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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.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
A. Generation of random samples.
B. Acceleration of convergence.
C. Longstaff-Schwartz technique.
D. Calculation of sensitivities.
a. Pathwise differentiation.
b. Calculation of sensitivities for Monte-Carlo with optimal control.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Pathwise differentiation.

e consider the original SDE MATH and a perturbed SDE MATH We introduce the difference MATH The difference $\delta X_{t}$ is given by the following SDE MATH We assume that the functions $a$ and $b$ are such that the solution of the original SDE $X_{t}$ changes slightly when the perturbation of the initial condition is slight. Hence, the $\delta X_{t}$ is small if $\delta x$ is small. Consequently, MATH We substitute these approximations into the SDE for $\delta X_{t}$ MATH and note that such SDE scales with the magnitude of $\delta x$ . We introduce the process $Y_{t}$ : MATH We arrive to the following system of SDEs: MATH The derivative of interest may be expressed in terms of $X_{t}$ and $Y_{t}$ : MATH

Notation. Index. Contents.

Copyright 2007