= Solution
{c}
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 is
$$
H_U=\frac{sF_s}{2U}.
$$
The <replicating portfolio> in the new asset and bank account therefore holds
$$
\boxed{\Gamma_t=\frac{S_tF_s(t,S_t)}{2U_t},\qquad
\beta_t=\frac{F(t,S_t)-\tfrac12S_tF_s(t,S_t)}{B_t}.}
$$
The wealth is $F$. The diffusion coefficient of $\Gamma\,dU$ is $2\sigma U\Gamma=\sigma SF_s$, agreeing with that of $dF$. Its drift together with the bank holding is
$$
(2\mu-r)\Gamma U+r\beta B
=\mu SF_s+r(F-SF_s),
$$
which 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.
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