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Tunnel options 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:

#### V = \frac{dP}{dσ}

where V is vega and \sigma is volatility.

### Evaluating Tunnel Vega

Tunnel Vega = Binary Call Vega(K_{1}) ― Binary Call Vega(K_{2})

where the first term and second terms are the call vega with strikes K_{1} and K_{2} 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 16.81 at 100.00. If the volatility drops to 16%, then 14%, the tunnel becomes worth 18.87 and 21.51 respectively.

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

Vega = (16.81/100 – 21.51/100)/(18% – 14%) = -1.1749

This tunnel vega is -1.1578. 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 18.8835 and 18.8603 respectively. Substituting:

Vega = (18.8603/100 – 18.8835/100)/(16.01% – 15.99%) = -1.1578 to 4dp

**Tunnel Vega and Volatility**

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

Between the strikes the tunnel vega is negative. But a look at the scale suggests that a change in volatility at any ‘vol’ level is going to have little influence on the price.

By: Hamish Raw