Why Dollar Gamma Scales with the Square of Spot Changes
Summary
The document explains why gamma exposure is expressed using the square of a change in the underlying price. It uses dimensional analysis: option-price changes are measured in dollars, while gamma measures the second derivative of option value with respect to spot and therefore has inverse-price-squared units.
To make the gamma contribution comparable to an option-price change, multiply gamma by the squared spot move. The resulting dollar gamma has dollar units, just as delta’s contribution becomes dollar delta when delta is multiplied by the spot move. The explanation is conceptual and gives no empirical test or trading rule; it focuses on making terms in a price-change expansion dimensionally consistent.
Key ideas
- Gamma has units of option value per squared unit of spot price.
- Multiplying gamma by the squared spot move gives a contribution measured in dollars.
- Delta also needs to be multiplied by the spot move to express its price contribution in dollars.
- Terms added in a price-change expansion must have compatible dimensions.
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# Why is gamma exposed through the square of spot prices?
# Why is gamma exposed through the square of spot prices?
As per this article, "the mathematically intuitive way to expose gamma is through the square of the underlying price": https://llllvvuu.dev/blog/unbundling-gamma
Can someone explain this? Thank you!
## Answer by user34971 (score 9, accepted)
https://quant.stackexchange.com/a/66488
Dimensional analysis is the key:
The change in option price is in dollars. The change in option price is of course the sum of its changes (partial derivatives) with respect to its underlying risk factors. However you cannot add terms with different dimensions, that would literally be trying to add apples and oranges.
Let's look at delta, which (in finite difference notation) is $\frac{\Delta C}{\Delta S}$ where $S$ is the spot price. The contribution of delta to the change in option price is not delta, but "dollar delta", which is $\frac{\Delta C}{\Delta S} \Delta S$, because then the product is in the same units as the change in option price $\Delta C$.
Now Gamma is the second order change in the option price with respect to spot, which is $\frac{\Delta^2 C}{\Delta S^2}$. In order to be able to include this in the series that gives as sum the change in option price (units in dollars), the Gamma has to be multiplied by $\Delta S^2$. The product of the two $\frac{\Delta^2 C}{\Delta S^2} \Delta S^2$ is called "dollar gamma".
Summary: when adding quantities the quantities must have the same dimension in order for the sum to make sense.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.