 Solution

The hedge is pathwise, so it works for other continuously evolving stock models, including stochastic or time-dependent volatility, as long as the interest rate and dividends are zero and trading/monitoring are continuous and frictionless. It does not require lognormal terminal prices.

A downward jump can overshoot the barrier: liquidation then leaves $S_{\tau }-K<0$ instead of the required zero, creating gap risk. Discrete monitoring or delayed execution can likewise destroy the stopping replication. Nonzero interest, dividends, transaction costs, funding differences and trading constraints require a changed portfolio or pricing model. Thus continuous barrier hitting and zero carry are the key assumptions, not constant volatility.