medium · Quantitative Finance stochastic
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 equals exactly twice the variance of its terminal value at time T.
- The drift of the reflected process equals the exact negative of the original process's drift under this reflection map.
- 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, since In plus Out together replicate a vanilla.
Sign up free to see the explanation and track your rank →
More Quantitative Finance stochastic practice
- Under Girsanov's Theorem, what does a change of probability measure primarily alter in a s
- In the context of the HJM framework, what is the primary lesson regarding the drift of the
- In the Vasicek short-rate model dr_t = κ(θ - r_t) dt + σ dW_t, what happens to the drift w
- For a standard Brownian motion W_t, what is the expected value of W_t^2?
- Under the geometric Brownian motion model with μ = 0.12, σ = 0.30, and S_0 = 100, what is
- According to the lognormal property, what is the expected stock price at time T = 1 year i
- A stock follows geometric Brownian motion dS = μ S dt + σ S dW. Using Itô's Lemma, find th
- Under Girsanov's theorem, if a stock follows dS_t = μ S_t dt + σ S_t dW_t under the real-w