Guaranteed terminal stock floor in the Black-Scholes model ($V=S\Phi (d_1)+K{e}^{-r(T-t)}\Phi (-d_2)$)

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 $V=S_t\Phi (d_1)+K{e}^{-r\tau }\Phi (-d_2)$, 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 $S_t{e}^{r\tau }\Phi (d_1)$ and the probability of ending below the floor is $\Phi (-d_2)$, proving the price. Its option delta is $\Phi (d_1)$, since $S_t\varphi (d_1)=K{e}^{-r\tau }\varphi (d_2)$ cancels the derivative terms. The replicating strategy holds that many stock units and $K{e}^{-r\tau }\Phi (-d_2)/B_t$ bank-account units.