# Is gamma always positive for American call/put options under Black-Scholes framework?

Most reference I could find only consider European options, but I would like to know whether this also holds for American options in general (with continuous dividend yield and/or discrete dividends)?

• Properties of American option prices by Erik Ekström: "assume that the pay-off function g is convex. Then the American option price P(s,t) is convex in the underlying s" so therefore $\Gamma$ is always >=0. – noob2 Mar 16 '20 at 21:34
• please see the discussion here: quant.stackexchange.com/questions/48813/… – Magic is in the chain Mar 16 '20 at 23:24
• @noob2 I read that paper. It doesn't consider dividends. – user69818 Mar 17 '20 at 1:22
• @Magicisinthechain I've read that question. It's different from what I ask. – user69818 Mar 17 '20 at 1:23

This is a good question.

See my answer to a question here

The point is that under Black-Scholes (and also many SV models) not only European prices but also American options prices are homogeneous of degree 1 in strike and spot as the optimal exercise time does not affect the homogeneity property in strike and spot price.

Hence also for American options the dollar gamma is the risk-neutral probability density (where maturity date $$T$$ is replaced by optimal exercise date $$\tau$$), which is always positive. So gamma for Americans is always positive.

• Great answer. This sounds to me as if dollar gamma (= $\Gamma\cdot S_t^2$) is the same for Euopean and American options. Is this true? As a consequence, dollar delta (= $\Delta\cdot S_t$) would also be the same? Can we say something about ''normal'' Delta and Gamma? Like do American puts have a more negative delta than European puts (prior to exercise when the delta becomes constant)? What about normal gamma? – Alex Mar 17 '20 at 13:30
• I am not sure how to make a comment out of an answer actually. But to answer your question: No I do not think the gamma and/or delta of American and European options are equal. My answer is to argue that the gamma of American calls/puts is always greater than zero. – Frido Rolloos Mar 17 '20 at 16:07
• Thank you for your response! If we stick with the simpler delta which is the amount of stocks I need to trade in order to hedge, right? Shouldn’t the European put delta be lower (more negative) than an American put delta? – Alex Mar 17 '20 at 16:14
• Hmm, never really thought about this, and another good question. But are you sure about your statement? For a dividend paying stock the delta of a put option would be $-e^{-q (T-t)} N(-d_1)$, but when the American put is exercised before expiry date $T$ its delta would already be -1, whereas the European put would be less neative, e.g. -0.95. Or am I missing something? – Frido Rolloos Mar 17 '20 at 16:22
• So in your example, the European delta is lower! Before expiry and early exercise, would we hold the same amount of stocks in our hedge portfolio or would we the possibility of early exercise affect the number of shares we hold? – Alex Mar 17 '20 at 16:45