A down-and-out put option is a put option that becomes void if the asset price falls below the barrier \(S_{b}\). We must have \(S_{b}

where \(P\) is the price of the vanilla put, and \(P_{\mathrm{di}}\) and \(P_{\mathrm{do}}\) are the prices of the down-and-in and the down-and-out options, respectively. Sometimes a rebate is paid to the option holder if the barrier is crossed and the option is canceled; in such a case the above parity relationship is not correct.

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P = Pdi + Pdo;