American and European Put Gamma Near the Exercise Boundary

Summary

The document asks whether an American put must have Gamma at least as large as the Gamma of a matching European put when early exercise is not optimal. The response challenges that proposed ordering by appealing to continuity across the American option’s free boundary. If the Greeks vary continuously there, a claim that one Gamma is always no smaller in the continuation region can be tested at points approaching the boundary; the answer says a diagram makes a counterexample apparent.

The excerpt is incomplete: it supplies no diagram, formula, numerical example, or rigorous proof, and ends before detailing the argument. It therefore raises a useful caution about assuming Greek comparisons from the early-exercise feature alone, but does not establish the precise conditions or extent of the failure. Readers seeking a formal result would need the omitted material or an independent derivation.

Key ideas

• The question concerns Gamma ordering for otherwise identical American and European puts in the continuation region.
• The response uses continuity near the free boundary to dispute a universal greater-than-or-equal claim.
• The excerpt refers to a diagram but does not include it or provide a complete mathematical proof.
• Any conclusion should be limited because the argument and its conditions are only partially presented.

Tags

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


# 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.

## Answer by dm63 (score -1)

https://quant.stackexchange.com/a/75521

I think the argument of continuity as suggested by the deleted post does apply. American options should be continuous in all their Greeks across “boundaries”, because it is a free boundary. Given continuity , the statement fails. For example the following diagram makes it pretty obvious. Not a math proof I

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