= European power put formula
{title2=$P_n=e^{-r\tau}\sum_{j=0}^n(-1)^j\binom njK^{n-j}S^j e^{jr\tau+j(j-1)\sigma^2\tau/2}\Phi(-d_j)$}
In the dividend-free <Black-Scholes model>, let $\tau=T-t>0$, $K>0$, $S>0$, $\sigma>0$ and
$$
d_j=\frac{\log(S/K)+[r+(j-\tfrac12)\sigma^2]\tau}{\sigma\sqrt\tau}.
$$
Then the displayed <finite sum> prices the <European option> with <power put option> payoff. Here $\Phi$ is the <standard normal distribution function>. Indeed, $\log S_T$ conditionally has a <normal distribution> with mean $\log S+(r-\sigma^2/2)\tau$ and <variance> $\sigma^2\tau$. Expand $(K-S_T)^n$ by the <binomial theorem> on $S_T<K$ and apply the lower <truncated lognormal moment> to each term. Equivalently,
$$
P_n=e^{-r\tau}\mathbb E[(K-SY)_+^n],\qquad Y=\exp[(r-\sigma^2/2)\tau+\sigma\sqrt\tau Z],\quad Z\sim N(0,1).
$$
This <expectation> representation proves monotonicity and convexity in $S$. It also gives $P_n(0,t)=K^ne^{-r\tau}$ and $P_n(S,t)\to0$ as $S\to\infty$ by the <dominated convergence theorem>. For positive $S$ the <lognormal distribution> smoothing makes the price smooth and strictly positive.
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