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

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

$$\Delta_{t + 1} \approx \Delta_t + \Gamma_t \left( S_{t + 1} - S_t \right)$$

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 $t$.

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.

• The option is near ATFM and has only 1 month left, so the curve of Gamma has a peak and you are near that peak.. The Gamma curve decreases sharply on either side (higher or lower F), towards "more usual" values. You might try to plot it, it is interesting to see. – noob2 Sep 1 '16 at 12:04
• @LocalVolatility – Ultimate LaForsch Sep 1 '16 at 15:26
• @LocalVolatility Thank you for the explanation. Maybe one follow-up question if I may, how would one define a reasonable small change, so I could use the gamma values for my calculation (or would know when I would have to use gamma values for different price levels to calculate an estimated delta change for a larger price move of the underlying) ? – Ultimate LaForsch Sep 1 '16 at 16:04
• @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