# Binary Options Tunnel Gamma definition and profile

Binary Options Tunnel gamma is the metric that describes the change in the delta of a Binary Options Tunnel due to a change in the underlying price, i.e. it is the first derivative of the Binary Options Tunnel fair value with respect to a change in underlying price and is depicted as:

Γ=dΔ/dS

where  Δ is the Binary Options Tunnel Delta.

Options gamma is crucial in risk management and Binary Options Tunnel gamma is no different.  The blue 25-day profile is pretty much flat which is no surprise since the 25-day profile of Figure 1 Binary Options Tunnel Delta is so flat. The 10-day and 4-day profiles turn negative between the strikes and positive outside the strikes although the 1-day gamma reverts to zero midway between the strikes. And as ever the 0.2-day gamma is off the map and provides such switchback risk it would be pointless even attempting sustainable risk management.

Figure 2 provides Binary Options Tunnel gamma over a range of implied volatilities. The Binary Options Tunnel gamma is not as volatile as the very short-term gamma of Figure 1 but it swings sufficiently to cause major gamma hedging issues when volatility is low.

Evaluating Binary Options Tunnel Gamma

Binary Options Tunnel Gamma = Binary Call Option Gamma(K1) ― Binary Call Option Gamma(K2)

where the first term and second terms are the binary options call gamma with strikes K1 and K2 respectively.

Finite Gamma

Figure 1of the Binary Options Tunnel Delta page shows an 4-day, 25% implied volatility 1150/1250 Binary Options Tunnel delta profile. At the underlying soybeans price of 1130 this tunnel delta is 1.0673, while at underlying prices of 1125 and 1135 the deltas are 0.9414 and 1.1750 respectively. Using the finite difference method:

Binary Options Tunnel Gamma = (Δ1―Δ2)/(S1―S2)

where:

S1=The higher underlying price

S2=The lower underlying price

Δ 1=Binary Options Tunnel delta with the higher underlying price

Δ 2=Binary Options Tunnel delta with the lower underlying price

so that the above numbers provide a 4-day 25% implied volatility Binary Options Tunnel gamma of:

Binary Options Tunnel Gamma = (1.1750‒0.9414)/(1135‒1125) = 0.0234

If the implied volatility increment was reduced from 5 to 0.5 then:

S1=1130.5

S2=1129.5

P1=1.0790

P2=1.0554

so that the 4-day 25% implied volatility soybean Binary Options Tunnel gamma becomes:

Binary Options Tunnel Gamma = (1.0790‒1.0554)/(1129.5‒1130.5) = 0.0236

Doing the same again for δS = 0.00001:

S1=1130.00001

S2=1129.99999

P1=1.067258

P2=1.067257

so that the 4-day 25% implied volatility soybean Binary Options Tunnel gamma becomes:

Binary Options Tunnel Gamma = (1.067258‒1.067257)/(1129.99999‒1130.00001) = 0.023639

Which is the equivalent gamma should the delta be differentiated at that point.