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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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 $Y_{T}$ is tracked via a volatility smile. We will utilize the notation of the formula ( Black Scholes formula ) for the undiscounted call price: MATH where MATH We use the notation MATH Assume that $\omega$ has dependence on $K$ derived from dependence of $\sigma$ on $K$ . This is the "volatility smile" practice. The Black-Scholes formula is fitted into marked quotes by introducing dependence of the parameter $\sigma$ on $K$ and $T$ . We are going to calculate the implied distribution of the random variable $Y_{T}$ via double differentiation of $C$ and use the formula ( Fair Variance vs Log contract ).

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

MATH

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

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

Summary

Suppose that marginal distributions of the diffusion process $Y_{t}$ are given by the volatility smile $\Sigma$ parametrized by $d_{2}$ : MATH MATH where the notations are taken from ( BlackScholesUndiscountedCall ). Then MATH





Notation. Index. Contents.


















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