Put-call gamma parity

Question

A European call and a European put are written on the same underlying with the same strike $K$ and the same expiry. The call option has a gamma of $0.02$. What is the gamma of the put option?

Accepted Answer

For European options on the same underlying asset with the same strike $K$ and time to expiry $T$ , put-call parity gives:

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

where $C$ is the call price, $P$ the put price, $S$ the spot price, and $r$ the risk-free rate. Gamma is the second derivative of the option price with respect to $S$ :

\Gamma = \frac{\partial^2 V}{\partial S^2}

Differentiating put-call parity twice with respect to $S$ :

\frac{\partial^2 C}{\partial S^2} - \frac{\partial^2 P}{\partial S^2} = \frac{\partial^2}{\partial S^2}\left(S - K e^{-rT}\right) = 0

because $S$ is linear in $S$ (second derivative zero) and $K e^{-rT}$ is constant. Hence:

\Gamma_C - \Gamma_P = 0 \quad \Rightarrow \quad \Gamma_C = \Gamma_P

Given $\Gamma_C = 0.02$ , the put gamma is also $0.02$ .

Put-call gamma parity

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