# Binary Options Tunnel Vega definition and profile

Binary options tunnel vega is the metric that describes the change in the fair value of a binary options tunnel due to a change in implied volatility, i.e. it is the first derivative of the binary options tunnel fair value with respect to a change in implied volatility and is depicted as:

V=dP/dσ

Binary options tunnel vega is displayed against time to expiry in Figure 1. The 0.2-day vega (black) clearly depicts the profile of the long binary call theta at the 1150 strike and the short binary call theta (long binary put theta) at the 1250 strike. The 0.2-day binary options tunnel vega profile is otherwise zero which reflects the fact that the fair value profile, Figure 2 of Binary Options Tunnel, is predominantly 0 or 100. The 1-day binary options tunnel vega profile shows that midway between the strikes at 1200 the vega turns slightly negative which reflects that the increase in time to expiry has increased the probability of the soybean price moving outside the 1150/1250 range sufficiently to lower the price from 100 to 99.8529. From 1-day to 4-days the binary options tunnel vega inverts so that instead of the profile having two troughs just inside the strikes, there is now just one smooth depression. This trend continues out to the 10-day profile but then the absolute value of the binary options tunnel vega starts to fall again. This is because the price profile is becoming increasingly shallow as the probability of the soybeans price being between the strikes becomes less dependent on where the current underlying price is.

Figure 2 provides binary options tunnel vega over a range of implied volatilities. Between the strikes the binary options tunnel vega is negative although the 15% profile suggests that the vega midway between the strikes is heading back to zero vega. The actual difference between vegas at implied volatility of 20% and above is very little reflecting the uniform increase in the price of Figure 3 of the Binary Options Tunnel page.

Evaluating Binary Options Tunnel Vega

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

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

Finite Vega

Figure 3of the Binary Options Tunnel page shows a 5-day, 25% implied volatility 1150/1250 binary options tunnel price profile. At the underlying soybeans price of 1200 this tunnel is worth 84.5746, while at 24.5% and 25.5% implied volatilities their values are 85.3976 and 83.7510 respectively. Using the finite difference method:

Binary Options Tunnel Vega = (P1―P2)/(σ1―σ2)

where:

σ1=The higher implied volatility

σ2=The lower implied volatility

P1=Binary tunnel fair value with the higher implied volatility

P2=Binary tunnel fair value with the lower implied volatility

so that the above numbers provide a 5-day 25% implied volatility binary tunnel vega of:

Binary Options Tunnel Vega = (83.7510‒85.3976)/(25.5‒24.5) = ―1.6465

If the implied volatility increment was reduced from 0.5 to 0.00001 then:

σ1=25.00001

σ2=24.99999

P1=84.574547

P2=84.574580

so that the 5-day 25% implied volatility soybean binary options tunnel vega becomes:

Binary Options Tunnels Vega = (84.574547‒84.474580)/(25.00001‒24.99999) = ―1.646971 