# Binary Put Options Gamma

Binary put options gamma is the metric that describes the change in the delta due to a change in the underlying price. Binary put gamma is the first derivative of the binary put delta with respect to a change in the underlying price and is depicted as:

#### $\Gamma = \frac{d\Delta}{dS}$

where  $\Gamma$ is gamma and $\Delta$ is the tunnel delta.

The binary put gamma is subsequently the gradient of the delta profiles of Figures 1 & 2 on Binary Put Delta.

What is the relevance of binary put gamma? Gamma, whether discussing binaries or conventionals, gives the rate of change of exposure to the underlying.

For example: If one is long an out-of-the-money put the delta may be ―0.30. A long 100 contract put position would then be equivalent to short 30 futures. This position will be long gamma which means if the underlying falls as far as the strike the delta will become increasingly negative. Just prior to the strike, if the delta is now ―0.60 then the position would be equivalent to short 60 futures.

On the other hand, if the underlying price rose then the position would remain long gamma until it approaches zero. This would mean that as the underlying rose the delta would fall to, maybe ―0.10, so that the equivalent position is now short just 10 futures. If the position was established as delta-neutral then 30 futures would also be bought along with the puts. Then at the lower underlying the net delta would have been ―0.30 (instead of ―0.60) and at the higher underlying the net delta would have been +20.

So, on the way down the position would be equivalent ―30≤Futures≤0, i.e. profitable. On the way up it would have been 0≤Futures≤20, also profitable.

Negative delta indicates that if the underlying rises one would become progressively shorter futures, while on the way down progressively long futures.

### Binary Put Gamma Over Time

Binary put gamma is displayed against time to expiry in Figure 1. As with binary call gamma, the binary put gamma is positive when out-of-the-money and negative when in-the-money. Gamma is zero when at-the-money.

The amount of time to expiry has a major influence on the absolute value of the gamma. With very short-term options gamma tends to ±∞. Alternatively, the example of Figure 1 shows gamma to be pretty much flat at zero for the 25-day option. Figure 1 – Binary Put Options Gamma w.r.t. Time to Expiry

### Binary Put Gamma Over Time

Figure 2 provides binary put gamma over a range of implied volatilities. As the implied volatility falls from 45% to 5% the peak and trough of the gamma profile close in on the strike reflecting the steepening of the delta which in turn signals the increasingly higher gearing of the asset price.