Attributing Option P&L to Gamma and Theta
Summary
The document explains gamma and theta profit or loss by approximating an option’s price change with a Taylor expansion. The approximation includes a delta term for the underlying price move, a gamma term proportional to the squared price move, and a theta term for elapsed time. Gamma P&L is thus the second-order price contribution, while theta P&L is the time contribution.
It also connects gamma P&L with the effect of delta rebalancing. For a long option, positive gamma makes the local gamma contribution positive for either direction of underlying move, but this does not make the position an assured profit: time decay can offset it. The explanation is a local approximation that omits higher-order terms and other possible sources of realized P&L, so it is a conceptual attribution rather than a full accounting method.
Key ideas
- A Taylor expansion separates approximate option P&L into delta, gamma, and theta contributions.
- Gamma P&L is the second-order contribution from the underlying price move.
- Theta P&L represents the option price change associated with elapsed time.
- Positive gamma can benefit a long option across either direction of price move, while theta decay can offset that benefit.
- The expansion is an approximation and leaves out higher-order effects.
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Full text
# What does "Gamma profit/loss" mean?
# What does "Gamma profit/loss" mean?
I understand the Greeks as derivatives, but I'm very confused with terms like "Gamma profit/loss""Theta profit/loss". What do these terminologies mean? I've searched online but can't find a proper definition.
## Answer by Magic is in the chain (score 4)
https://quant.stackexchange.com/a/47154
Think of this in terms of Taylor series. Let's say the option price today is $C\left(S,t\right)$ where S is the underlying price and t time. Let's say the underlying price changes by $\Delta S$ in a time interval $\Delta t$, so your P/L will be:
$\mathrm{P/L}=C\left(S+\Delta S,t+\Delta t\right)-C\left(S,t\right) $
Use Taylor series to first order in t and second order in S to approximate this P/L or to attribute it to factors:
$C\left(S+\Delta S,t+\Delta t\right)-C\left(S,t\right) \approx \frac{\partial C}{\partial S}\Delta S+\frac{1}{2} \frac{\partial^2C}{\partial S^2} \left(\Delta S\right)^2+\frac{\partial C}{ \partial t}\Delta t$
The second term on the right hand side is the gamma P/L and the last term is the theta P/L.
You will also hear the gamma P&L being called the P/L resulting from delta rebalancing. And please also google Cash Delta and Cash Gamma!
Aside: If you are long option, the gamma will be positive, which when multiplied by the square of the change means the gamma P/L will be positive. Pretty much similar to Bond convexity. So what gives? Theta decay, option maturity shrinks as time progresses.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.