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.

  • 2
    $\begingroup$ The Greeks like delta, vega, rho, theta, have obvious pnl at attributions. Gamma is a little more involved - it's the pnl due to the extra delta you have accumulated as a result of the gamma position you hold. $\endgroup$
    – will
    Aug 16 '19 at 18:57
  • $\begingroup$ See any options textbook where they discuss Profit & Loss on an option position. Assuming only S and t change, the P&L can be approximated by the sum of three terms: The Delta P&L, the Gamma P&L and the Theta P&L. $\endgroup$
    – Alex C
    Aug 16 '19 at 19:10
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    $\begingroup$ quant.stackexchange.com/questions/41052/… $\endgroup$
    – Sanjay
    Aug 16 '19 at 19:37

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.

  • 1
    $\begingroup$ Short, precise and informative. Great! answer $\endgroup$
    – Sanjay
    Aug 17 '19 at 1:26
  • $\begingroup$ Thanks Sanjay, very kind of you! $\endgroup$ Aug 17 '19 at 12:55

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