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.
source: http://www.optiontradingtips.com/greeks/gamma.html
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?
Answer
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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