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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
1. Conditional probability.
2. Normal distribution.
3. Brownian motion.
4. Poisson process.
5. Ito integral.
6. Ito calculus.
A. Example: exponential of stochastic process.
B. Example: integral of t_dW.
C. Example: integral of W_dW.
D. Example: integral of W_dt.
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.

Example: integral of W_dt.

e aim to derive distribution of the random variable MATH . According to the formula ( Ito_derivative_of_product ) and rules ( Ito calculus ) MATH We calculate MATH The integral MATH was previously calculated, see the formula ( Int t_dW ): MATH for some normal variable $\xi$ . The variables $W_{T}$ and $\xi$ are jointly normal (because these are sums of linearly dependent components) with zero mean. It remains to calculate the standard deviation of MATH . We have MATH The only part that we did not consider yet is MATH We introduce a uniform mesh MATH , $t_{p}=p\Delta t$ , $p=0,...,P$ , $P\Delta t=T$ and represent the integral as pre-limit sums with intention to pass back to the limit after some transformations: MATH where the MATH is a collection of independent standard normal variables, MATH . We continue MATH We now pass it back to the integral limit MATH Hence, MATH

Notation. Index. Contents.

Copyright 2007