= Solution
{c}
For this stock-dependent payoff, replace the stock drift by $r$ under a <risk-neutral measure>. Conditional lognormal moments then give, with $\tau=T-t$,
$$
\boxed{V_t=e^{-r\tau}\mathbb E^Q[S_T^3\mid\mathcal F_t]
=S_t^3e^{(2r+3\sigma^2)\tau}.}
$$
This is the <Power payoff in the Black-Scholes model>. The <delta hedge> is $\Delta_t=\partial V/\partial S=3V_t/S_t$ shares, so the stock holding is worth $3V_t$. The rest is a bank position worth $-2V_t$, or $-2V_t/B_t$ units of the bank asset. This gives a <replicating portfolio>; its self-financing identity follows from Itô's formula and the Black-Scholes pricing equation, and its terminal value is $S_T^3$.
The full two-Brownian filtration contains an additional orthogonal noise direction, but this particular payoff is replicated using the traded stock and bank asset, so that extra noise does not affect its price.
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