Barrier hitting times under geometric Brownian motion

Stock starts at $100. Exact inverse Gaussian, risk-neutral drift of log S = r − q − σ²/2. Every slider recomputes both maps.

1 · Hitting-time density — over time and barrier, at the volatility set here

2 · Probability of knock-out before maturity — over barrier and volatility

The dot on map 2 is the contract: the barrier set at the top and the volatility of map 1. On map 1 the dotted line is that barrier, the teal line the most likely hitting time (mode) and the orange dashes the median. Where the drift points away from the barrier the distribution is defective: the barrier may never be hit, and the median exists only while that chance is above one half.