# Pricing A Variance Swap

In this article I examine the risk-neutral pricing of a variance swap, which can be constructed with this elegant formula:

# Static Portfolio Replication

** proposition** Let be twice continuous differentiable so that for any and any , then,

** proof** We will prove the case in which ; similar proof can be sketched for the other case. When , we have . The the second integral can be simplified as:

Using integration by part:

And,

The above formula shows that `any`

twice-differentiable continuous time-T payoff can be replicated using a `static portfolio`

of:

In practice, discrete set of strike can be used to approximately replicate. For example, given and payoff function . Here we choose to replicate with (Though we can theoretically replicate the payoff with any , a close to is typically chosen) and a discrete set of option strikes with increment of capping at . In this replicating portfolio we will hold:

# Variance Swap

A `variance swap`

is an over-the-counter financial derivative that allows a party to trade on the future variance of a given underlying security. For example, a trader would pay the `realized variance`

of log-price changes in exchange of a fixed payment called `variance strike`

, normalized by the `vega notional`

into dollar terms. The payoff of the VS is:

We can calculate the variance strike which results in a zero discounted expectation of the payoff (per unit of vega notional).

Assuming that the underlying process follows a geometric Brownian motion with local volatility :

Therefore the `realized variance`

of is:

Furthermore, if we combine the SDE of and ,

Taking integral on both sides,

Combining with the formula for the realizead variance of ,

Using the proposition proven above, we can create a static replication portfolio and replicate the payoff.

If we choose the cutoff as the forward price , we can largely simplified the formula as follow:

In conclusion, if we assume a GBM underlying process, the fair-value variance strike can be calculated as the sum of calls and puts across a continuum of strikes.

# Practical Consideration

In practice, variance swap is costly to implement, requires constant hedging and an entire array of options. The advantage of a variance swap is that it is `purely exposed to volatility risk`

, as oppose to an option which contains directional risk.

The P&L of a variance swap depends directly on the difference between realized and implied volatility. Since historically the implied volatility has been above realized volatility, a.k.a. `variance risk premium`

, volatility arbitrage (rolling short variance trade) can be carried out with variance swaps.

Reference

*Variance and Volatility Swaps*, FinancialCAD Corporation, http://docs.fincad.com/support/developerFunc/mathref/VarianceSwaps.htm