# How to prove Gamma is the same for a European call and European put with the same inputs?

I saw from a text "From put-call parity, call and put with the same inputs have the same gamma", but I don't see how put-call parity is related to Gamma. Can someone explain? Thanks!

Put-call parity says that a call and put (worth $$C$$ and $$P$$ respectively) with the same strike $$K$$ have the following relationship with the spot rate $$S$$, risk-free rate $$r$$, and time to maturity $$T$$ --

$$C - P = S - e^{-rT} K$$

Taking the first derivative with respect to $$S$$,

$$\frac{\partial C}{\partial S} - \frac{\partial P}{\partial S} = 1$$

which relates the delta of the call and put. Taking the second derivative,

$$\frac{\partial^2 C}{\partial S^2} - \frac{\partial^2 P}{\partial S^2} = 0$$

which implies that the call and put have the same gamma.

• LOL it's so easy. Thanks. Sep 2 '19 at 14:33