I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 2 Gauss-Hermite Integration.
 3 Asymptotic expansions.
 A. Asymptotic expansion of Laplace integral.
 B. Asymptotic expansion of integral with Gaussian kernel.
 C. Asymptotic expansion of generic Laplace integral. Laplace change of variables.
 D. Asymptotic expansion for Black equation.
 4 Monte-Carlo.
 5 Convex Analysis.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Asymptotic expansion for Black equation.

e seek an asymptotic expansion for solution of the problem for integrable function

We integrate the equation in from to : and then we substitute the same expression into the RHS: and simplify for ,

We now provide alternative derivation of the same result.

According to the section ( Backward Kolmogorov equation ), the solution of the problem may be represented as where is the standard Brownian motion. We perform the following standard calculations. We make the change of variable then

We arrive to the situation of the proposition ( Asymptotic of integral with Gaussian kernel ). We introduce the notation and arrive to The proposition ( Asymptotic of integral with Gaussian kernel ) requires regular -power series for the function . We do not have those in case of the call payoff. However, we can consider a domain of away from the singularity or restrict attention to a regular or approximate a call payoff with a smooth function. In either of such situations, we have uniformly converging Taylor series: in some bounded domain. We apply the proposition ( Asymptotic of integral with Gaussian kernel ) and find thus and we confirm our calculation.

 Notation. Index. Contents.