= Guaranteed terminal stock floor in the Black-Scholes model
{title2=$V=S\Phi(d_1)+Ke^{-r(T-t)}\Phi(-d_2)$}
The claim $\max(K,S_T)$ for $K>0$ combines a guaranteed cash floor with a <European call option>, equivalently a stock with a put. In the <Black-Scholes model>, for $\tau=T-t>0$, its price is $V=S_t\Phi(d_1)+Ke^{-r\tau}\Phi(-d_2)$, where $d_1=[\log(S_t/K)+(r+\sigma^2/2)\tau]/(\sigma\sqrt\tau)$ and $d_2=d_1-\sigma\sqrt\tau$. Under the risk-neutral lognormal law, the truncated stock moment is $S_te^{r\tau}\Phi(d_1)$ and the probability of ending below the floor is $\Phi(-d_2)$, proving the price. Its <option delta> is $\Phi(d_1)$, since $S_t\phi(d_1)=Ke^{-r\tau}\phi(d_2)$ cancels the derivative terms. The <replicating strategy> holds that many stock units and $Ke^{-r\tau}\Phi(-d_2)/B_t$ bank-account units.
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