Chapter 6 Volatility and variance swaps

In this section, we will see in detail how volatility and variance swaps work. We will go through pricing and replication, but also what they are used for. ## Definition Volatility swaps are forward contracts on the realized volatility of an underlying. AS for a classic forward contract, the buyer will lose or earn money depending on the realized volatility on the elapsed period. The payoff of a volatiliy swaps of maturity T and strike price K on an underlying is: \[ \boxed{(\sigma-K)*N_{vega}} \] Variance swaps work the same but on the variance of the underlying instead of the volatility:

\[ \boxed{\frac{\sigma^2-K^2}{2K} * N_{vega} = (\sigma^2-K^2)*N_{variance}} \] * Here, \(\sigma\) is the realized vol computed as \(\sigma^2 = \frac{T}{252}\sum\limits ^{T}_{i=i} \left[ ln(\frac{S_i}{S_{i-1}})\right]^2\) with \(T\) in business days. * We also deduce that \(\boxed{\frac{N_{vega}}{2K}}\)