Why is it that Gamma is always positive

CFA Level III Derivatives
#1

Does anyone know why gamma is always positive for call and puts?

#2

Gamma is positive for “long” calls and puts (when you buy options). Perhaps someone can explain it better than me, but here goes:

  1. Gamma = measure of how Delta changes when the price of the underlying stock moves up or down. It is a change measurement for Delta. (Remember that Delta = change in the option value due to a change in the underlying stock price).

  2. When you buy options, you’re adding positive Gamma. When you sell options, you’re adding negative Gamma. You can be Gamma neutral if you combine buying and selling options into one strategy.

  3. Long story short - positive Gamma for long calls and puts means: (a) the Delta of “long” calls will become MORE positive (approaching +1) when the stock price rises, and LESS positive (approaching 0) when the stock price falls; and (b) the Delta of “long” puts will become MORE negative (approaching -1) if the stock price falls, and LESS negative (approaching 0) when the stock price rises.

That’s as I understand it anyway, someone might have a better explanation.

Cheers - good luck - you got this :+1:

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#3

Makes sense. Thanks.

#4

The price curves for call options and put options are convex (mathematicians would say, "concave upward). The second derivative of a convex (concave upward) function is positive; the slope of the tangent line (delta) is increasing from left to right.

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#5

Thanks.

#6

Don’t ignore the “squared” in the second order derivative.

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