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 generic Laplace integral. Laplace change of variables.

n the calculation of the section ( Asymptotic expansion of Laplace integral ) we want to place the limits of integration so that the difference term of the integration by parts would grab the most significant value. To be precise, let us consider the integral and assume that the function has exactly one local maximum on the interval at point . We assume that is continuously differentiable and is twice continuously differentiable, thus, We cannot directly apply the same technique of the section ( Asymptotic expansion of Laplace integral ) because and we have a pole in the denominator. Instead, we do the following change of variable.

There is a neighborhood where and for . In such neighborhood we introduce a variable according to the relationship In addition, we introduce the inverse relationship functions : Then

Next, we investigate the behavior of around (and drop the notation ): where the numerator and denominator vanish as . Thus, at where is negative. Then Whether the is positive or negative depends on orientation of variable . Consider : if increases from to then becomes negative and we may have or . We choose .

We have and we apply the proposition ( Asymptotic of integral with Gaussian kernel ),

We summarize our findings.

Proposition

(Asymptotic of generic Laplace integral) Let are functions , and . Let be the only local maximum of the function on and .

Assume that there is an such that the integral exists. Then as .

 Notation. Index. Contents.