# Eachway Tunnel Vega

Eachway tunnel vega is the metric that reveals how much the eachway tunnel’s value changes owing to a change in volatility.

Eachway tunnel vega describes the change in the fair value of the eachway tunnel 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:

#### $V = \frac{dP}{dσ}$

where  $V$ is vega, $P$ is eachway tunnel value and $\sigma$ is volatility.

### Evaluating Eachway Tunnel Vega

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

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

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

where R1 + R2 = 1 and R1 < R2.

On this page R1  = 0.4  and R2 = 1 – R1 = 0.6

### Eachway Tunnel Vega Over Time

The eachway tunnel vega is displayed against time to expiry in Figure 1. With volatility was set to 10%, time to expiry set by the legend, how much impact on the eachway tunnel value does a given change in volatility.

On checking out the two extremes in the legend a couple of observations can be made:

• The 0.1 day to expiry has the lowest absolute vega. This means that when there is a given change in volatility, with very little time to expiry, the eachway tunnel value will gain  or lose little value.
• The 25-day expiry is always negative. This compares with all the other profiles which are both positive and negative. Within this asset price range, 10% ‘vol’ and 25 days to expiry, a rise in volatility will lower the value of the eachway tunnel. And, of course, a fall in ‘vol’ will increase the value of the eachway tunnel.

At 100.00 the 25-day vega falls to where the 8-day and 2-day profiles sit. The 8-day and 2-day profiles of eachway tunnels Figure 2a show the values to be either side of 50.0, 43.33 & 68.43 to be exact. The value at 50 would generate the highest absolute vega and would be below the 2-day and 8-day vegas. As the value rises there is less absolute ‘elbow room’ as the updise has decreased leading to a higher 0.5-day vega. At 0.1 days to expiry the eachway tunnel is now valued at nearly 100.0 so the vega is now nearly zero.

### Eachway Tunnel Vega and Volatility

Figure 2 provides eachway tunnel vega over a range of implied volatilities. Here the time to expiry is set at 5 days with volatility determined by the legend.

As with Figure 1 only at the wings do the higher volatilities have positive vega. At the wings the asset price is out-of-the-money and generally out-of-the-money options will have a positive vega.

As volatility falls the valley becomes steeper as the vega tends towards the individual binary call vega of the different strikes.

The ultra low volatility of 2% creates the usual gyrating issues of the profile.

For example: there are just 5 days to expiry, volatility has collapsed to a historically low 2%, but there’s just  one snag. The only liquid instrument is the eachway tunnel. What to do? At 99.30 and 100.70 the vega is over 7.0 and, remarkably, 99.30 is where the underlying is currently trading. You get lucky and pay fair value of 51.67 for the eachway tunnel. The delta is 71.28 so the trader wants the asset to go up.

Scenario 1:

Asset  price very slowly drifts down to 98.70.

• Problem 1: the positive delta creates a loss.
• Problem 2: the vega is now –4.63 so the trader is now hoping for a further fall in volatility, not a rise. The fall from 99.30 to 98.70 is quite likely to generate a rise in ‘vol’ so another loss.
• Problem 3: The theta is 0.0093, marginally positive but not much of a crumb of comfort.

In aggregate the ‘vol’ has risen to 6% and the eachway tunnel is now worth 31.66, a loss of 31.66 –  51.67 = –20.01.

Scenario 2:

Asset  price very slowly rises to 100.00.

• Long delta creates a profit so eachway tunnel is now worth 98.00
• Long theta a marginal benefit.
• Vega now -5.22 and the 70 tick rise from 99.30 has increased volatility to 7%. Eachway tunnel value now only 60.03.

In aggregate a small profit. At 100.00 the theta is healthily positive although a negative vega would create a big problem in the eevent of another 70 tick move up or down.

In summary, the eachway tunnel vega is a useful tool but yet again the proviso has to be added that only useful when time to expiry and volatility are not extremely low.

By: Hamish Raw

error: Content is protected !!