trade volatility. 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: Analytically pricing variance swaps with discrete-sampling One might find discrete-sampling of the realized variance, says, as defined earlier, more practical in valuing the variance strike since, in reality, we are only able to observe the underlying price discretely in time. This is even more persuasive since there is an assertion that converges in probability to the actual one as the number of price's observation increases. Suppose that in the risk-neutral world with a martingale measure , the underlying asset price solves the following SDE: where: imposes the swap contract expiry date, is (time-dependent) risk-free interest rate, is (time-dependent) price volatility, and is a Brownian motion under the filtered probability space where is the natural filtration of . Given as defined above by the payoff at expiry of variance swaps,