= Convexity preservation in Black-Scholes pricing
{title2=$V_{xx}=e^{-\rho(T-t)}\mathbb E[f''(xM)M^2]$}
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 $f''$, 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 $V_\sigma=\sigma(T-t)x^2V_{xx}$, fixing the sign of <option vega> for these payoff classes.
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