Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 28J / d

At $B=K$, homogeneity givesThe price is therefore $C(S_0,K)-P(S_0,K)$. Zero-rate put-call parity yields

The zero-rate barrier-strike call hedge givesHold one share and borrow $80$. Stop and liquidate when the price first hits $80$: continuity makes the portfolio value exactly zero then, so it meets the knocked-out payoff. If no hit occurs before maturity, the stock remains above $80$ and the portfolio pays $S_T-80$, exactly the call payoff. There is no dependence on the stated volatility or maturity in this special contract.

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.