 Solution

If the value is independent of spot, both spatial derivatives vanish. The Black-Scholes equation reduces to $V_t-rV=0$, so $\boxed{V(S,t)=A{e}^{r(t-T)}}$, where $A$ is an arbitrary constant. A constant terminal payoff is therefore discounted at the interest rate $r$; its separation constant is $\lambda =0$.