= Solution
{c}
For a <call option on a power payoff>, the replicating asset $V$ from part (c) has $V_T=S_T^3$, volatility $3\sigma$, and risk-neutral drift $r$. The new payoff is therefore a <European call option> on $V$, with strike $K$. Applying the given <Black-Scholes formula> gives
$$
\boxed{C_t=V_t\Phi(D_1)-Ke^{-r\tau}\Phi(D_2),\qquad
D_1=\frac{\log(V_t/K)+(r+9\sigma^2/2)\tau}{3\sigma\sqrt\tau},\quad
D_2=D_1-3\sigma\sqrt\tau.}
$$
Here $V_t=S_t^3e^{(2r+3\sigma^2)\tau}$. Equivalently $D_1=[\log(S_t^3/K)+(3r+15\sigma^2/2)\tau]/(3\sigma\sqrt\tau)$. It is the price process $V_t$, rather than the raw power $S_t^3$, that serves as the self-financing underlying for this application.
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