# Binary Options Put Gamma definition and profiles

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

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

The binary put options gamma is subsequently the gradient of the delta profiles of Figs. 1 & 2 on Binary Put Options Deltas.

What is the relevance of binary put options gamma? Gamma, whether discussing binaries or conventionals, gives the rate of change of exposure to the underlying. If one is long the gold put illustrated then if it is out-of-the-money 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 so that 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 absolute 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, i.e. 30 futures were also 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, while 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 options gamma is displayed against time to expiry in Figure 1. As with binary call options gamma, the binary put options gamma is positive when out-of-the-money and negative when in-the-money, while 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 having gamma that tends to ±∞ while the example of Figure 1 shows gamma to be pretty much flat at zero for the 25-day 25% option.

Figure 2 provides binary put options gamma over a range of implied volatilities. The absolute value of the binary put options gamma is fairly static over the range of implied volatility. As the implied volatility falls the peak and trough of the options close in on the strike reflecting the steepening of the delta which in turn signals the increasingly higher gearing the strategy offers.

Binary put options 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 an options book as a market maker is fraught with difficulties as binaries bough to hedge pass through the strike and become a liability in their own right.

There are a plethora of on-line platforms now offering very short-term trading opportunities and the marketing strategy of these platforms is to stress the high returns available. These high returns are based on the extremely high binary call options deltas and binary put options deltas, supported by the high gammas.

Finite Gamma

Figure 2of the Binary Put Options Delta page shows 5-day $1700 binary put options delta profiles. At the underlying gold price of$1725 and 25% implied volatility the delta is worth ―0.702929. The binary put options delta at the underlying  of $1720 and$1730 are ―0.736039 and ―0.664852 respectively. Using the finite difference method:

where:

so that the above numbers provide a 5-day binary put options gamma of:

Binary Put Options Gamma = (―0.664852‒(―0.736039))/(1730‒1720) = 0.007119

If the underlying price increment was reduced from 5 to 0.00001 then:

so that the 5-day gamma becomes:

Binary Put Options Gamma = (―0.70292907‒(―0.70292922))/(1725.00001‒1724.99999) = 0.007152

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.

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