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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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 MATH for complex $x$ and assume that the function $\phi$ is such that MATH exists for some $x_{0}$ . Then the integral exists for all $x$ s.t. MATH . For such $x$ we obtain asymptotic expansion for MATH via integration by parts MATH and we obtained a combination MATH as MATH MATH . We apply the same operation again: MATH Thus MATH where the expansion may be continued as long as $\phi$ is continuously differentiable.

To confirm that the precise values of $\phi$ for $t>0$ really are of no relevance to the asymptotic expansion, split the interval of integration into two pieces MATH for some $\varepsilon>0$ . Then, for $x$ s.t. MATH , MATH for any $n>0$ .

We summarize our findings.

Proposition

(Asymptotic expansion of Laplace integral) Let $\phi$ be an $N$ -times continuously differentiable function and the integral MATH exists for some $x=x_{0}$ . Then for any complex number $x$ s.t. MATH , MATH





Notation. Index. Contents.


















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