The digital call gamma is the first derivative of the digital call options delta with respect to a change in the underlying price.
where 
is gamma and 
is the digital call delta.
The gamma is the slope of the delta profile. The digital call gamma reflects whether a digital call’s equivalent futures position will become longer or shorter as the underlying price rises or falls.
The gamma of conventional calls is always positive. A long conventional call’s equivalent underlying position will always increase as the underlying rises, irrespective of whether the option is in-the-money or not. Digital calls do not behave in the same manner. The out-of-the-money digital call will always have positive gamma. The in-the-money digital call will always have negative gamma.
Digital Call Gamma Over Time
Figure 1 illustrates the 100.00 digital call gamma against days to expiry. When the digital call is deep in-the-money or far out-of-the-money the gamma tends to zero, just like the conventional call. When the digital call is at-the-money the gamma is zero. In contrast a conventional call, when at-the-money, has gamma is at its highest.
What is apparent from the above illustration is how the gamma can soar and plunge as time to expiry approaches zero:
- Figure 1 has the peak and trough of the 0.1-day profile at +8.4947 and ―8.4633 at asset prices of 99.80 and 100.20 respectively.
- At the bottom of the legend’s days to expiry the 25-day profile has a high of 0.0351 and -0.0337 at 97.80 and 102.20 respectively. The peak of the 25-day gamma is somewhere lower than 97.80 as the profile is still rising. Likewise the gamma is still descending at 102.20.
Strange things these digital ‘greeks’. In practical terms the gamma is zero with 25-days to expiry.
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Digital Call Gamma and Volatility
Figure 2 illustrates the digital call option gamma with respect to different implied volatilities. At the underlying of 99.80 the 2.0% gamma has a high of 4.3266; at 100.20 the gamma is -4.3102. If 25% volatility was included it would offer a profile as flat as a pancake.
Here we see the bottom left and top right quadrants empty. If the market heads off north then the negative gamma indicates that the position is going to get even shorter as gamma is negative at all volatilities. The rate it gets shorter is dependent on the volatility. So the 18.0% gamma profile indicates that the short underlying asset equivalent position will only become a little shorter. The 6.0% profile indicates that the delta will initially turn negative more aggressively until 100.70 is reached. From then on, as the asset price still heads upwards, the position will still get shorter deltas but at a slower rate than when the asset price was 100.70.
Summary
A professional conventional options trader might describe their style of trading as ‘long gamma’ or ‘short gamma’. This nomenclature would in no way work for a digital trader. Figures 1 & 2 show that a trader with a long gamma position one minute could quite literally be holding a short gamma position next minute.
Traded options market-makers will, bye-and-large, quickly scan their position for two pieces of information; firstly their current delta, secondly a look across a two-dimensional array of price v volatility to see where the negative gamma positions are. Those areas of negative gamma are the market-maker’s pothole in the road. The only saving grace of the negative gamma scenario is that even the gamma returns to zero at the extreme market moves. But this can have limited attraction if the book is now short 1,000 equivalent deltas in a market heading to the moon.