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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
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 MATH and assume that the function $h\left( t\right) $ has exactly one local maximum on the interval $\left[ a,b\right] $ at point MATH . We assume that $g$ is continuously differentiable and $h$ is twice continuously differentiable, thus, MATH We cannot directly apply the same technique of the section ( Asymptotic expansion of Laplace integral ) because MATH and we have a pole in the denominator. Instead, we do the following change of variable.

There is a neighborhood MATH where MATH and MATH for $t\not =t_{0}$ . In such neighborhood we introduce a variable $y$ according to the relationship MATH In addition, we introduce the inverse relationship functions $t_{\pm}$ : MATH Then MATH

Next, we investigate the behavior of MATH around MATH (and drop the notation $\pm$ ): MATH where the numerator and denominator vanish as $y\rightarrow0$ . Thus, at MATH MATH where MATH is negative. Then MATH Whether the MATH is positive or negative depends on orientation of variable $y$ . Consider $\left( \#\right) $ : if $t$ increases from $t_{0}$ to $t>t_{0}$ then MATH becomes negative and we may have MATH or MATH . We choose MATH .

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

We summarize our findings.


(Asymptotic of generic Laplace integral) Let $g,h$ are functions MATH , MATH and MATH . Let $t_{0}$ be the only local maximum of the function $h$ on MATH MATH and MATH .

Assume that there is an $x_{0}$ such that the integral MATH exists. Then MATH as MATH .

Notation. Index. Contents.

Copyright 2007