Tunnel accumulator gamma describes the change in the value of a tunnel accumulator delta due to a change in the underlying price. Firstly, the tunnel accumulator delta is the first derivative of the tunnel accumulator w.r.t. a change in underlying price. This leads to the tunnel accumulator gamma being the second derivative of the tunnel accumulator w.r.t. the underlying price. This gamma can be depicted as:

where Δ is the tunnel accumulator delta and ** S** is the underlying price. The tunnel accumulator gamma is the gradient of the slope of the tunnel accumulator delta.

### Evaluating Tunnel Accumulator Gamma

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

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

Tunnel Accumulator Gamma = R_{1} * Digital Call Gamma(K_{1}) + R_{2} * Digital Call Gamma(K_{2}) + R_{3} * Digital Call Gamma(K_{3}) + R_{4} * Digital Call Gamma(K_{4}) –

R_{4} * Digital Call Gamma(K_{5}) – R_{3} * Digital Call Gamma(K_{6}) – R_{2} * Digital Call Gamma(K_{7}) – R_{1} * Digital Call Gamma(K_{8})

where:

R_{1} + R_{2} + R_{3} + R_{4} = 1

and

K_{1} < K_{2} < K_{3} < K_{4} < K_{5} < K_{6} < K_{7} < K_{8}

### Tunnel Accumulator Gamma Over Time

An already profitable long accy position midway between the central strikes requires some thought. Why? Because the position has now entered negative gamma territory. This means that if the underlying goes down the delta gets equivalent long. Alternatively, the trader holding the long tunnel accy gets equivalent short the asset if the asset price rises.

What compensates for this exposure is that the tunnel accumulator theta is in the long tunnel accy’s favour. If the asset price remains ‘unch’ at 100.00 the time decay is in the long’s favour as the tunnel accy heads up to 1.00 settlement price.

In Figure 1 with 0.1 days to expiry the gamma is already heading back up to 0 midway between the strikes as the tunnel accy value approaches 1.00. Apart from this feature, the only other point is that its manic gyrations make it a worthless risk management tool so short a time to expiry.

Elsewhere the 25-day profile is as flat as a pancake meaning that tunnel accy delta is flat too.

European Digitals | Tunnel Accumulator | Tunnel Accumulator Delta | Tunnel Accumulator Theta | Tunnel Accumulator Vega |

### Tunnel Accumulator Gamma and Volatility

Figure 2 displays the gamma against ‘vol’. Here one can see the higher the ‘vol’ the flatter the gamma. This inverse relationship between gamma and vega is common. Subsequently, whether conventional options or digital options, the higher the implied volatility of an option the lower its gamma.

### Summary

- The tunnel accumulator is of little use when there is very little time to expiry or implied volatility is very low. In these scenarios the gamma oscillated so wildly it becomes meaningless.
- Otherwise, the rule of thumb is that between the centre strikes the gamma is always negative.
- At the wings, when the tunnel accumulator is out-of-the-money, the gamma is positive.
- The point at which the gamma moves from negative to positive is dependent on both time to expiry and impled volatility.