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- Sensitivity to Skew and Convexity
Sensitivity to Skew and Convexity
Skew is commonly thought of as an important component of variance swap prices, with put skews seen as having a much greater impact on prices than call skews. Although, in practice, this is a useful framework for thinking about how variance swap prices behave, it is not theoretically correct.
Example:
underlying has 3-month ATM IV = 20%, a linear put skew of 5% (per 10 volatility points), with put IVs capped at 35% and all OTM call volatilities flat at the level of the ATM IV (Fig. 106). The theoretical 3-month variance strike can then be calculated to be 23.05.
Now consider a situation where the skew is a mirror image of that in Figure 106. ATM IV = 20%, and all OTM put volatilities are flat at 20%, but call volatilities increase linearly by 5% per 10 points as they become more OTM, capped at 35% (Fig. 107). In this case the theoretical variance swap price is virtually identical at 23.15.
1) Why does this symmetry exist?
Since the 1/strike-squared replicating portfolio has a much higher weighting of puts, it would be natural to assume that their associated implied volatilities should have a proportionally greater effect on the variance swap strike.
2) Also, in practice, variance swaps can effectively be priced and hedged with ATM volatility plus a contribution from the skew.
1) The exposure to the skew curve is symmetrical. That is, the contribution to the variance swap price of the volatility of an OTM call is exactly the same as from an OTM put with same (risk-neutral) probability, N(d2), of ending ITM. The OTM puts have a greater weighting in the replicating portfolio, simply because their dollar gamma is lower and so they must be scaled up in order to provide constant dollar gamma across the range of strikes.
2) In reality, equity skews look much more like Fig. 106 than Fig. 107, at least for short dated index volatility. This explains why Derman’s approximation assuming a linear skew is most successful in these conditions. Directionality of volatility, at least in the case of indices, makes volatility skews highly asymmetric, with OTM calls trading much closer to ATM volatility than the corresponding OTM puts.
So what exactly determines the contribution of volatilities across the skew curve to the variance swap price?
Clearly a very OTM option should make a relatively small contribution to the variance swap price. (This must be true in order to replicate variance swaps in practice, otherwise the value of the replicating portfolio will be very sensitive to the exact choice of replicating portfolio.) It then seems sensible that a variance swap represents a kind of weighted average of volatilities across the skew curve, with the closer-to-the-money volatilities higher weighted.
In fact, this is exactly the case, with the average being taken over the variances rather than the volatilities, and the weighting function simply being the risk-neutral probability density function, N′(d2), that the corresponding OTM option ends up ITM.
If we define the variable z to be the standard Black-Scholes parameter: 
then it can be shown that 
The N′(d2) term is the probability density function for the underlying at expiry, T. That is, the cumulative distribution N(d2) gives the (risk-neutral) probability that the underlying will be trading above z at time T. Thus the parameter z, simply represents the ‘moneyness’ of the corresponding OTM option. This means that the variance swap price is a weighted sum of squared option implied volatilities weighted by the probability that the (OTM) option will end ITM.
Furthermore, if the skew curve is quadratic in the variable z (the moneyness of the option) of the form:
then substituting and integrating gives 
, i.e. in ‘d2-space’ the variance swap price is not affected by the linear component of the skew α, but only on the base level of volatility σ0, and the convexity parameter β.
This explains the phenomena observed previously: very different (linear) skews, but similar convexities give almost identical variance swap strikes. The reason that there is any difference at all in variance swap strikes, is because in d2-space, the convexity in Fig. 107 is slightly greater, since the maximum volatility is at a strike of 130, which is slightly less OTM than the 70 strike where the maximum volatility is achieved in Fig. 106.
This also helps to explain why the convexity has a greater effect on longer maturity variance. As maturity increases, the probability of far OTM (in relative strike terms) options ending ITM increases. Therefore the relative weight of e.g. the 160-strike call will increase with maturity. At shorter maturities perhaps only the 80-120 portion of the skew surface significantly affects the variance swap price, and this part of the skew is relatively linear (at least for indices). At 5 years, strikes out to say 50 -180 become relevant to the pricing and the convexity of these strikes can be much higher.
In practice, skew is most often thought about in terms of relative strikes, rather than moneyness – but the point remains: the contribution of a point on the skew curve to the variance swap price depends on the (risk-neutral) probability N(d2) of the associated OTM option ending ITM.
Therefore ATM volatility will provide the greatest contribution to variance swap prices – particularly for short maturities.
Both high put skews, and high call skews (where OTM calls have higher volatilities than ATM) will increase variance swap prices.
In fact there is some kind of feedback effect here because the contribution of each volatility is determined by the probability of an option being ITM, but, for example high put skews will increase the probability of OTM puts ending ITM so both the volatility and the weighting factor will increase.
To sum up:
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Given a flat skew, variance will price at the same level as ATM volatility.
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Positive convexity will always act to increase variance swap strikes.
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With a negatively convex skew – OTM volatilities are (on average) less than ATM volatility – it is theoretically possible that variance could price below ATM volatility.