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 Laplace integral. onsider the integral for complex and assume that the function is such that exists for some . Then the integral exists for all s.t. . For such we obtain asymptotic expansion for via integration by parts and we obtained a combination as  . We apply the same operation again: Thus where the expansion may be continued as long as is continuously differentiable.

To confirm that the precise values of for really are of no relevance to the asymptotic expansion, split the interval of integration into two pieces for some . Then, for s.t. , for any .

We summarize our findings.

Proposition

(Asymptotic expansion of Laplace integral) Let be an -times continuously differentiable function and the integral exists for some . Then for any complex number s.t. , Notation. Index. Contents.