Why Option Gamma Can Exceed One
Summary
The document clarifies the meaning of gamma in an option hedging simulation. Gamma is the rate at which delta changes as the underlying price changes, but multiplying gamma by a price move gives only a local approximation to the new delta. That approximation works best when both the price change and elapsed time are small.
Gamma itself changes with the underlying price and time, so a large move cannot be extrapolated by treating the initial gamma as fixed. A delta that is bounded between zero and one can still have a derivative greater than one at a point; boundedness of a differentiable function does not require its derivative to share the same bounds. The example involves an at-the-money natural gas futures option and reports a gamma above one, but the answer focuses on interpretation rather than validating the specific calculation or its units. The practical lesson is to use local Greeks cautiously for large moves and account for their changing values.
Key ideas
- Gamma measures the local rate of change of option delta with respect to the underlying price.
- A gamma-based delta estimate is most accurate for small price and time changes.
- Gamma varies with the underlying price and time, so it should not be held constant over a large move.
- A bounded delta function can have a derivative, gamma, that exceeds one.
- The explanation does not independently check the example calculation or its units.
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# Interpretation of an option gamma larger than one
# Interpretation of an option gamma larger than one
I am working on an option hedging simulation. In this context, I wanted to expand the simulation to include gamma. For testing purposes, I used among others the natural gas futures. When I calculate the greeks for an ATM-option for the current October contract
(f=2.86, st=2.90, days=30, vol=0.4, r=0.005),
I get a gamma of 1.21.
As far as I know, the gamma as the second partial derivative of the option price with respect to the underlying price, gives you the rate of change for the delta with respect to a change in the underlying. So if the natural gas futures would change by one point (which obviously would be a very large percentage move), the delta would change by more than one even if it has the boundaries of [0, 1]. Is my definition, calculation or interpretation of gamma not correct or am I missing something else ? (Before posting this, I looked gamma up in some option textbooks like Hull, Sinclair and Natenberg but could not find an explanation regarding this)
Thanks for help (and sorry for my trivial question ;-)
## Answer by LocalVolatility (score 3, accepted)
https://quant.stackexchange.com/a/29933
You are correct in saying that gamma represents the rate of change of the delta with respect to changes in the underlying asset price. However, the approximation
\begin{equation} \Delta_{t + 1} \approx \Delta_t + \Gamma_t \left( S_{t + 1} - S_t \right) \end{equation}
is only accurate when the changes $S_{t + 1} - S_t$ and $\Delta t$ are small. The reason is that gamma itself is not constant but a function of $S$ and $t$.
In general, a bounded and differentiable function (delta in your case) does not need to have a bounded derivative (gamma in your case). See for example this question.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.