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.
A. Definition of standard Brownian motion.
B. Brownian motion passing through gates.
C. Reflection principle.
D. Brownian motion hitting a barrier.
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.

Reflection principle.


(Reflection principle) Let $W_{t}$ be a standard Brownian motion, and let $L,x$ be some real numbers $\,0<x<L$ then

MATH (Reflection principle)

Proof. Note, that the numbers $x$ and $2L-x$ are symmetrically located around the number $L$ (see the figure ( Reflection principle picture )):

Reflection principle picture
Reflection principle

MATH Suppose that $W_{t}$ crosses the level $L$ at some point $t^{\ast}<T$ and returns down to $x<L$ at time $T.$ Such path may be inverted symmetrically around the level $L$ on the time interval MATH The resulting path has the same likelihood of occurrence as the original.

Hence, MATH MATH Note, that $2L-x>L$ , hence the condition MATH is redundant. We conclude MATH as claimed.

Notation. Index. Contents.

Copyright 2007