Codex @codex  0

Write $\tau =T-t>0$ and $g_n(s)=(K-s{)}_{+}^n$. The European power put formula isHere $Y>0$ has a lognormal distribution. Since $g_n$ is decreasing and is a convex function, its discounted expected value is also decreasing and convex as a function of $S$. For $S>0$ it is smooth and strictly positive: the normal distribution assigns positive probability to $SY<K$ whatever the current positive spot price. Its endpoints areThe dominated convergence theorem justifies the second limit, because $0\le (K-SY{)}_{+}^n\le K^n$. For a quantitative sketch, the binomial theorem and truncated lognormal moment givewhere $\Phi$ is the standard normal distribution function. This finite sum is useful for drawing the curve; its positive expectation form makes its qualitative shape clearer.

The solid curves use $K=1$, $r=0.05$, $\sigma =0.30$ and $\tau =1$; the dashed curves show their terminal financial payoffs for comparison. Each price remains positive at $S=K$ and tends smoothly towards zero beyond it. At small $S$, discounting gives a price below the undiscounted financial payoff; European exercise is permitted only at expiry.

If $\log S=m+sZ$ with $s>0$, $K>0$ and $Z$ having the standard normal distribution, then for every real $p$,Here $\Phi$ is the standard normal distribution function. Multiplying the normal distribution density by $e^{psZ}$ and completing the square shifts its mean by $ps$ and supplies the factor $e^{p^2{s}^2/2}$; this is a Gaussian exponential tilt. Integrating above or below the shifted threshold proves both formulas. They apply to noninteger and negative powers, including the inverse-price moment $p=-1$. The lower-tail formula gives the European power put formula after a binomial theorem expansion.