Tunnel accumulator vega describes the change in the fair value of a tunnel accumulator due to a change in volatility. The tunnel accy vega is the first derivative of the tunnel accumulator w.r.t. a change in implied volatility. This vega can be depicted as:

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

where V is the tunnel accumulator vega and ** P** is the tunnel accumulator value and \sigma is implied volatility. The tunnel accy vega is the gradient of the slope of the tunnel accumulator value versus implied volatility.

**Evaluating Vega**

The tunnel accy vega is the sum of all the eight different strike binary call vega.

Alternatively, the tunnel accy vega can be considered as:

Tunnel Accumulator Vega = (Call Accumulator Vega + Put Accumulator Vega)/2

Going down the binary call vega route:

Tunnel Accumulator Vega = R_{1} * Binary Call Vega(K_{1}) + R_{2} * Binary Call Vega(K_{2}) + R_{3} * Binary Call Vega(K_{3}) + R_{4} * Binary Call Vega(K_{4}) –

R_{4} * Binary Call Vega(K_{5}) – R_{3} * Binary Call Vega(K_{6}) – R_{2} * Binary Call Vega(K_{7}) – R_{1} * Binary Call Vega(K_{8})

where:

R_{1} + R_{2} + R_{3} + R_{4} = 1

and

K_{1} < K_{2} < K_{3} < K_{4} < K_{5} < K_{6} < K_{7} < K_{8}

### Vega Over Time

Figure 1 shows that with volatility 10% and 0.1 days to expiry a change in volatility will have very little effect on the accy value. If the asset price is at 100.00 the illustration shows that a lengthening time to expiry has the vega most negative at 8 days. The 25-day vega has now risen again so there is a specific day when absolute vega is at its highest.

This day proves to be 4.5 days before expiry when the vega is -3.8849

**Vega and Volatility**

Figure 2 displays the vega against ‘vol’. Here one can see the higher the ‘vol’ the flatter the vega.

Between the strikes a rise in implied volatility would mean a higher probability the asset price will move outside the inner strikes. Thereby the tunnel accy will lose value and the vega is subsequently negative.

At the wings the vega is positive as an increase in ‘vol’ increases the probability of the asset price moving to an in-the-money status.

by: Hamish Raw