# Tunnel Delta

Tunnel delta describes the change in the fair value of a digital options tunnel due to a change in the underlying price. The tunnel delta is the first derivative of the tunnel fair value with respect to a change in underlying price. It is depicted by:

where P is the tunnel value and S is the underlying price.

Options delta, generally, is the most used of the Greeks, especially by options market-makers. Options market makers need to hedge their directional exposure immediately on having traded. They also need to hedge their portfolio directional risk in the event of asset price and/or volatility changing. Plus of course the passage of time can effect the delta significantly. Nowadays most DMA options trading software has the facility to automatically delta-neutralise an individual trade. It’s also possible to automatically delta-neutralise a portfolio of options should the delta exceed a certain parameter.

### Evaluating Tunnel Delta

Tunnel Delta = Digital Call Delta(K1) ― Digital Call Delta(K2)

where the first term and second terms are the digital call delta with strikes K1 and K2 respectively.

### Tunnel Delta Over Time

Digital options tunnel delta is displayed against time to expiry in Figure 1. Immediately the black 0.1-day profile clearly reflects the long and short digital call deltas that make up the tunnel.

The 25-day tunnel delta is flat at less than ±0.1. This reflects the value of tunnel option Figure 2 remaining in a narrow range between 0.209 and 0.298 over a 4.40 asset range.

At the other extreme, the 0.1-day value peaks at 0.243 (at 99.00) and falls to -0.239 at 101.00.

#### Switchback Tunnel Delta

At 100.00 the tunnel delta is zero for all the delta profiles since the asset price is midway between the strikes. The asset price is at the optimum asset price (100.00). A short delta value of the lower digital call is exactly offset by the long delta position of the upper strike digital call delta.

Example: With 0.1 days to expiry a trader purchases a 99.00/100.00 tunnel when the underlying is 100.00. The trader has bought the tunnel in-the-money, in fact, at the underlying of 100.00 it couldn’t be any more ‘in-the-money’. The 100.00 underlying means a zero delta position. The delta position gets longer on the way down to 99.00 and shorter on the way up to 101.00.

Both asset price changes are undesirable as the tunnel loses value as the underlying asset gets closer to expiring out-of-the money.  A move downwards will lose money so a hedge would involve selling the underlying asset. Unfortunately a move upwards creates a negative delta thereby requiring a purchase of the underlying asset to remain delta neutral. The trader can be whipsawed buying and selling the underlying at a loss in order to hedge this position. This is not a good strategy. Better to buy the tunnel and walk away.

### Tunnel Delta and Volatility

Figure 2 provides the tunnel delta over a range of implied volatilities. When implied volatility is 18% with 5-days to expiry the profile is smooth which provides an easier ride for the hedger. The tunnel with high volatility is a fairly placid instrument as the long and short call deltas that make up the tunnel deltas cancel each other out.

At 101.00 the green 10% profile has a deeper dip followed by an even deeper dip from the red 6% profile. The red 6% profile shows the delta on a steeply declining gradient as it passes through the strikes mid-point at 100.00. This would create the hedger’s nightmare.

At 2% volatility the gradient at the aforementioned midpoint is collapsed to zero so here we see the 99.00 and 101.00 deltas acting quite independently of each  other.

### Summary

Any delta-neutral trader that has held a naked short at-the-money straddle or strangle will understand the issues involved in hedging the delta. The long tunnel position (which would equate to the short straddle or strangle) can become an extremely position if the trader hedges delta either side of the strike’s midpoint. Using Figure 1 & 2, if the asset price rose to 100.50 the delta-neutral trader would buy the underlying to hedge against a further rise. Yet if the underlying fell back to 100.00 then the traders position is long the futures just purchased at  100.50. Say the trader bought 10 futures with each 0.01 tick worth $10 then she has racked up a loss of: Delta Hedge Loss = (100.00 – 100.50) x$10 x 10

= \$5,000

Unless time decay offset that loss (it is possible but unlikey a fall in volatility will) that is a waste of good money and  the trader will be kicking herself.