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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
A. Finite difference basics.
B. One dimensional heat equation.
C. Two dimensional heat equation.
D. General techniques for reduction of dimensionality.
a. Stabilization.
b. Predictor-corrector.
c. Separation of variables for Crank-Nicolson scheme.
E. Time dependent case.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.


et $A\,$ be a space-direction finite difference operator, MATH Assume that $A$ is independent from $t$ . We seek for efficient ways to convert to a finite difference scheme in $t$ -direction. We integrate over $t$ : MATH and approximate the integral MATH where the $\tau$ is the step $t^{j+1}-t^{j}$ . Hence, MATH MATH If the operator $A$ is positive definite the the norm of MATH is less then $1$ and the scheme is stable.

The predictor-corrector way to improve scheme's efficiency is the following. We find a separation MATH for the implicit part of the scheme. Observe that the Crank-Nicolson may be written in two steps as MATH MATH We transform further MATH MATH MATH The last equation is equivalent to MATH We aim to replace the last equation with the equation MATH To see that we preserve the second order of approximation we compute MATH MATH The resulting scheme is MATH MATH MATH These should be more efficient because we split the inversion into two, presumably, simpler components. The last step is explicit.

To explore stability we put all steps together MATH MATH Hence MATH Make the change of function MATH then MATH and the stability follows from the stabilization scheme considerations.

Notation. Index. Contents.

Copyright 2007