Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / B / 5 / c

For this stock-dependent payoff, replace the stock drift by $r$ under a risk-neutral measure. Conditional lognormal moments then give, with $\tau =T-t$,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.