# Eachway Call 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. It is depicted as:

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

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

### Evaluating Eachway Call Delta

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

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

R1 and R2 are the payouts at expiry assuming between strikes (R1) and above upper strike (R1 + R2).

Furthermore, R1 + R2 = 1 and R2 > R1.

### Eachway Call Delta Over Time

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

The two humps of the deltas of the individual strikes is evident. The ratio of the eachway call is 40% and 60% of each individual binary call option then plainly the delta at the upper strike is considerably higher than at the lower strike as expiry approaches.

The 0.5 (red) profile clearly show the individual deltas of each binary call option. Only with 8 days to expiry do the two separate deltas take a shape without humps which in turn makes the eachway call much easier to hedge.

Example: With 0.5 days to go and the asset trading around 99.00 the trader buys 10 99.00/101.00 eachway calls. The trader then sells the asset to get delta-hedged. As the price moves up the delta falls so the short assets required to hedge falls. This persists until the asset price reaches 100.00. During this time the trader has been buying back the asset (at a loss) to remain delta neutral.

The asset price then continues its upward momentum so the trader starts selling the asset again until the underlying price is 101.00. So having bought the asset back at a loss between 99.00 and 100.00 what to do now? The trader is in exactly the same position as when the asset was trading at 99.00. The difference now is that the trader has to be short more asset than at 99.00 to be delta neutral.

Should the asset price keep going up from 101.00 to 102.00 then the trader will lose more than in the rally from 99.00 to 100.00 as the trader now has more asset to buy back.

Maybe better to just buy the eachway call naked?

### Eachway Call Delta Over Time

Figure 2 provides the Eachway Call delta over a range of implied volatilities. As implied volatility falls from 18% the profiles only gradually reflect the positions of the individual strike prices. The featureless profiles continue until the volatility falls to 2% where the humps at the strikes are pronounced.

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

In summary, the eachway call feature of returning 40% for a ‘place’ provides a strategy with a fair value that moves far less aggressively than the outright binary calls. This may well attract the more conservative trader as risk is reduced.

By: Hamish Raw

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