 Solution

For the linear spatial factor, $f^{\prime \prime }=0$ and $S{f}^{\prime }=f$. Hence $\lambda =r-q$ and $g^{\prime }=qg$. Absorbing the constant $k$ into the time-factor amplitude gives $\boxed{V(S,t)=AS{e}^{q(t-T)}}$. In particular a terminal claim equal to $A{S}_T$ is worth $AS{e}^{-q(T-t)}$, as expected for an asset paying a continuous dividend yield.