (or 90%-110%) skew is correct measure for fixed strike 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. We have shown mathematically that an absolute difference (eg, 90%-110% or 90%-100%) is the correct measure of skew, but we can also show it empirically. There is no correlation between volatility and skew (90%-110%) for any European stocks that have liquid equity derivatives. If skew were to be divided by volatility, it would introduce a negative correlation between this measure and volatility. 3. DELTA SKEW IS VIRTUALLY IDENTICAL TO STRIKE SKEWArguably the best measure of skew is delta skew, where the difference between constant delta puts and calls is divided by 50 delta implied. An example of skew measured by delta is [25 delta put – 25 delta call] / 50 delta. As this measure widens the strikes examined as vol rises, in addition to normalizing (i.e. dividing) by the level of volatility, it is a ‘pure’ measure of skew (i.e. not correlated to the level of volatility). While delta skew is theoretically the best measure, in practice it is virtually identical to strike skew. As there is a R ^{2} 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). |

### Measuring Skew

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