# Tunnel Accumulator Vega

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   =   R1 * Binary Call Vega(K1) + R2 * Binary Call Vega(K2) + R3 * Binary Call Vega(K3) + R4 * Binary Call Vega(K4) –

R4 * Binary Call Vega(K5) – R3 * Binary Call Vega(K6) – R2 * Binary Call Vega(K7) – R1 * Binary Call Vega(K8)

where:

R1 + R2 + R3 + R4 = 1

and

K1 < K2 < K3 < K4 < K5 < K6 < K7 < K8

### 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.