= 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 \b[<continuous> barrier hitting and zero carry are the key assumptions], not constant volatility.
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