Dollar Gamma, Delta-Hedging P&L, and the One-Percent Convention
Summary
The document clarifies competing conventions for dollar gamma. It distinguishes dollar gamma, commonly expressed as gamma multiplied by the square of the underlying price, from the half-dollar-gamma coefficient used in the second-order approximation of delta-hedged P&L. The latter appears because the Taylor or Itô expansion has a one-half factor on the gamma term. For a price change, the gamma contribution to hedged P&L is proportional to half the gamma times the squared price move.
A second answer defines a one-percent dollar gamma measure by applying that quadratic P&L term to a one-percent relative price move. It notes that contract multipliers matter for instruments that are not one share per contract and connects the measure to P&L from a change in variance under a geometric Brownian motion assumption. The explanations expose a convention mismatch behind the question, rather than establishing one universal reporting definition. The variance relation and resulting P&L expression rely on the stated model assumptions and small-move approximation.
Key ideas
- Dollar gamma is commonly defined as gamma multiplied by the square of the underlying price.
- The gamma contribution to delta-hedged P&L includes a one-half factor from the second-order expansion.
- A one-percent dollar gamma convention evaluates the quadratic P&L term for a one-percent relative price move.
- Contract multipliers must be included when translating option gamma into dollar exposure.
- The connection to variance-based P&L relies on a geometric Brownian motion framework.
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Full text
# Dollar gamma formula and its derivation
# Dollar gamma formula and its derivation
I am seeing two formulas:
- $gamma = 0.5 * gamma * (stock price ^ 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?
- Change in dollar delta for 1% move in the underlying price move.
- Additional additional dollar amount needed to remain in delta hedged for 1% move in the underlying price move.
Thanks
## Answer by nbbo2 (score 5, accepted)
https://quant.stackexchange.com/a/70071
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 by Dorian B. (score 3)
https://quant.stackexchange.com/a/75095
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://ibkr.co.uk/download/TWSGuide.pdf
Derivation using Ito's Lemma. Considering an option with risk-neutral pricing $ V=V(t,S) $ for a GBM process $ dS=r Sdt + \sigma S dW_t$, it's total derivative is:
$$ dV=\frac{\partial{V}}{\partial{t}}dt+\frac{\partial{V}}{\partial{S}}dS+\frac{1}{2}\frac{\partial{V^2}}{\partial^2{S}}{dS}^2 $$
Where by definition, gamma is:
$$ \Gamma = \frac{\partial{V^2}}{\partial^2{S}} = \frac{\partial{\delta}}{\partial{S}} $$
The change in portfolio value from the gamma term (the last term in the Ito expansion) for a small change in price $ \Delta S $ is:
$$ \Gamma_{DV$} = { 1 \over 2 } \Gamma {\Delta S}^2 = { 1 \over 2 } \Gamma S^2 \left(\frac{\Delta S}{S}\right)^2 $$
By definition gamma dollars is the change for a 1% change ie $ \frac{\Delta S}{S} = 1\% $. Thus the formula in the beginning. QED.
The definition with 1% is useful in practice in that it can be multiplied by the change in term variance directly to get the PnL number for a bump in volatility. From the GBM process we have:
$$ \left(\frac{\Delta S}{S}\right)^2 = \sigma^2 \Delta t$$
Resulting finally in:
$$ \Delta PnL = \Gamma_{DV$} \left(\sigma_1^2-\sigma_0^2\right)(T-t)$$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.