# What does it mean to be long gamma?

When you are "long gamma", your position will become "longer" as the price of the underlying asset increases and "shorter" as the underlying price decreases.

My intuition tells me that if you're long gamma, all that means is that if gamma increases, so does the value of your portfolio. Correct me if I'm wrong, but this seems to conflict with the quoted definition above (it is possible for gamma to decrease while the value of your portfolio goes up). Am I totally wrong? Does being long gamma simply mean your portfolio has a positive gamma as the quoted definition suggests?

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Long gamma means that the gamma of your portfolio is positive. Your gamma could get shorter (i.e. smaller, but still positive) while you make money and vice versa. – Tal Fishman Dec 14 '11 at 1:19
I see, that makes perfect sense now. I guess my intuition was wrong (wouldn't be the first time, hehe). – sooprise Dec 14 '11 at 1:26

## 2 Answers

Gamma is the second partial derivative of the change in the price of the option wrt to the change in the underlying. Said another way, it is the change in delta. If you write down the Black-Scholes pricing formula, you's see the gamma term:

$$...\frac{1}{2}\frac{\partial^2C}{\partial S^2}(\Delta S)^2...$$

Notice that the $\Delta S$ (change in stock price) term is squared, meaning that the gamma term is positive when long regardless if $\Delta S$ is positive or negative. (This comes from the derivation of BS using Ito's Lemma.) What this means is that if you are long gamma (long a call or put option) then the P/L attributed to your position from gamma will increase regardless of the direction the stock moves.

Gamma (convexity) is a gift from God in this regard when the payoff is nonlinear, but remember there is no free lunch. The theta of a long option position is negative and will erode your P/L at the same time - faster than you will accumulate P/L from gamma if you are not careful.

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Very good explanation! – sooprise Dec 14 '11 at 17:04
Theta is not always negative: quant.stackexchange.com/questions/16525/… – emcor Jun 14 '15 at 18:20

When you're betting on the value of a stock to go up using an option (long call, short put), you're going to have a positive delta and positive gamma position -- because you're betting that if the underlying will rise, then the value of your option will increase.

When you're short the stock with an option (short call, long puts), you're betting on the value of the stock to decline. Because of this relationship, you're short delta and gamma because a change in the underlying price causing an opposite change in the value of the option.

So when you think of "long gamma" just think that you're betting on the underlying to go up. When you're "short gamma" just think that you're betting against the underlying.

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This is all wrong, by the way. Short puts have negative gamma, not positive! Long option => long gamma, short option => short gamma – frickskit Oct 15 '13 at 22:11