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 expansions.

symptotic expansions are useful for obtaining high performance analytical formulas, for validation of solutions, acceleration of Monte-Carlo convergence, obtaining preconditioners and adaptive grids. The references for this section are [Erdelyi] and [Sveshnikov] . The presentation here is partial.

Definition

(O symbols) Let be a Banach space.

1. For functions we write iff is bounded around and as .

2. The series is "asymptotic expansion" of at iff We will use notation

3. The expansion are called "asymptotic power series".

Proposition

(Differentiation of asymptotic power series) If is differentiable and both and possess asymptotic power series then such power series are connected by termwise differentiation.

Proof

The proof is based on assertion that there is similar result for integration, see the proposition ( Dominated convergence theorem ) or less generic results from calculus. We have and Thus, we must have Then and we match terms with the expansion for .

 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.
 Notation. Index. Contents.