Replicating and Hedging in Practice

A variance swap can be statically hedged with a portfolio of OTM European options, weighted according to the inverse squares of their strikes.

In theory, it makes it easy to calculate the fair value of a variance swap, assuming option prices are available across the entire range of strikes.

In practice, traded strikes are not continuous, although for major liquid indices they are closely spaced. A more serious limitation is the lack of liquidity in OTM strikes, especially for puts, as these provide a relatively large component of the variance swap price in the presence of steep put skews. 

The problem with the lack of OTM puts can be seen from following through the practical example of setting up the replicating portfolio above.

The long futures position is used to create a pay-out which is equivalent to a long log contract plus realised variance.

In opposition the long options/short forward position is used to create a short log contract and pay the fixed strike.

Supposing the market falls significantly, the delta-hedge will be long the log contract (a will hence lose), while the options should counteract this by being short the log contract. However, if not enough downside puts were used, the options portfolio will not fully reflect the short log-contract exposure needed and hence the overall hedge will lose money.

This lack of liquidity at the wings has led to the development of conditional variance swaps which can remove exposure to volatility once the underlying moves into areas where vanilla options are illiquid. 

 

In practice, market-makers will not attempt to hedge with the entire strip of options but typically will use only two or three – including one close to the money and one or more OTM, but liquid, puts.

 

Alternatively they could approach the replicating by hedging the vega with an OTM put whose implied volatility coincides with the variance swap strike – close to the money for 1-month maturity, 95% for 3-6 months, 90% for 1-year, 85% for 2-3 years etc. In this case they would also look to buy back the wings/convexity separately. 

One problem with this kind of approach is that the partial hedge is no longer static, and must be dynamically managed. For example if the market sells off towards the strike, the market maker will have to trade further OTM puts to ensure that their exposure to volatility, in the form of dollar-gamma, remains constant.

 

This makes the actual variance swap replication more akin to a combination of alternatives 2 and 3 (previously listed). Here the constant dollar gamma would be maintained by a combination of holding a portfolio which has roughly constant dollar gamma if the underlying does not move too much, and re-hedging by trading more options if the underlying does move significantly.

 

Another limitation comes from the discrete nature of adjusting the delta hedge on the close, which introduces possible errors due to large daily moves. However these moves are actually of order the cube of the move and hence are negligible for all but very large moves. Also as previously noted if interest rate changes are related to changes in the spot underlying, this can also have an impact on the ability to accurate replicate the log-contract and realised volatility.

 

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