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I am seeing two formulas:

  1. $gamma = 0.5 * gamma * (stock price ^ 2)

  2. $gamma = gamma * (stock price ^ 2)

Not sure where this 0.5 term is coming from.

And also, what is the correct definition of dollar gamma?

  1. Change in dollar delta for 1% move in the underlying price move.
  2. Additional additional dollar amount needed to remain in delta hedged for 1% move in the underlying price move.

Thanks

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Here is how I remember it: In the famous paper by Carr and Madan Towards a theory of volatility trading the term $\frac{\Gamma S^2}{2}$ is referred to as "half the dollar gamma" so the dollar gamma is $\Gamma S^2$. Carr was the world's foremost expert on volatility trading (RIP) and the main result in that paper is worth memorizing.

What is the definition? The Dollar Gamma comes in if we consider the P&L on a hedged position when the stock price changes by dS. It can be shown that this P&L is proportional to (dS)^2. The constant of proportionality is half the Dollar Gamma, and the Dollar Gamma (as mentioned) can be shown to be $\Gamma S^2$.

For a derivation and a clear description of how it all ties together see this web page Delta Hedging, Gamma and Dollar Gamma.

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The correct formula is: $$ \Gamma_{DV$} = { 1 \over 2 } \Gamma (S * 1 \%)^2 $$

Gamma dollars is the change in the delta dollars for a 1% change in underlying around price S. Depending on what you're trading, you will need to include the contract multiplier next to S as only for stocks it's 1:1.

Source: TWS Guide - Report Metrics - Page 1010 of 1742 https://www.interactivebrokers.com/download/TWSGuide.pdf

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  • $\begingroup$ It is interesting to see how IB defines it. (I upvoted). But note that they refer to it as "gamma dollars" rather than "dollar gamma" which is a slightly different term FWIW. $\endgroup$
    – nbbo2
    Apr 2, 2023 at 13:38
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    $\begingroup$ It is not IB's definition. Just they know it very well as it's their bread and butter. This can also be deduced from first principles using differential forms. Then if you replace the dS/S with 1% you end up with exactly this formula. Also, dollar gamma and gamma dollars are interchangeable. $\endgroup$
    – Dorian B.
    Oct 17, 2023 at 0:24

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