Digital put gamma is the metric that describes the change in the delta due to a change in the underlying price. Digital put gamma is the first derivative of the digital put delta with respect to a change in the underlying price and is depicted as:
where 
is gamma and 
is the tunnel delta.
The digital put gamma is subsequently the gradient of the delta profiles of Figures 1 & 2 on Digital Put Delta.
Gamma Relevance
What is the relevance of digital put gamma? Gamma, whether discussing digitals 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.
Digital Put Gamma Over Time
Digital put gamma is displayed against time to expiry in Figure 1. As with digital call gamma, the digital 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.
| European Digitals | Digital Put Options | Digital Put Delta | Digital Put Theta | Digital Put Vega |
Digital Put Option Gamma and Volatility
Figure 2 provides digital put gamma over a range of implied volatilities. As the implied volatility falls from 18% to 2% 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.
Digital put gamma is zero when at-the-money so that as the underlying passes through the strike the position will change from long gamma to short gamma, and vice versa. Trading a digital options book as a market maker is fraught with difficulties as options pass through the strike and the option takes on completely new characteristics.
Gamma is an extremely important measure of trading in the options trading industry. Traders define themselves by their attitude to gamma referring themselves as a long gamma or short gamma player. Irrespective of market conditions it is possibly irrational; should not a short gamma trader adopt to volatile markets and become a long gamma trader? Possibly. But never on this site will you get support for the rational expectations theory! As a ‘local’ options trader for numerous years, the rational behaviour of markets has never, and probably never will, exist.
Summary
- This gamma is positive when above the strike, out-of-the-money, and negative below the strike when in-the-money.
- At-the-money gamma is always zero.
- If the corresponding binary put delta can go to negative infinity you can be sure the gamma can go to both +ve and -ve infinity.