Pricing and Valuation
The variance swap may be hedged and hence priced using a portfolio of European call and put options with weights inversely proportional to the square of strike.
Any volatility smile model which prices vanilla options can therefore be used to price the variance swap. For example, using the Heston model, a closed-form solution can be derived for the fair variance swap rate. Care must be taken with the behaviour of the smile model in the wings as this can have a disproportionate effect on the price.
We can derive the payoff of a variance swap using Ito's Lemma. We first assume that the underlying stock is described as follows:
Applying Ito's formula, we get:
Taking integrals, the total variance is:
We can see that the total variance consists of a rebalanced hedge of and short a log contract.
Using a static replication argument, i.e., any twice continuously differentiable contract can be replicated using a bond, a future and infinitely many puts and calls, we can show that a short log contract position is equal to being short a futures contract and a collection of puts and calls:
Taking expectations and setting the value of the variance swap equal to zero, we can rearrange the formula to solve for the fair variance swap strike:
Where:
is the initial price of the underlying security,
is an arbitrary cutoff,
is the strike of the each option in the collection of options used.
Often the cutoff is chosen to be the current forward price, in which case the fair variance swap strike can be written in the simpler form:
Read more about this topic: Variance Swap