# Is American put Gamma always greater than the European one in the non-early-exercise domain?

Consider a pair of American and European puts with the same specifications except the former has the continuous early exercise right. Has anyone plotted the Gamma's of both as functions of the underlying price and time to expiry for the underlying greater than the critical exercise price? Is the American put Gamma necessarily greater than or equal to that of the European counterpart in this domain? I would like a mathematical proof if it is true. I suspect the negative answer may predominantly come from the region where the underlying is close to and above the critical exercise price.

• Does this question help? May 11, 2023 at 14:48
• @Kevin: Yes, it does. Thank you. I am now more interested in a proof if this inequality is always true.
– Hans
May 11, 2023 at 16:00

• This is false. $\frac{\partial^2 P}{\partial S^2}$ is not necessarily continuous across the exercise, or free, boundary. In fact, in most cases it is discontinuous across the boundary. Your diagram just shows the continuity of $\frac{\partial P}{\partial S}$.