Tunnel Vega | Tunnel Option | Digital Option

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:

$V = \frac{dP}{d\sigma}$

where $V$ is vega and $\sigma$ 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.

Figure 1 – Tunnel Vega w.r.t. Time to Expiry

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.

Figure 2 – Tunnel Vega w.r.t. Volatility

Between the strikes the tunnel vega is negative, unless it’s zero. Outside the strikes the vega is positive, unless it too is zero.

The 2% profile clearly shows the long 99.00 digital call vega profile and the short 101.00 digital call vega profile. The zero vega at 100.00 shows the two vegas working independently of each other.

The next absolutely smallest vega at 100.00 is the 18% vega so between 6% and 2% the vega profile has completely inverted.

Summary

As time to expiry (Fig.1) and volatility (Fig.2) decrease the wild oscillations of the vega do not overly help the risk manager make confident decisions. Only once the vega of the vega is incorporated into a global risk management analysis incorporating all derivatives based on the underlying, including the underlying itself, does the value of the individual vega is disputable.