Variance swap pricing.

ariance swap is a path-dependent contract that pays at maturity an amount depending on the quantity MATH called realized variance. The times $t_{i}$ are daily closure prices (or something reasonable, so that $t_{i+1}-t_{i}$ are small for all ), $t_{0}=0$ , $t_{N}=T$ . We assume everywhere in this section that the process $X_{t}$ is a diffusion (does not jump): MATH and is either considered in its martingale measure or being discounted as in the section ( optimal utility ). Under this assumption we proceed to show that $\Omega_{T}^{X}$ is the measure of volatility during time interval $\left[ 0,T\right]$ . Indeed, MATH MATH Hence, MATH If the payoff of the contract is a linear function then we are interested in the risk neutral expectation MATH We next show how the last quantity could be approximated with a linear combination of European claims (static hedge).

Since we assume the diffusion process, MATH Hence, MATH We also assumed "no-drift", hence, MATH We obtain MATH The last expectation is the European payoff and was considered in the section ( Log contract ).

Summary

If the stochastic process is given by the SDE MATH then the variance $\Omega_{T}^{X}$ defined by MATH has the properties MATH where the and are defined in section ( Recovery of implied distribution ) and the is defined by MATH

We consider next the process with deterministic drift and dividends.

Variance swap pricing for drifting price process.

Volatility smile formula for fair variance.

Notation.

Index.

Contents.