Long Gamma, Convexity, and the Cost of Theta Decay
Summary
The explanation clarifies that being long gamma means holding a position with positive sensitivity of delta to the underlying price, rather than simply benefiting whenever gamma itself rises. In an option’s local price expansion, the gamma contribution is proportional to the squared change in the underlying. This makes that contribution positive for either an upward or downward move when the position has positive gamma, assuming other effects are set aside.
The discussion uses the Black–Scholes expansion to motivate this convexity interpretation and points out the main trade-off: long options generally carry negative theta, so time decay can offset gains from movement. It does not quantify the balance or account for changes in implied volatility, rates, transaction costs, or larger-move effects. The explanation is conceptual and should not be read as a guarantee that a long-gamma position will have positive total profit.
Key ideas
- Gamma measures how an option’s delta changes as the underlying price moves.
- A positive-gamma position has a positive second-order price contribution for moves in either direction.
- Long calls and puts can provide positive gamma exposure.
- Long options generally incur negative theta, which can erode gains during quiet markets.
- Gamma’s contribution alone does not determine total profit because other pricing effects also matter.
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# What does it mean to be long gamma?
# 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. 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 by strimp099 (score 23, accepted)
https://quant.stackexchange.com/a/2591
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.