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The binary call options gamma is the first derivative of the binary call options delta with respect to a change in the underlying price.

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

where \Gamma is gamma and \Delta is the binary call delta.

The gamma is the slope of the delta profile. The binary call gamma reflects whether a binary 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. Binary calls do not behave in the same manner. The out-of-the-money binary call will always have positive gamma. The in-the-money binary call will always have negative gamma.

### Binary Call Gamma Over Time

Figure 1 illustrates the 100.00 binary call gamma against days to expiry. When the binary call is deep in-the-money or far out-of-the-money the gamma tends to zero, just like the conventional call. When the binary 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 +1.7622 and ―1.7483 at asset prices of 99.60 and 100.40 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 binary ‘greeks’. In practical terms the gamma is zero with 25-days to expiry.

### Binary Call Gamma and Volatility

Figure 2 illustrates the binary 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.

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 binary 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.

A conventional options trader that adopts a particular gamma-based options strategy would be completely bemused if they tried to implement a similar strategy while trading binaries.

By: Hamish Raw