Why American Option Gamma Remains Positive Under Black–Scholes

Summary

The answer addresses whether gamma is positive for American calls and puts under Black–Scholes, including settings with continuous or discrete dividends. Its reasoning is that American option values retain homogeneity of degree one in spot and strike because the optimal exercise time does not break that scaling property.

Using this homogeneity, the answer relates dollar gamma to a risk-neutral probability density evaluated at the optimal exercise date rather than a fixed maturity date. Since the density is positive, it concludes that American option gamma is positive. The excerpt gives a conceptual argument, not a derivation, numerical evidence, or a detailed treatment of dividend and model assumptions; its claim is framed within Black–Scholes and notes related stochastic-volatility models.

Key ideas

• American option values preserve degree-one homogeneity in spot and strike under the stated framework.
• The optimal exercise time does not remove that homogeneity property.
• Dollar gamma is linked to a risk-neutral density evaluated at the optimal exercise date.
• The answer concludes that American option gamma is positive under these assumptions.

Tags

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


# 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)?

## Answer by user34971 (score 3, accepted)

https://quant.stackexchange.com/a/51673

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.

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