 Solution

For a long position in the claim, hold $-V_S$ shares and bank value $S{V}_S-V$ so that claim plus hedge has zero initial value. The Black-Scholes equation and Itô's formula show that the total position is self-financing and its Brownian and drift terms cancel. Equivalently, the positive replicating portfolio holds $V_S$ shares and bond value $V-S{V}_S$.

Under the usual growth, integrability and admissibility conditions, no-arbitrage forces the claim price to be $V$: a different price could be traded against the replicating portfolio. The qualification matters because an arbitrary formal PDE solution without a suitable financial class need not be a valid price.