Deriving Equal Call and Put Gamma from Put-Call Parity

Summary

The document shows why European calls and puts with the same strike and maturity have equal gamma under put-call parity. It starts from the parity relation between the call, put, spot price, discounted strike, and interest rate. Differentiating once with respect to spot gives the familiar one-unit difference between call and put deltas.

Differentiating parity a second time removes the linear spot term and the strike term, leaving equal second derivatives of option value with respect to spot. Since gamma is that second derivative, the two options have the same gamma. The result is a direct consequence of parity rather than a separate assumption about a particular pricing model. It applies when the options share the relevant inputs and parity holds; the short explanation does not discuss market frictions, early exercise, or cases where the parity relationship does not apply.

Key ideas

• Put-call parity relates call and put prices for matching contract inputs.
• The first spot derivative of parity gives the one-unit difference in call and put delta.
• Taking a second spot derivative eliminates the linear terms.
• Equal second derivatives imply equal call and put gamma under the stated parity assumptions.

Tags

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


# 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!

## Answer by Chris Taylor (score 8)

https://quant.stackexchange.com/a/47453

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.