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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
1. Conditional probability.
2. Normal distribution.
A. Definition of normal variable.
B. Linear transformation of random variables.
C. Multivariate normal distribution. Choleski decomposition.
D. Calculus of normal variables.
E. Central limit theorem (CLT).
3. Brownian motion.
4. Poisson process.
5. Ito integral.
6. Ito calculus.
7. Change of measure.
8. Girsanov's theorem.
9. Forward Kolmogorov's equation.
10. Backward Kolmogorov's equation.
11. Optimal control, Bellman equation, Dynamic programming.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Calculus of normal variables.

uppose $X_{1}$ and $X_{2}$ are two jointly normal random variables: MATH with correlation MATH . We introduce two jointly normal variables $\xi_{1}$ and $\xi_{2}$ according to the relationships

MATH (Orthogonal normal variables)
and claim that $\xi_{1}$ , $\xi_{2}$ are iid MATH . To verify such claim we calculate MATH MATH

Similarly to the relationships ( Orthogonal normal variables ) we may represent any collection of jointly normal variables as a linear combination of iid standard normal variables. We consequently treat the jointly normal variables as vectors in the sense of elementary geometry: MATH In the above calculation, the $\xi_{1}$ and $\xi_{2}$ act as orthogonal basis and MATH acts like scalar product.

Notation. Index. Contents.

Copyright 2007