BarrierAnalytic

Reiner-Rubinstein closed forms for a single-barrier European option with continuous monitoring, zero rebate, and a constant cost of carry b and volatility sigma.

This is the exact reference for the smoothed, per-step-monitored barrier in ExoticProducts.barrier: the Monte-Carlo payoff there trades a small bias (set by the smoothing width and the discrete monitoring) for a differentiable price, and this class is what that bias is measured against. Note that a discretely monitored barrier converges to the continuous price only after the Broadie-Glasserman-Kou exp(0.5826 sigma sqrt(dt)) shift, so a like-for-like check uses many monitoring dates.

The eight in/out x up/down x call/put combinations are assembled from the standard six terms A, B, C, D (the E, F rebate terms are zero here) with a call/put sign phi and an up/down sign eta.