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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.
a. Antithetic variables.
b. Control variate.
c. Importance sampling.
d. Stratified sampling.
C. Longstaff-Schwartz technique.
D. Calculation of sensitivities.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Stratified sampling.

e introduce sets $A_{k}$ and separate our sampling of $X$ into MATH : MATH Let $q_{k}$ be allocation rates, $\sum_{k}q_{k}=1$ and $p_{k}$ be probabilities of $A_{k}$ : $\sum_{k}p_{k}=1$ . We proceed with identification of parameters. We want to keep estimation unbiased and reduce variance.

First, we seek weights $\alpha_{k}$ such that MATH MATH MATH Hence, we need to take $\alpha_{k}$ so that MATH to have an unbiased estimate.

We calculate the variance as follows MATH MATH MATH MATH MATH MATH

It remains to calculate two claims:

1. In case $p_{k}=q_{k}$ we have MATH (follows from formula ( Jensen inequality )).

2. Optimal $q_{k}$ is $p_{k}\sigma_{k}$ up to normalization constant (direct verification).

Notation. Index. Contents.

Copyright 2007