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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 integral with Gaussian kernel.


roposition

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

Proof

Split the integral MATH into two pieces MATH for some number $\delta.$ For the second integral we estimate MATH

We substitute the expansion for $\phi$ into the first integral MATH We make a change $xt^{2}=\tau$ , MATH , MATH . MATH At this point we bring in the gamma function

MATH (Gamma function primer)
For the first term we have MATH We evaluate the second term MATH We evaluate the third term MATH We collect the results MATH It remains to note that the term $\frac{1}{x}$ disappears from expansion of $\Phi_{1}$ by symmetry $t\rightarrow-t$ .





Notation. Index. Contents.


















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