= Black-Scholes parameter sensitivities
{c}
{title2=$p_\sigma=\sigma(T-t)x^2p_{xx},\quad p_\rho=(T-t)(xp_x-p)$}
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 $p_{xx}\geq0$, so increasing <spot volatility> cannot lower the price. The <interest rate> sensitivity depends on the sign of the cash position $p-xp_x$. Calendar-time sensitivity is $p_t=\rho(p-xp_x)-\tfrac12\sigma^2x^2p_{xx}$. Thus a convex payoff with a negative cash position has a nonincreasing price at fixed spot when $\rho\geq0$. Negative rates invalidate that conclusion: $f(x)=x-K$ gives $p=x-Ke^{-\rho(T-t)}$ and $p_t=-\rho Ke^{-\rho(T-t)}>0$ when $\rho<0$.
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