# Tunnel Accumulator Delta

The tunnel accumulator delta describes the change in the fair value of a tunnel accumulator due to a change in the underlying price. The tunnel accumulator delta is the first derivative of the tunnel accumulator with respect to a change in underlying price and is depicted as:

where P is the tunnel accumulator value and S is the underlying price. The delta is the gradient of the slope of the tunnel accumulator price against the underlying asset price.

### Evaluating Tunnel Accumulator Delta

The tunnel accumulator can be thought of as a set of embedded tunnel options with different strikes. The tunnel delta is made up of a long and short digital call delta. Alternatively, the tunnel accumulator delta can be the aggregate of the call accumulator delta and the put accumulator delta.

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

R4 * Digital Call Delta(K5) – R3 * Digital Call Delta(K6) – R2 * Digital Call Delta(K7) – R1 * Digital Call Delta(K8)

where:

R1 + R2 + R3 + R4 = 1

and

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

### Tunnel Accumulator Delta Over Time

The tunnel accumulator delta is always positive below the midpoint of the central strikes and negative above. This suggests that the strategy could be used as a directional play when out-of the money.

For example: The underlying is trading at 98.00, ‘vol’ is 10%, two days to expiry and Figure 1’s tunnel accy is on offer. The trader fancies the market up over the next two days. Buying the tunnel accy would prove an interesting ploy. At this point the tunnel accumulator is worth 0.1320 and (ignoring market maker bid/ask, trading costs) this is the maximum loss possible. Initially the delta (green) is +0.23, positive so a bullish play. The trader puts up $1,000. #### Scenario 1: The trader is wrong, the asset price dumps! Tunnel Accy settles at 0.00 Trader’s P&L = (0.00 – 0.1320)/0.1320 x$1,000 = –$1,000 #### Scenario 2: Trader is right but not right enough as the asset crawls up to 98.40. Tunnel accy settles at 0.10 Trader’s P&L = (0.10 – 0.1320)/0.1320 x$1,000 = –$242.42 #### Scenario 3: After 1.5 days the asset price is at 99.20 So with 0.5 days (red profile) to go the delta is at its maximum value of +0.63. If the asset keeps rising the delta will reduce to 0.00 at 100.00. So if the asset keeps rising the trader now has a tunnel accumulator theta play. The trader will not profit from directional exposure but will henceforth profit from theta exposure. The trader ‘sticks’ and the asset price remains between 99.00 and 99.50 so the accy settles at 0.60. Trader’s P&L = (0.60 – 0.1320)/0.1320 x$1,000 = –$3,545.45 #### Scenario 4: The trader does not ‘stick’ but runs the position The asset price has one final move up of 31 ticks to 99.51. The trader’s settlement is now max’d out at 1.00 as the asset price is between the two inner strikes. Trader’s P&L = (1.00 – 0.1320)/0.1320 x$1,000 = –$6,575.76 #### Scenario 5: The trader underestimates the correctness of their forecast The trader, like a paralysed rabbit, is caught in the headlamps of an oncoming truck. The trader rides the long delta through the asset price of 100.00. The delta is now negative and remains negative until it reaches zero at an asset price level above 102.20. The tunnel accumulator is now worthless and settles at 0. This scenario is the same outcome as Scenario 1. Trader’s P&L = (0.00 – 0.1320)/0.1320 x$1,000 = –\$1,000

N.B. I am personally sympathetic to this trader’s plight because I was, as a market maker, always long gamma. In this case tunnel accumulator gamma. Being long gamma means that if the underlying kicks off you don’t have to constantly check your delta every five seconds. If the market steams upwards you get longer, if it dumps you get shorter. The above example is a trader that started off with a long gamma position that morphed into a short gamma position. At 100.00 the trader needed cojones to run the position and hope the asset remains where it is to profit from a tunnel accumulator theta. Never did when I was a market maker!

### Tunnel Accumulator Delta and Volatility

Figure 2 displays the delta against ‘vol’ where it can be seen that the higher the ‘vol’ the flatter the delta across the range of underlying.

The tunnel accumulator provides an interesting alternative to the short straddle. It enables a trader to short ‘vol’ with an automatic limited loss built in.

### Summary

• The tunnel accumulator delta does not have the same gyrating delta as we have seen in other digital options with fewer strikes.
• The delta is always zero midway between the centre strikes.
• Only between the strikes of the central two digital options (100.00) is there a huge shift in delta and that is when time to expiry is short or  volatility  is  extremely low.