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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.

For $f=k{S}^{\alpha }$, $S{f}^{\prime }=\alpha f$ and $S^2{f}^{\prime \prime }=\alpha (\alpha -1)f$. Thus $\lambda ={\lambda }_{\alpha }={\sigma }^2\alpha (\alpha -1)/2+(r-q)\alpha$, and all values with this fixed spatial factor areThis Power payoff in the Black-Scholes model calculation remains valid for any real $\alpha$ on $S>0$; the specified nonzero exponent excludes the constant-payoff subpart. It also supplies the moment term in power put-call parity above.

Let $\tau =T-t$. Conditional on ${\mathcal{F}}_t$ under the risk-neutral measure,Using the moment-generating function of a normal distribution in the risk-neutral expectation gives the Power payoff in the Black-Scholes model valueIn particular,The delta hedge holdsunits of the risky asset.

Risk-neutral valuation givesSince $W_T^Q\sim N(0,T)$,Hence the Power payoff in the Black-Scholes model has price