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 ;-)

• I did the calculation with your input values and Gamma of about 1.21 appears to be correct. – noob2 Sep 1 '16 at 11:58

$$\Delta_{t + 1} \approx \Delta_t + \Gamma_t \left( S_{t + 1} - S_t \right)$$
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 $t$.
• @Ultimate LaForsch: This again depends on the sensitivity of the gamma w.r.t. to changes in the underlying asset price. This is often referred to as the "speed". A second order approximation of the change in delta would thus be $\Delta_{t + 1} \approx \Delta_t + \Gamma_t \left( S_{t + 1} - S_t \right) + \frac{1}{2} \mathcal{S}_t \left( S_{t + 1} - S_t \right)^2$ where I used $\mathcal{S}_t$ for the speed. – LocalVolatility Sep 4 '16 at 11:35