hard · Quantitative Finance
For a standard Brownian motion W_t started at W_0 = 0, the Reflection Principle states that P(max_0 ≤ t ≤ T W_t ≥ a) = 2P(W_T ≥ a) for a > 0.
Why does the factor of 2 appear?
- The variance of the maximum of a Brownian motion is exactly twice the variance of the terminal value.
- The drift of the reflected process is the negative of the original drift.
- Paths that hit a are equally likely to end above or below a due to the symmetry of Brownian motion after the first touch.
- It accounts for the 'In' and 'Out' components of a barrier option simultaneously.
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