What does "Gamma profit/loss" mean?

Question

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 1

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.