Assume that a non-dividend-paying security with price \(S_t\) at time \(t\) can move to either \(S_t u\) or \(S_t d\) at time \(t + 1\). The continuously compounded rate of interest is \(r\), and \(u > e^r > d\). A financial derivative pays \(\alpha\) if \(S_{t+1}\) = \(S_t u\) and \(\beta\) if \(S_{t+1} = S_t d\).

A portfolio of cash (amount \(x\)) and the underlying security (value \(y\)) at time \(t\) exactly replicates the payoff of the derivative at time \(t + 1\).