= Solution
Let $T=t_0$, $\tau=T-t$, and let $N$ have the <standard normal distribution>. For a twice <differentiable> payoff with enough growth control for the following <expectations> and derivatives, <risk-neutral pricing> in the dividend-free <Black-Scholes model> gives
$$
\boxed{V(x,t)=e^{-\rho\tau}\mathbb E[f(xM)],\qquad
M=e^{(\rho-\sigma^2/2)\tau+\sigma\sqrt\tau N}.}
$$
The conditional discounted payoff is a <martingale> under the <risk-neutral measure>. Applying the <Itô formula> to $e^{-\rho s}V(S_s,s)$ and setting its <drift> to zero therefore gives the <Black-Scholes equation>
$$
\boxed{V_t+\frac12\sigma^2x^2V_{xx}+\rho xV_x-\rho V=0,\qquad V(x,T)=f(x).}
$$
The <delta hedge> holds $\Delta=V_x$ shares; the remaining bond value is $\beta=V-x\Delta$, or $h=\beta/e^{-\rho(T-t)}$ units of the <zero-coupon bond>. Substitution in the <Itô formula> and the displayed equation gives $dV=\Delta\,dS+h\,dB$, proving replication and <self-financing>.
Differentiating the pricing expectation in the current <stock> price gives its <option delta> and <option gamma>:
$$
\Delta=e^{-\rho\tau}\mathbb E[f'(xM)M],\qquad
\Gamma=e^{-\rho\tau}\mathbb E[f''(xM)M^2].
$$
Thus a nondecreasing payoff has a nondecreasing price; a nonincreasing payoff has a nonincreasing price. A <convex> payoff has $\Gamma\geq0$, and a <concave> payoff has $\Gamma\leq0$. This is <convexity preservation in Black-Scholes pricing>.
The <Black-Scholes parameter sensitivities> follow directly from the same expectation. Differentiation in the <interest rate> gives the <option rho>
$$
\boxed{V_\rho=\tau(x\Delta-V)=-\tau\beta.}
$$
For <spot volatility>, differentiation first gives
$$
V_\sigma=e^{-\rho\tau}\mathbb E[f'(xM)xM(\sqrt\tau N-\sigma\tau)].
$$
The <Gaussian integration by parts> identity $\mathbb E[N\psi(N)]=\mathbb E[\psi'(N)]$, applied to $\psi(N)=f'(xM)xM$, cancels the term containing $f'$ and gives the <option vega>
$$
\boxed{V_\sigma=\sigma\tau x^2\Gamma.}
$$
Consequently increasing <spot volatility> increases a <convex> payoff's value and decreases a <concave> payoff's value, with weak inequalities in affine or degenerate cases. A short bond position gives positive <option rho>, while a long bond position gives negative <option rho>. The physical <stock> <drift> $\mu$ does not enter these prices: the <risk-neutral measure> replaces it by $\rho$.
The <option theta> is the calendar-time derivative at fixed current <stock> price. The <Black-Scholes equation> gives
$$
\boxed{V_t=\rho\beta-\frac12\sigma^2x^2\Gamma.}
$$
For $\rho\geq0$, a <convex> payoff with $\beta<0$ therefore has $V_t\leq0$, while a <concave> payoff with $\beta>0$ has $V_t\geq0$. The signs are strict when $\rho>0$. At $\rho=0$, an affine payoff can give equality. Remaining-maturity sensitivity is $V_T=-V_t$ when the payoff is independent of $T$. All these time comparisons hold at fixed current <stock> price.
The stated time-monotonicity conclusion needs the nonnegative-rate condition. For example, the <convex> affine payoff $f(x)=x-K$ has
$$
V=x-Ke^{-\rho(T-t)},\qquad \beta=-Ke^{-\rho(T-t)}<0,\qquad
V_t=-\rho Ke^{-\rho(T-t)}>0\quad(\rho<0).
$$
Thus with negative <interest rates> it provides an explicit counterexample even though the replica is short in bonds.
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