= Solution
Let $T=t_0$, $\tau=T-t$, $x=S_t>0$, and $\sigma>0$. Use a dividend-free <Black-Scholes model> with constant <interest rate> $\rho$. The terminal payoff is assumed independent of the parameters being varied. In addition to twice differentiability, impose enough <integrability> to differentiate the pricing formula; for example, $f,f',f''$ of polynomial growth suffice. Twice differentiability alone is not enough: the smooth payoff $f(x)=e^{x^2}$ has infinite <expectation> under every nondegenerate <lognormal distribution>.
Under the <risk-neutral measure> $Q$, future <stock> values conditional on $S_t=x$ have the form
$$
S_T=xM,\qquad M=\exp\left((\rho-\sigma^2/2)\tau+\sigma\sqrt\tau Z\right),\qquad Z\sim N(0,1).
$$
<Claim replication>, or equivalently the <martingale> property of discounted value in the <complete market>, gives
$$
\boxed{p(x,t)=e^{-\rho\tau}\mathbb E[f(xM)]
=\frac{e^{-\rho\tau}}{\sqrt{2\pi}}\int_{\mathbb R}f\left(xe^{(\rho-\sigma^2/2)\tau+\sigma\sqrt\tau z}\right)e^{-z^2/2}\,dz.}
$$
The <replicating strategy> holds the <option delta> $p_x$ shares and bank-account value $\beta=p-xp_x$. No physical <drift> parameter appears: the <Girsanov theorem> replaces the physical <drift> by $\rho$ under the pricing measure, and the replication argument cancels it directly. The <risk-neutral pricing> operator is linear in the payoff and preserves pointwise payoff inequalities.
Here is a verification of the <Black-Scholes equation> using the integral itself. Differentiation in spot gives
$$
p_x=e^{-\rho\tau}\mathbb E[Mf'(xM)],\qquad
p_{xx}=e^{-\rho\tau}\mathbb E[M^2f''(xM)].
$$
Differentiating with respect to $\tau$ gives
$$
p_\tau=-\rho p+e^{-\rho\tau}\mathbb E\left[S_Tf'(S_T)
\left(\rho-\frac{\sigma^2}{2}+\frac{\sigma Z}{2\sqrt\tau}\right)\right].
$$
For $G(Z)=S_Tf'(S_T)$, the <Gaussian integration by parts> identity $\mathbb E[ZG]=\mathbb E[G']$ yields
$$
\mathbb E[Z S_Tf'(S_T)]
=\sigma\sqrt\tau\,\mathbb E[S_Tf'(S_T)+S_T^2f''(S_T)].
$$
Substitution cancels the $-\sigma^2/2$ term and proves
$$
p_\tau=-\rho p+\rho xp_x+\tfrac12\sigma^2x^2p_{xx}.
$$
Since $p_t=-p_\tau$, this is precisely the <Black-Scholes equation>. As $\tau\downarrow0$, $M\to1$ and growth control permits passage to the limit, giving $p(x,T)=f(x)$. The <delta hedge> then supplies the admissible replicating gains under the same regularity assumptions.
The <Black-Scholes parameter sensitivities> describe the dependence on the model parameters without assuming a particular payoff. If $f$ is increasing, then $p_x\geq0$. If $f$ is <convex>, then $p_{xx}\geq0$, so its price is <convex> in spot. The <option gamma> $p_{xx}$ measures the response of the <option delta> to spot changes.
For the <interest rate>, the distribution of the future <stock> also changes with $\rho$, so looking only at discounting gives the wrong general answer. Differentiation gives the <option rho>
$$
\boxed{p_\rho=-\tau p+\tau e^{-\rho\tau}\mathbb E[S_Tf'(S_T)]
=\tau(xp_x-p)=-\tau\beta.}
$$
Thus a short bank-account position gives positive rate sensitivity, while a positive bank-account position gives negative rate sensitivity. For a cash payoff the result is negative; for the payoff $f(x)=x$ it is zero.
For <spot volatility>, differentiate the lognormal multiplier and then use <Gaussian integration by parts>:
$$
\begin{aligned}
p_\sigma&=e^{-\rho\tau}\mathbb E[S_Tf'(S_T)(\sqrt\tau Z-\sigma\tau)]\\
&=\sigma\tau e^{-\rho\tau}\mathbb E[S_T^2f''(S_T)]
=\boxed{\sigma\tau x^2p_{xx}}.
\end{aligned}
$$
Consequently a <convex> payoff has nonnegative <option vega>. This monotonicity can also be understood by a mean-preserving increase of dispersion in the discounted <stock>. Strict positivity requires nonzero curvature seen with positive probability; an affine payoff has zero <option vega>.
Time enters through the remaining maturity $\tau$. Holding spot fixed, the <Black-Scholes equation> gives
$$
\boxed{p_t=\rho\beta-\tfrac12\sigma^2x^2p_{xx},\qquad p_T=-p_t.}
$$
Under the conventional assumption $\rho\geq0$, a <convex> payoff and a short bond position $\beta<0$ imply $p_t\leq0$. The inequality is strict if $\rho>0$, or if $p_{xx}>0$ with positive volatility. Thus the price at fixed <stock> value is nonincreasing in calendar time, and nondecreasing in maturity, under these hypotheses. It is not a claim that the realized stochastic price path decreases: that path still has Brownian exposure $\sigma S_tp_x\,dW_t^Q$.
The printed conclusion needs the nonnegative-rate qualification; it is false for arbitrary negative rates. For $K>0$, the twice-differentiable <convex> payoff $f(x)=x-K$ has
$$
p=x-Ke^{-\rho\tau},\qquad\beta=-Ke^{-\rho\tau}<0,\qquad
p_t=-\rho Ke^{-\rho\tau}>0\quad\text{if }\rho<0.
$$
This is an exact counterexample, not a failure of the pricing formula. Also, with $\rho=0$ and an affine payoff, “decreasing” can only mean nonincreasing. Without payoff monotonicity or the specified bond-position sign, neither the spot nor rate nor time sensitivity has a universal sign. Any additional payoff parameter, such as a strike, is differentiated through $f$; pointwise payoff ordering, rather than a generic parameter rule, determines its price effect.
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