Tunnel accumulator theta describes the change in the fair value of a tunnel accumulator due to the passage of time. 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:
where P is the tunnel accumulator value and t is time.
Evaluating Tunnel Accumulator Theta
The tunnel accumulator theta at its most simple method of evaluation is the sum of all the eight different strike digital 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 * Digital Call Theta(K1) + R2 * Digital Call Theta(K2) + R3 * Digital Call Theta(K3) + R4 * Digital Call Theta(K4) –
R4 * Digital Call Theta(K5) – R3 * Digital Call Theta(K6) – R2 * Digital Call Theta(K7) – R1 * Digital 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 Accumulator Theta Over Time
In Figure 1 with just 0.1 days to expiry the theta is positive. Where the depression sits between the two pinnacles is where the tunnel accy has already (almost) reached the value of 1.00. 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.
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Tunnel Accy Theta and Volatility
Figure 2 is more interesting; it depicts how the 2-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 1.00.
Summary