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.

## Volatility smile formula for fair variance.

ollowing the common business practice we will assume in this section that the future distribution of a traded instrument is tracked via a volatility smile. We will utilize the notation of the formula ( Black Scholes formula ) for the undiscounted call price: where We use the notation Assume that has dependence on derived from dependence of on . This is the "volatility smile" practice. The Black-Scholes formula is fitted into marked quotes by introducing dependence of the parameter on and . We are going to calculate the implied distribution of the random variable via double differentiation of and use the formula ( Fair Variance vs Log contract ).

We proceed with calculation of the derivative: By the formula ( Black Scholes property 1 ) Also, hence, Consequently, where is a dependent function. We use the notation introduced in ( Black Scholes formula ). Hence,

We now invoke the formulas ( Fair Variance vs Log contract ) and ( Distribution density via Call ): Therefore, We perform the change of variables and continue The function is rapidly decaying at infinity. We use such property to perform the following integration by parts We use this result to remove the term in the expression for : We perform the similar integration by parts: and use it to remove the term in the expression for : Note that since then hence

Recall that where the is the Black-Scholes volatility. Hence we arrive to the following summary.

Summary

Suppose that marginal distributions of the diffusion process are given by the volatility smile parametrized by : where the notations are taken from ( BlackScholesUndiscountedCall ). Then

 Notation. Index. Contents.