 Black-Scholes parameter sensitivities

For a fixed twice-differentiable terminal payoff with sufficient growth control, differentiate its lognormal distribution pricing integral. Gaussian integration by parts gives the displayed identities. Convexity implies $p_{xx}\ge 0$, so increasing spot volatility cannot lower the price. The interest rate sensitivity depends on the sign of the cash position $p-x{p}_x$. Calendar-time sensitivity is $p_t=\rho (p-x{p}_x)-\frac{1}{2}{\sigma }^2{x}^2{p}_{xx}$. Thus a convex payoff with a negative cash position has a nonincreasing price at fixed spot when $\rho \ge 0$. Negative rates invalidate that conclusion: $f(x)=x-K$ gives $p=x-K{e}^{-\rho (T-t)}$ and $p_t=-\rho K{e}^{-\rho (T-t)}>0$ when $\rho <0$.