# Tunnel Accumulator Theta

Tunnel accumulator theta describes the change in the fair value of a tunnel accumulator due to a change in the underlying price. The tunnel accumulator theta is the first derivative of the tunnel accumulator w.r.t. a change in time to expiry. This theta can be depicted as:

$\Theta = \frac{dP}{dt}$

where P is the tunnel accumulator value and t is time. The tunnel accumulator theta is the gradient of the slope of the tunnel accumulator price over time.

### Evaluating Tunnel Accy Theta

The tunnel accumulator theta at its most simple method of evaluation is the sum of all the eight different strike binary call theta.

Alternatively, the tunnel accumulator can be considered as a long call accumulator and long put accumulator. This would lead to the tunnel accumulator theta being the sum of the call accumulator theta and the put accumulator theta.

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

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

where:

R1 + R2 + R3 + R4 = 1

and

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

Below the example are:

R1 = 10%, R2 = 20%, R3 = 30% and R4 = 40%

### Tunnel Accy Theta Over Time

In Figure 1 with just 0.1 days to expiry the theta is positive. Where the crevass sits between the two pinnacles is where the tunnel accy has already (almost) reached the value of 100.00. in this case further time decay will not lead to any further (large) upside movement in the tunnel accy..

At 25 days to expiry the profile is pretty much flat.

### Tunnel Accy Theta and Volatility

Figure 2 is more interesting; it depicts how the 5-day tunnel accumulator will change in price when trading at different volatilities.

The 2% profile is so low that between the centre strikes the tunnel accy is already almost trading at 100.

The adjacent volatility, 6%, now has the highest theta. The 6% theta is a shade higher than the 10% profile and higher than the 14% and 18% which are now falling. So, is there an optimal volatility that one should buy this tunnel accumulator when there are 5 days to expiry and the underlying is trading at 100.00?

Figure 3 shows there is. The asset price and time to expiry are constrained to 100.00 and 5 days respectively. The highest theta is at an implied volatility of 9.5%.

At this point the tunnel accumulator theta has risen to 0.03885 before it tails off as ‘vol’ moves higher. The problem is that although the underlying asset may stick rigidly to 100.00, we can be assured that time will not stay at 5 days!

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