Variance Swaps and Option Volatilities

Variance swap strikes are well correlated with BS IV derived from options prices.

--> not surprising --> both can be interpreted as market estimates of future volatility. 


Similar but different:

- ATM IV reflects the market estimate of future volatility realised around the current level. 

- VS represents the market estimate of variance, independent of future market level. 


The table below shows ATM volatility and variance swap levels for a number of European stocks and indices.

Whilst all VS price > ATM volatility --> variation in the spread --> partly due to the differing skews.

This spread becomes larger and more unpredictable at longer maturities --> effect shape of skew becomes more important. 


Table vs vs atm iv


At 6m maturity, index variance appears rich in comparison to its ATM IV -->  index having higher put skews than single-stocks.

At 3y maturity, they seem more in line --> greater upside convexity of single-stocks becomes more important. 


Why does VS invariably trade above ATM IV of the corresponding maturity? 

  • Convexity premium: gain more from increase in volatility than corresponding loss from decrease in volatility. 
  • Theoretical price of VS is calculated from prices of a replicating ptf of options.


With skew and skew convexity, average volatilities will usually be above ATM volatility, making the VS more expensive. 


Ultimately it comes down to the fact that a long VS is also long vol of vol, due to the convex nature of the payoff.

Convex payoff is paid through a replicating ptf of options across the skew surface, which in effect price in this vol of vol. 


For short dates: 

- skew relatively linear around ATM volatility

- put skews more significant than call skews 


--> the price can be thought of as a function of ATM volatility level and the slop of the skew. 


In practice, contribution of skew component means that VS strikes tend to trade a similar levels to OTM puts. 

- 95-100 strikes for maturities of around 1-3 months

- 90-95 strikes for maturities of about a year


Skew and convexity become more important factors at longer dates --> probability of reaching more OTM strikes increases. 

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