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I assume you work in the Black Scholes framework. Then, \begin{align*} P(S_0,K,T) = Ke^{-rT}\Phi(-d_2)-S_0\Phi(-d_1), \end{align*} where \begin{align*} d_1 &= \frac{\ln\left(\frac{S_0}{K}\right)+\left(r+\frac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}, \\ d_2 &= \frac{\ln\left(\frac{S_0}{K}\right)+\left(r-\frac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}= ...


overall gamma is second derivative of whole portfolio over underlying. adding any function (such as underlying*constant) which second derivative is 0 does not alter overall second derivative.

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