# Tunnel Options Delta

Tunnel options delta describes the change in the fair value of a binary 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:

#### $\Delta = \frac{dP}{dS}$

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 = Binary Call Delta(K1) ― Binary Call Delta(K2)

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

### Tunnel Delta Over Time

Binary 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 binary 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 20.91 and 29.76 over a 4.40 asset range.

At the other extreme, the 0.1-day value peaks at 2.43 (at 99.00) and falls to -2.39 at 101.00.

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 binary call is exactly offset by the long delta position of the upper strike binary 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 profiles are smooth which provides an easier ride for the hedger. The 25% implied volatility is relatively high, the profile has a shallow dip and the tunnel gamma is low.

The green 10% profile has a deeper dip followed by an even deeper dip from the red 6% profile. The red 6% profile now has two shoulders outside the strikes which are the preceding stage to the 2% profile. The individual lower and upper strike binary call vega make clear the construction of the tunnel vega.

The tunnel with high volatility is a fairly placid instrument as the long and short call that make up the tunnel cancel each other’s risk out.

By: Hamish Raw

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