Solution

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

{title2=$B=K,\quad C_{\rm out}=S-K$}

For a <down-and-out European call> with barrier equal to strike, zero interest and no dividends, hold one share and borrow the strike until barrier hitting or maturity. At continuous hitting the share price is exactly the debt, so liquidating leaves zero; on survival the portfolio pays $S_T-K$. Hence the contract has value $S-K$ and delta one before knock-out. The pathwise replication works beyond constant-volatility <Black-Scholes model>, but can fail through downward jumps or execution gaps, and nonzero carry, transaction costs or discrete monitoring require changes.

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 Incoming links: Zero-rate barrier-strike call hedge