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.

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(Reflection principle) Let be a standard Brownian motion, and let be some real numbers then

 (Reflection principle)

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

Reflection principle

Suppose that crosses the level at some point and returns down to at time Such path may be inverted symmetrically around the level on the time interval The resulting path has the same likelihood of occurrence as the original.

Hence, Note, that , hence the condition is redundant. We conclude as claimed.

 Notation. Index. Contents.