I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 1 Black-Scholes formula.
 2 Change of variables for Kolmogorov equation.
 3 Mean reverting equation.
 4 Affine SDE.
 5 Heston equations.
 6 Displaced Heston equations.
 7 Stochastic volatility.
 A. Recovering implied distribution.
 B. Local volatility.
 C. Gyongy's lemma.
 D. Static hedging of European claim.
 E. Variance swap pricing.
 a. Variance swap pricing for drifting price process.
 b. Volatility smile formula for fair variance.
 8 Markovian projection.
 9 Hamilton-Jacobi Equations.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Variance swap pricing.

ariance swap is a path-dependent contract that pays at maturity an amount depending on the quantity called realized variance. The times are daily closure prices (or something reasonable, so that are small for all ), , . We assume everywhere in this section that the process is a diffusion (does not jump): 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 is the measure of volatility during time interval . Indeed, Hence, If the payoff of the contract is a linear function then we are interested in the risk neutral expectation 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, Hence, We also assumed "no-drift", hence, We obtain 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 then the variance defined by has the properties where the and are defined in section ( Recovery of implied distribution ) and the is defined by

We consider next the process with deterministic drift and dividends.

 a. Variance swap pricing for drifting price process.
 b. Volatility smile formula for fair variance.
 Notation. Index. Contents.