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A down-and-out European call pays the usual European call option payoff only if a lower barrier has not been hit before maturity. In the zero-interest, zero-dividend Black-Scholes model, for $0<B\le K<S$, its value is $C(S,K)-(S/B)C(B^2/S,K)$. This follows from reflection of the drifted logarithmic price killed at the barrier.

Assume a positive barrier $B\le K<S_0$. Under the zero-rate risk-neutral Black-Scholes model, $X_t=\log (S_t/B)$ starts at $x=\log (S_0/B)>0$ and is Brownian motion with variance rate ${\sigma }^2$ and drift $\mu =-{\sigma }^2/2$. The drifted transition density killed at zero isTo see the factor, multiply the zero-drift reflected Gaussian difference by $\exp [\mu (y-x)/{\sigma }^2-{\mu }^{2}t/(2{\sigma }^2)]$; its second term is the ordinary drifted Gaussian from starting point $-x$ times $e^{-2\mu x/{\sigma }^2}$.

Integrate the payoff $(B{e}^y-K{)}_{+}$ against this density. The first term is $C(S_0,K)$. Since $K\ge B$, the payoff is zero for $y\le 0$, so the second integral may be extended to all real $y$ and equals the ordinary call price from reflected spot $B{e}^{-x}=B^2/S_0$. The reflection weight is $e^x=S_0/B$. Thus the down-and-out European call value isThis assumes continuous monitoring and knock-out on first hitting, with no rebate.

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.