This post ist published by Hamish Raw of https://hamishraw.com/

The binary call options gamma is the first derivative of the binary call option delta with respect to a change in the underlying price. The gamma is the slope of the delta profile. What the binary call options gamma does is reflect whether a binary call option’s equivalent futures position will become longer or shorter as the underlying price rises or falls.

Example: The binary call option delta page provides an example of a long 100 call position with a delta of 0.30 having an equivalent underlying position of 100 x 0.30 = 30. The binary call options gamma is a number that states at that particular point whether the equivalent underlying position will be more or less than 30 as the underlying rises. If the equivalent position increases from 30 as the underlying price rises then the position has positive gamma; if the equivalent position drops to say 25 as the market rises then the position is said to have negative gamma.

The gamma of conventional calls is always positive so that on buying conventional calls the equivalent position will always increase as the underlying rises, irrespective of whether the option is in the money or not. Binary call options do not behave in the same manner in that out-of-the-money binary call options always have positive binary call options gamma, while in-the-money binary call options always have negative binary call options gamma.

Figure 1 illustrates the binary call options gamma with respect to different implied volatilities and from the scale, it is clear that the gamma in these circumstances is not heavy. In fact, as it reflects the slopes of the delta profiles used in the examples of binary call option deltas it is evident how shallow the deltas actually are. This extremely low gamma is a function of the high implied volatility but is also constrained by the fact that the binary call options gamma has to pass through zero when at the money.

The 20% profile is markedly higher than the 40%, even 30% gamma, and should the implied volatility fall to 5% the peak and trough of the gamma are ±0.69, which is still modest compared with conventional option gamma.

Figure 2 offers the binary call options gamma with respect to time to expiry.

What is apparent from the illustration is how the gamma can soar and plunge as a time to expiry approaches zero; in Fig.2 the peak and trough of the 0.2 Day profile are +0.49 and ―0.48 respectively. Yet these numbers are still constrained by the overall limited risk to buyer and seller of the binary call option which in turn holds down the binary call options gamma.