# Eachway Tunnel Delta

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

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

where P is the eachway call value and S is the underlying price. The delta is the gradient of the slope of the eachway call price.

### Evaluating Eachway Tunnel Delta

Eachway Tunnel Delta = R1 x Binary Call Delta(K1) + R2 x Binary Call Delta(K2)

– R2 x Binary Call Delta(K3) – R1 x Binary Call Delta(K4)

where K1, K2, K3 & K4 are lowest strike to highest strike, and

where R1 + R2 = 1 and R1 < R2.

### Eachway Tunnel Delta Over Time

The eachway tunnel delta is displayed against time to expiry in Figure 1.

The two humps and troughs of the deltas of the individual strikes is evident with 0.1 days to expiry. The ratio of the eachway tunnel is R1 = 40% and R2 = 60%. This creates absolute deltas at the middle strikes considerably higher than at the outer strikes as expiry approaches.

The 0.5 (red) profile is beginning to show the individual deltas of each binary call option. Only with 8 days and more to expiry does the eachway tunnel delta take a shape without the humps and troughs which makes the eachway tunnel easier to hedge.

The eachway tunnel delta is always positive (or zero) below the underlying price midway between the central strikes. Above this midpoint the eachway tunnel delta is always negative or zero. The delta is always zero midway between the middle strikes. Conventional options traders will recognise the similarities between being a short at-the-money straddle and long the eachway tunnel. If the underlying moves away from 100.00 in the above example should one start hedging with the underlying?

Example: There are two days to expiry and a trader buys the eachway tunnel for 68.33. Just say that the trader bought 10 contracts and each contract is worth $68.33 then the trader has$683.30 at risk. If the asset price is between 99.50 and 100.50 the trader makes a profit of $316.70. If the underlying is between 98.50 and 99.50 or between 100.50 and 101.50 the trader loses$683.30 – $400 =$283.30. Outside 98.50 and 101.50 the trader loses the $683.30. The asset price drifts down to 99.50 where the delta is 0.275. 10 contracts x 0.275 means the trader should sell$2.75 of the underlying in order to neutralise the directional risk. But as the reader can see, the 2-day eachway tunnel delta continues to rise as the underlying falls further from 99.50. At 98.90 the delta is now 0.39 which would mean further losses accrued by the eachway tunnel although the hedge at 99.50 would be up. Should the trader sell more of the underlying at 98.90?

If the trader sold more at 98.90, 0.39 – 0.275 more, then if the asset price continued south then the trader’s underlying makes more than the eachway tunnel loses. This is because the delta is now declining in value. The underlying is falling and the position is getting shorter as the trader is now overhedged. This is now a long eachway tunnel gamma position.

But what if the asset rallies as soon as the trader hedged at 99.50. If the underlying rose back to 100.00 the delta is zero. The short underlying would then be required to be bought back if the trader is assuming a delta neutral stance. The trader is back where they started but with a  loss of \$275.00, nearly as much as the potential profit on the initial trade. And there are  still two days to go……

Binary options in general are a nightmare to hedge as too often the instrument can have both positive and negative Greeks. Conventional options do not suffer these gyrating Greeks, e.g. a long call position is always long delta, long vega, long gamma and short theta. A long binary call option is long delta but the gamma, theta and vega can be long or short.

If you buy or sell the eachway tunnel be ready to wear it to expiry.

### Eachway Tunnel Delta and Volatility

Figure 2 provides the eachway tunnel delta over a range of implied volatilities. Yet again the delta is always negative or zero above the midpoint of the inner strikes. The delta is always positive below that midpoint.

As time to expiry and implied volatility rise Figures 1 & 2 both reflect the more conservative nature of the eachway tunnel. Even when both time to expiry and implied volatility fall markedly the maximum absolute delta is still well below the deltas of the individual outright calls.

In summary, if one had to choose between selling a conventional straddle or going long an eachway tunnel the risk averse trader has to choose the latter. The straddle seller is naked short whereas the eachway tunnel, as with the simple tunnel, is a  limited risk proposition.

By: Hamish Raw

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