This post is published by Hamish Raw of https://hamishraw.com/
Binary options tunnel theta is the metric that describes the change in the fair value of a Binary Options Tunnel due to a change in time to expiry, i.e. it is the first derivative of the binary options tunnel fair value with respect to a change in time to expiry and is depicted as:
Binary options tunnel theta is displayed against time to expiry in Figure 1. The 25-day binary options tunnel theta profile (blue) is very flat at a very low theta while the 10-day binary options tunnel theta profile (orange) shows theta outside the strikes as negative with between the strikes theta is positive. This trend is more pronounced for the 4-day profile (green) whereas the 1-day profile (red) changes shape with theta midway between the strikes returning to zero. The 0.2-day profile exaggerates this trend with a wider incidence of the binary options tunnel theta being zero between the strikes. This is because the soybeans tunnel fair value, Figure 2 of Binary Options Tunnel, has already reached the maximum value of 100 and is therefore static at that price meaning that theta falls to zero.
Figure 2 provides binary options tunnel theta over a range of implied volatilities. The 30% and 35% implied volatilities are almost mapping each other but as implied volatility falls to 20% and then 15% the profiles between the strikes firstly levels off, then falls back towards zero in the case of the 15% profile. Once the implied volatility has fallen low enough to drive the 15% profile of Figure 3 on Binary Options Tunnel then the binary options tunnel theta profile will be zero between the strikes.
In figure 1 above the profiles are zero at the strikes whereas the binary options tunnel theta profiles of Figure 2 do not necessarily pass through the strike.
Selling the soybean binary options tunnel at a price of close to 100 at a low implied volatility could be compared to buying a conventional strangle at a very low premium.
Evaluating Binary Options Tunnel Theta
Binary Tunnel Options Theta = Binary Call Option Theta(K1) ― Binary Call Option Theta(K2)
where the first term and second terms are the binary call options theta with strikes K1 and K2 respectively.
Figure 2of the Binary Options Tunnel page shows a 4-day 1150/1250 binary tunnel price profile. At the underlying soybeans price of 1200 this tunnel is worth 88.8768, while at 3.5-day and 4.5-days their values are 91.1344 and 86.6818 respectively. Using the finite difference method
Binary Options Tunnel Theta = ―(P1―P2)/(T1―T2)
T1=The greater number of days to expiry
T2=The lesser number of days to expiry
P1=Binary Options Tunnel fair value with greater number of days to expiry
P2=Binary Options Tunnel fair value with lesser number of days to expiry
so that the above numbers provide a 4-day binary options tunnel theta of:
Binary Options Tunnels Theta = ‒(86.6818‒91.1344)/(4.5‒3.5) = 4.4527
If the day increment was reduced from 0.5 to 0.00001 then:
so that the 4-day soybean binary options tunnel theta becomes:
Binary Options Tunnels Theta = ‒(88.876785‒88.876875)/(4.00001‒3.99999) = 4.464282