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

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.