# Eachway Tunnel Vega

Eachway tunnel vega describes the change in the fair value of a eachway tunnel (ET) option due to a change in implied volatility. Eachway tunnel vega is the first derivative of the eachway tunnel fair value with respect to a change in implied volatility. It is depicted as:

where  is vega and is volatility.

### Evaluating ET Vega

The eachway tunnel vega can be constructed from the digital call vega.

Eachway Tunnel Vega = R1 x Digital Call Vega(K1) + R2 x Digital Call Vega(K2)

– R2 x Digital Call Vega(K3) – R1 x Digital Call Vega(K4)

where K1, K2, K3 & K4 are lowest strike to highest strike, and

where R1 + R2 = 1 and R1 < R2 and R2 = 1 – R1.

### Eachway Tunnel Vega Over Time

Eachway tunnel vega is displayed against time to expiry in Figures 1a & 1b. In both views the vega at 100.00 is negative; the 0.1 day vega profiles are nearly zero as the eachway tunnel has almost reached 1.00 so is unable to rise any further.

Another observation shows that the vega at 100.00 initially falls in value from 25 days to expiry. In Figure 1a the 8 day expiry has the most negative vega while in Figure 1b this award goes to the 2-day expiry profile. The change of vega due to a change in time to expiry is known as  DVegaDtime or Veta. As can be seen from both Figs.1a & 1b the veta would be throwing out some very odd numbers.

In the above illustrations both 25-day ET vega profile are always negative within the asset price range of 97.80 to 100.20. This can be construed as the 25 days to expiry leading to the probability of the asset price being in any specific 100-tick range has averaged out. Those 100-tick ranges far out-of-the-money now have a higher probability of seeing the settlement price within the range’s boundaries. Correspondingly, the range straddling 100.00 has a lower probability of housing the winning settlement price.

### Eachway Tunnel Vega and Volatility

Figures 2a & 2b provide eachway tunnel vega over a range of implied volatilities.

Between the central strikes the eachway tunnel vega is negative or zero. Outside the strikes the ET vega oscillates which makes it almost impossible to use vega as a metric for hedging against adverse volatility.

With both rebates (0.40 and 0.25) the 10% – 18% profiles are smooth enough to allow some sensible vegas to enable practical volatility hedging. But bearing in mind that the only way to hedge is with other options (precious few volatility futures exist) those options will too have quirky vega profiles (if digitals).

### Summary

It has already been postulated in the Summary of tunnel vega that an oscillating metric provides little worthwhile information in order to hedge if there is excessively low implied volatility and/or very little time left to expiry. The eachway tunnel magnifies this lack of efficacy.