# Tunnel Vega Tunnel vega describes the change in the fair value of a tunnel option due to a change in implied volatility. Tunnel vega is the first derivative of the tunnel fair value with respect to a change in implied volatility. It is depicted as: where is vega and is volatility.

### Evaluating Tunnel Vega

Tunnel Vega = Digital Call Vega(K1) ― Digital Call Vega(K2)

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

### Tunnel Vega Over Time

Tunnel vega is displayed against time to expiry in Figure 1.

The 0.1-day tunnel vega outlines the profile of the long call vega at the 99.00 strike and the short call vega at the 101.00 strike.

The 25-day tunnel vega profile is always negative in the range of asset price. This means that with so much time to expiry the probability of the asset price being between the strikes at expiry have been reduced. At 18% volatility, 25 days to expiry the tunnel is worth 0.168073 at 100.00. If the volatility drops to 16%, then 14%, the tunnel becomes worth 0.188719 and 0.215068 respectively.

Using the above data one can calculate the 16% vega at 100.00 so:

Vega = (0.168073/100 – 0.215068/100)/(18% – 14%) = -0.011749

This tunnel vega is -0.011578. By replacing 14% and 18% with 15.99% and 16.01% one can achieve a more accurate vega approximation. At 15.99% and 16.01% the tunnel is worth 0.188835 and 0.188603 respectively. Substituting:

Vega = (0.188603/100 – 0.188835/100)/(16.01% – 15.99%) = -0.011578 to 6dp

This method of establishing a Greek is known as the Finite Difference method.

### Tunnel Option Vega and Volatility

Figure 2 provides tunnel vega over a range of implied volatilities.