Touched but not closed 2

What this barrier-touching probability question tests

This is a medium-difficulty statistics question that appears frequently in quant trading interviews. It combines Brownian motion, barrier events, and conditional probability to test whether you can reason about the joint behaviour of a path and its final outcome.

The core challenge is understanding how much information the fact that a price "touched" a level tells you about where it will finish. Candidates must set up the conditioning carefully, reason about the properties of driftless Brownian motion, and often use reflection or symmetry arguments to avoid heavy integration. Firms value candidates who can articulate why certain simplifications are valid.

• Properties of driftless (standard) Brownian motion
• Barrier-crossing and reflection principle
• Conditional probability and independence under Brownian motion
• Symmetry and martingale properties