Black-Scholes parameter sensitivities ($p_{\sigma }=\sigma (T-t)x^2{p}_{xx},p_{\rho }=(T-t)(x{p}_x-p)$)

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 $p_{xx}\ge 0$, 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 $\rho <0$.

In the Black-Scholes model the terminal stock price conditional on current price $x$ is $xM$ for a positive lognormal multiplier $M$. Differentiation under the risk-neutral pricing expectation gives the displayed option gamma. Its sign follows that of $f^{\prime \prime }$, so convexity and concavity of the terminal payoff are preserved. An analogous first-derivative identity preserves monotonicity. With appropriate growth conditions, Gaussian integration by parts also gives $V_{\sigma }=\sigma (T-t)x^2{V}_{xx}$, fixing the sign of option vega for these payoff classes.

The derivative of a claim price with respect to the interest rate, holding the payoff and other model parameters fixed. For a dividend-free Black-Scholes model with remaining time $\tau$, it is $\tau (x{p}_x-p)$, so its sign is opposite the bank-account value in the replicating strategy.

The second derivative of an option price with respect to its underlying price. It measures the change of option delta. A convex terminal payoff in the Black-Scholes model has nonnegative option gamma before maturity.