= Call option on a power payoff
{title2=$C_t=V_t^{(p)}\Phi(d_1)-Ke^{-r\tau}\Phi(d_2),\quad d_1=[\log(V_t^{(p)}/K)+(r+p^2\sigma^2/2)\tau]/(p\sigma\sqrt\tau)$}
For $p>0$ in a non-dividend-paying <Black-Scholes model>, the replicating price of the terminal power payoff $S_T^p$ is
$$
V_t^{(p)}=S_t^p\exp\left[\left((p-1)r+\tfrac12p(p-1)\sigma^2\right)(T-t)\right].
$$
Its <self-financing portfolio> holds $pV_t^{(p)}/S_t$ shares and bank wealth $(1-p)V_t^{(p)}$. <Itô's formula> gives physical drift $p\mu-(p-1)r$ and volatility $p\sigma$ for this replicating asset; its risk-neutral drift is $r$.
Since $V_T^{(p)}=S_T^p$, the payoff $(S_T^p-K)^+$ is a <European call option> on this replicating asset. The <Black-Scholes formula> gives the displayed price with $\tau=T-t$ and $d_2=d_1-p\sigma\sqrt\tau$. Using the raw process $S_t^p$ as a non-dividend-paying underlying would miss its time-dependent replication factor.
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