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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 for Black equation.


e seek an asymptotic expansion for solution MATH of the problem MATH for integrable function $g\left( x\right) .$

We integrate the equation MATH in $t$ from $T$ to $\tau$ : MATH and then we substitute the same expression into the RHS: MATH and simplify for MATH , MATH

We now provide alternative derivation of the same result.

According to the section ( Backward Kolmogorov equation ), the solution MATH of the problem MATH may be represented as MATH where $W_{t}$ is the standard Brownian motion. We perform the following standard calculations. MATH MATH MATH MATH We make the change of variable MATH then MATH

We arrive to the situation of the proposition ( Asymptotic of integral with Gaussian kernel ). We introduce the notation MATH and arrive to MATH The proposition ( Asymptotic of integral with Gaussian kernel ) requires regular $z$ -power series for the function MATH . We do not have those in case of the call payoff. However, we can consider a domain of $x$ away from the singularity or restrict attention to a regular $g$ or approximate a call payoff with a smooth function. In either of such situations, we have uniformly converging Taylor series: MATH MATH MATH in some bounded domain. We apply the proposition ( Asymptotic of integral with Gaussian kernel ) and find MATH MATH thus MATH and we confirm our calculation.





Notation. Index. Contents.


















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