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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.
5. Heston equations.
6. Displaced Heston equations.
7. Stochastic volatility.
8. Markovian projection.
9. Hamilton-Jacobi Equations.
A. Characteristics.
B. Hamilton equations.
C. Lagrangian.
D. Connection between Hamiltonian and Lagrangian.
E. Lagrangian for heat equation.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Characteristics.


e would like to extend a solution $u$ of the equation ( First order PDE ) from a surface of boundary condition along some family of curves MATH We seek an effective way to define such a family. We hope to reduce the non linear PDE ( First order PDE ) to a system of ODEs.

Notation

We introduce the notations MATH MATH




We differentiate the definition (*) along the parameter $s$ : MATH We also differentiate the ( First order PDE ) along $x_{n}$ for every $n$ : MATH We would like to get rid of the derivative MATH , hence, we differentiate the definition (**) of MATH with respect to $s$ : MATH Therefore, if we would require MATH we would easily remove the MATH from (***): MATH We collect our results as follows.

Definition

The functions MATH that satisfy the system of ODE MATH are called characteristics of the PDE ( First order PDE ).

If a function $u$ solves the PDE and $x\left( s\right) $ solves (c) with MATH and MATH then p and z solve (a) and (b). One may always attempt to go in reverse: extend a solution of PDE from some (consistent) boundary condition using a family of solutions of (a),(b),(c).





Notation. Index. Contents.


















Copyright 2007