Exercise: Reiner-Rubinstein closed-form for DOC

For a down-and-out call with $B \le K$ (barrier below strike), under Black-Scholes:

where $\nu = (r + \sigma^2/2)/\sigma^2$, and $C_{\text{BS}}$ is the standard Black-Scholes call price.

(The exponent convention varies in the literature; some use $1 - 2(r/\sigma^2)$.)

Tasks

1. Implement the Reiner-Rubinstein formula. Verify against $S_0 = 100, K = 100, B = 80, T = 1, r = 0.05, \sigma = 0.2$. The result should match the FDM result from the previous exercise.

2. Reflection principle interpretation. The term $(S_0/B)^{1-2\nu} C_{\text{BS}}(B^2/S_0, K)$ is the "reflected" vanilla call. Argue heuristically why a Brownian motion's reflected paths give this formula.
3. Limiting cases.
   • $B \to 0$: the barrier is unreachable; DOC $\to$ vanilla. Verify.
   • $B \to K$: the barrier is at the strike; DOC $\to ?$. Explore numerically.
   • $B > K$: the formula doesn't apply directly; why?
4. Greeks. Compute the analytical delta of the DOC by differentiating the closed-form. Compare with FDM-based delta near the spot.