 Solution

Let $F(t,s)$ be the ordinary Black-Scholes price of the terminal claim $G(S_T)$. Denote the new asset by $U_t=a(t)S_t^2$. Since it is strictly increasing in $S_t>0$, invert it as $s=\sqrt{U/a(t)}$ and write $H(t,U)=F(t,\sqrt{U/a(t)})$. Its derivative isThe replicating portfolio in the new asset and bank account therefore holdsThe wealth is $F$. The diffusion coefficient of $\Gamma dU$ is $2\sigma U\Gamma =\sigma S{F}_s$, agreeing with that of $dF$. Its drift together with the bank holding iswhich equals the drift of $F$ by its Black-Scholes equation. This verifies self financing and terminal replication. Thus the square-payoff claim has nonzero volatility and can replace the original stock as the traded risky asset.