= Solution
{c}
Under the <risk-neutral measure>, write
$$
S_T/s=\exp\left((r-\sigma^2/2)\tau+\sigma\sqrt\tau Z\right),\qquad Z\sim N(0,1).
$$
Complete the square in the <normal density> after multiplying by the power of $S_T/s$. The <truncated lognormal moment> gives, for $K>0$,
$$
\boxed{V_p(t,s,K)=M_p(\tau)\Phi(d_p),\quad
M_p(\tau)=e^{[pr+p(p-1)\sigma^2/2]\tau},\quad
d_p=\frac{\log(s/K)+[r+(p-1/2)\sigma^2]\tau}{\sigma\sqrt\tau}.}
$$
All real powers have finite moments in this <lognormal distribution>. If $K=0$, the indicator is identically one and $V_p=M_p$. The function $V_p$ is an undiscounted conditional moment; the option prices below additionally include the <discount factor>.
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