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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.
1. Black-Scholes formula.
2. Change of variables for Kolmogorov equation.
3. Mean reverting equation.
4. Affine SDE.
A. Ricatti equation.
B. Evaluation of option price.
C. Laplace transform.
D. Example: CDFX model.
5. Heston equations.
6. Displaced Heston equations.
7. Stochastic volatility.
8. Markovian projection.
9. Hamilton-Jacobi Equations.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Laplace transform.


e start our consideration from the Fourier transform.

MATH (Direct Fourier transform)
MATH (Inverse Fourier transform)
We aim to expand these relationships to a transformation MATH where the $z$ is a complex number $z$ , $z=\sigma+i\gamma$ , $-m\leq\sigma\leq m$ for some $m>0$ . The idea is to split MATH and apply ( Direct Fourier transform ),( Inverse Fourier transform ) to MATH .

Suppose the integral MATH converges absolutely for the given interval of values $\beta$ . We proceed according to the stated above idea: MATH MATH We would like to recover the $f$ from $\psi$ . We use ( Inverse Fourier transform ) with the substitution MATH , $x\rightarrow\gamma$ , MATH : MATH We transform the last relationship: MATH Hence, we invert MATH with MATH





Notation. Index. Contents.


















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