The eachway call option pays out for a win and a place. This bet consists of two strikes so that at expiry, if the underlying is above the higher strike, the bet settles at 1.00. Between the strikes the settlement value is a variable, for example 0:25:100, 0:40:100, 0:50:100, etc., which would generate settlement prices of 0.25, 0.40 and 0.50 respectively.

### Eachway Call Valuation

The Eachway Call is calculated as:

Eachway Call = R_{1} x Digital Call(K_{1}) + R_{2} x Digital Call(K_{2})

where K_{1} is the lower strike and K_{2} the upper strike digital call option. R is the ratio: R_{1} + R_{2} = 1 and R_{2} ≥ R_{1}.

R_{1} is the settlement value between the strikes.

### Ratio Call Strips

In options parlance an eachway call is a two strike (K_{1} & K_{2}) ratio call strip. In effect this would mean that the 0:40:100 eachway call is a 2 x 3 ratio digital call strip divided by 5 to constrain the values to the range 0.0 → 1.0.

Example: A client calls a broker and wants a quote in a ’31’ eachway call, meaning that the middle settlement price is 0.31.

This means that the price difference between the losing and winning settlement values are P_{1}=0.31 and P_{2}=(1.0 – 0.31) = 0.69.

The number of the lower strike that need to be bought are R_{1} = 0.31/P_{1} = 1 and the number of the upper strike needed buying is R_{2} = 0.69/P1 = 0.2258065.

The eachway call is then [R_{1} x Digital Call(K_{1}) + R_{2} x Digital Call(K_{2})] / (R_{1} + R_{2})

The R1 and R2 are shown for the most likely common ratios.

Ratio | 0:20:100 | 0:25:100 | 0:33.3333:100 | 0:40:100 | 0:50:100 |
---|---|---|---|---|---|

R1 | 1 | 1 | 1 | 1 | 1 |

R2 | 4 | 3 | 2 | 1.5 | 1 |

R1+R2 | 5 | 4 | 3 | 2.5 | 2 |

The results of using the above digital ratio call strip arithmetic turns out to be the same as valuing the Eachway Call below.

### Eachway Call At Expiry

Figure 1a shows the underlying between the two strikes settling at 0.40, while if the underlying price is below the lower strike the eachway call settles at zero.

Figures 1b shows the eachway call with the intermediate settlement values of 0.25.

As usual, if the underlying price settled on either of the strikes then the mean of the adjacent settlement prices would be used for settling the strategy, i.e. 0.20 and 0.70 for the lower and higher strikes respectively for Figure 1a, while 0.125 and 0.625 for Figure 1b.

Therefore, this particular digital options strategy is certainly not an all-or-nothing bet since it is possible to have five different settlement prices. The eachway could be perceived as a strategy that pays out a secondary settlement price for, in gambling parlance, ‘a place’.

The eachway call provides a second bite at the cherry in the case where the speculator forecasts the market inexactly. In the example of Figure 1a, maybe the asset price is moving upwards as forecast by the eachway call buyer but not quite at the pace required to get it over the line for the strategy to settle at 1.0. The secondary settlement price provides the consolation of calling the market right but getting the momentum wrong.

### Eachway Call Over Time

Figures 2a and 2b provide the 99.00/101.00 eachway call over a range of time to expiry to illustrate how the price profiles behave over time. The 8-day profile at 98.00 possesses a shallow gradient, thereby defining a low delta which reflects low gearing.

In the above illustration at 98.00 the 0:40:100 eachway call is worth 0.1099 compared with the 0:25:100 eachway call below of 0.0764. Both options would would generate a handsome return if the asset price at expiry was above 99.00, i.e. (0.40 – 0.1099)/0.1099 = 264% return for the above eachway call and (0.25 – 0.0764)/0.0764 = 227% for the below eachway call.

If the asset price over the eight days rose to above 101.00 at expiry the returns jump to (1.00 – 0.1099)/0.1099 = 810% and (1.00 – 0.0764)/0.0764 = 1029%. The 25-day profiles are more of a ‘slow burner’ with very low gearing. Midway between the strikes at 100.00 and 25 days to expiry the 0:40:100 is worth 0.4661 and the 0:25:100 option is worth 0.4215. This means that the 0:40:100 option at 100.00 with 25 days to expiry can look forward to losing only 0.0661 if the asset price remained unchanged to expiry in contrast to the 0.1715 of the 0:25:100 eachway call.

As time to expiry falls to the last day the profile slowly becomes steeper and only with 0.1-days to expiry does the profile resemble the settlement price profile of Figure 1. This creates some interesting eachway call theta profiles.

European Digitals | Eachway Call Delta | Eachway Call Gamma | Eachway Call Theta | Eachway Call Vega |

### Eachway Call & Volatility

In this section the 0:40:100 and 0:25:100 ratio eachway calls with 5 days to expiry are compared over the same range of volatilities.

One of the interesting features of a single digital call is that if out-of-the-money then an increase in volatility increases the call value. This is referred to as positive vega. Correspondingly, if the call is in-the-money then an increase in volatility lowers the call value. This is because when out-of-the-money a ‘vol’ increase means a higher probability of the call becoming in-the-money. An in-the-money call finds that a ‘vol’ increase means a higher probability of the call becoming out-of-the-money.

In the above two examples the price profiles cross each other between the strikes, but with the 0.25.100 ratio eachway call having this conjunction closer to the higher weighted 101.00 strike than the 0.40.100 ratio eachway call.

In effect, one would expect the volatility to have no impact on the eachway call value if the ratio was 0.50.100 and the asset price was midway between the strikes. This would equate to a zero eachway call vega. As the ratio gets increasingly skewed toward the upper strike then a change in volatility has decreasing effect on the eachway call value.

When the asset price is above the upper strike, with the ratio 0.25.100 the volatility’s effect on the eachway call is now very much like a single 101.00 call.

### Summary

- This form of a digital call strip is like having a rebate on a digital call just so long as the asset price is not below a pre-defined level. This pre-defined level is obviously the lower strike price.
- Just like with ratio conventional call spreads the eachway call provides the trader with all sorts of weird and wonderful combinations. The example of the ‘rebate’ being 0.31 is probably about as odd a number as one would get.
- The time to expiry can have an unexpected impact on the value of the eachway call which is probably to be expected due to the number of different combinations of strikes and rebates possible.
- Volatility will also add a touch of quirkiness to the value of the eachway call, especially when the middle settlement value is 0.5 and vega is both positive and negative between the strikes.