This post is published by Hamish Raw of https://hamishraw.com/
Binary options tunnel delta is the metric that describes the change in the fair value 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:
where S is the underlying price.
Options delta, generally, is the most used of the Greeks, especially by options market-makers who will hedge their directional exposure immediately having had a price they’ve made ‘hit’ by a counterparty. Nowadays most options trading software has the facility to automatically delta-neutralise an individual trade at time of execution, plus automatically delta-neutralise a portfolio of options should the delta exceed a certain parameter.
Binary options tunnel delta is displayed against time to expiry in Figure 1. And immediately the black 0.2-day profile reflects the long and short binary call options deltas that make up the tunnel. This binary options tunnel delta is flat at zero between the strikes reflecting that the price of Figure 2 of Binary Options Tunnel does not alter from 100. At the other extreme, the 25-day profile is extremely flat at a very low absolute value in turn reflecting that the change of underlying has little impact on the low probability at that point of the underlying price being between the two strikes at expiry.
Figure 2 provides binary options tunnel delta over a range of implied volatilities. When the variable is implied volatility with 5-days to expiry the profiles are a great deal smoother, which provides an easier ride for the hedger. As has been pointed out, for binary options purposes the implied volatilities at 25% middle are relatively high so that the binary options tunnel delta profiles follow similar paths, i.e. the binary options tunnel gamma is low.
Evaluating Binary Options Tunnel Delta
Binary Options Tunnel Delta = Binary Call Option Delta(K1) ― Binary Call Option Delta(K2)
where the first term and second terms are the binary call options delta with strikes K1 and K2 respectively.
Figure 2of the Binary Options Tunnel page shows a 10-day, 25% implied volatility 1150/1250 binary tunnel price profile. At the underlying soybeans price of 1170 this tunnel is worth 60.1219, while at underlying prices of 1169.5 and 1170.5 their values are 59.8503 and 60.3899 respectively. Using the finite difference method:
Binary Options Tunnel Delta = (P1―P2)/(S1―S2)
S1=The higher underlying price
S2=The lower underlying price
P1=Binary options tunnel fair value with the higher underlying price
P2=Binary options tunnel fair value with the lower underlying price
so that the above numbers provide a 10-day 25% implied volatility binary options tunnel delta of:
Binary Options Tunnel Delta = (60.3899‒59.8503)/(1170.5‒1169.5) = 0.5397
If the implied volatility increment was reduced from 0.5 to 0.00001 then:
so that the 10-day 25% implied volatility soybean binary options tunnel delta becomes:
Binary Options Tunnel Delta = (60.121870‒60.121859)/(1170.00001‒1169.99999) = 0.539679
which is the same as the result for differentiating the fair value price with respect to the underlying price.