Dollar gamma formula and its derivation

Question

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

Answer 1

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.

Answer 2

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