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 of integral with Gaussian kernel.

roposition

(Asymptotic of integral with Gaussian kernel) Suppose for the function may be expanded and for some the integral exists. Then for the following expansions hold

Proof

Split the integral into two pieces for some number For the second integral we estimate

We substitute the expansion for into the first integral We make a change , , . At this point we bring in the gamma function

 (Gamma function primer)
For the first term we have We evaluate the second term We evaluate the third term We collect the results It remains to note that the term disappears from expansion of by symmetry .

 Notation. Index. Contents.