# Tunnel Accumulator

The tunnel accumulator is a volatility trade, as is the tunnel and eachway tunnel. The tunnel accumulator has further strikes providing nine different settlements levels, including ‘dead heats’. This is, yet again, another strategy whereby the trader receives increased returns the more accurate the asset price forecast. The difference between the tunnel accumulator and the conventional iron butterfly is that at expiry the ‘walls’ of the ‘fly are set at ±45° while the ‘walls’ of the various tunnels are vertical.

### Tunnel Accumulator at Expiry

Figure 1 illustrates the expiry price profile of the tunnel accumulator with its progressively increasing steps on the way up to and down from the midpoint of the central tunnel. Below the asset price of 100.00 we have a call accumulator while above 100.00 we have a put accumulator. It should be no surprise that the formula for the tunnel accumulator is:

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

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

R5 * DigitalCall(K5) – R6 * DigitalCall(K6) – R7 * DigitalCall(K7) – R8 * DigitalCall(K8)

where:

R1 + R2 + R3 + R4 + R5 + R6 + R7 + R8 = 2

When discussing tunnel accumulators we assume:

R1 = R8 = 0.1; R2 = R = 0.2; R3 = R6 = 0.3;  and R4 = R5 = 0.4

The strikes increase from the lowest strike K1 to the highest strike K8 so that:

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

There can be variations in both payouts and strikes relating to each other. For example:

R1 = R2 = R3 = R4 = R5 = R6 = R7 = R8 = 0.25.

Alternatively, the strikes could be defined thus:

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

which would be creating something like a European version of an up-and-out call option.

### Tunnel Accumulator Over Time to Expiry

Figure 2 illustrates the tunnel accumulator price profiles over time with 10% implied volatility. Midway between the central strikes the strategy could be considered a theta play when there are 25-days and 8-days to expiry since time decay has a positive effect on the price over a wide range of the underlying. When there are 2-days to expiry at 100.00 the tunnel accy has a fair value of 0.738 which would create a 2-day return of 35.5% should the asset price not drift more than 50 ticks either side, i.e. below 99.50, above 100.50 at expiry.

Although the 0.01-day profile is already worth 1.00 at the asset price of 100.00 the profile is still smooth, more like a short conventional strangle, as the averaging of the four different options on each side comes into play. With the increased number of strikes the tunnel accy price profile only assumes the expiry profile of Figure 1 (with 10% volatility) until the very last minute. The lower the volatility the sooner the expiry profile is reached [see below].

When there are 2-days or less to expiry, with the underlying outside the lowest and highest strikes the tunnel accumulator also provides a directional play with substantial deltas as measured by the gradients of those price profiles. So, at 101.80 the 2-day profile has a fair value of 0.1893 which on the underlying falling 180 ticks to 100.00 to rises to 0.738, a return of (0.738 – 0.1893)/0.1893 =  290%. If the underlying sat on 100.00 for the next 24 hours the tunnel accy would be worth 0.8466 generating a total return of (0.8466 – 0.1893)/0.1893 =  347%.

### Tunnel Accumulator and Volatility

Yet if the asset price ‘sat on 100.00’ as suggested above then it is quite feasible that the implied volatility would fall. At 100.00 with 2 days to expiry, volatility at 10%, the tunnel accy would be worth 0.7380. Figure 4 below shows the 2-day to expiry profiles where one can see the effect volatility has on the tunnel accy values.

Figure 5 offers the same except there is now only 1 day to expiry.

Midway between the centre strikes and with just 2% volatility, the tunnel accy is already worth 100.00, 24 hours before time. The green 10% profile is now up to 0.8466 at 100.00 from 0.7380 a day earlier.

### Summary

The tunnel accumulator provides an interesting alternative to the short straddle and enables a trader to sell premium with an automatic limited  loss built in. For this reason alone digital options should prove of great interest to retail.