# Eachway Puts

Eachway puts pay out for a win and a place. Eachway puts consist of two strikes so that at expiry, if the underlying is below the lower strike, the eachway put settles at 1.00. Between the strikes the eachway put settles at 0.25, but could be any number depending on the ratio, e.g. 100:40:0. Above the upper strike the bet settles at 0.

If the asset price is exactly the same as a strike price then the settlement price is the arithmetic mean of the above and below settlement price. So in the case of settlement prices of the ratio 100:40:0 the ‘dead heat’ settlement prices would be (1.00 + 0.40)/2 = 0.70 and (0.40 + 0)/2 = 0.20. Alternatively, 100:25:0 ratio would have dead heat settlement prices of 0.625 and 0.125.

### Eachway Put Valuation

The eachway put is calculated as:

Eachway Put    =    R1 x Digital Put(K1) + R2 x Digital Put(K2)

where K1 is the lower strike and K2 the upper strike.

R1 > R2 and R1 + R2 = 1

### Eachway Put at Expiry

Figure 1a shows the underlying between the two strikes settling at 0.50, while if the underlying price is below the lower strike the eachway put settles at 100. The at-the-money settlement values are now:

(1.00 + 0.50)/2 = 0.75 and (0.50 + 0)/2 = 0.25

Figures 1b shows the eachway put with the intermediate settlement values of 0.25. Here at-the-money settlements are:

(1.00 + 0.25)/2 = 0.625 and (0.25 + 0)/2 = 0.125

Digital options structures have now gone from the digital put options with three (not two binary) outcomes to an eachway put with five outcomes. To avoid semantics maybe these eachway puts should be referred to as ‘quinquery’ options. Yes, silly! Far  better to just acknowledge that digital options are not binary options.

### Eachway Puts over Time

Figures 2a and 2b provide the 98.50/101.00 eachway puts with ratios 100:50:0 and 100:25:0.

In both illustrations the 25-day profile shows the shallowest gradient and lowest gearing. The eachway put gamma will be constantly extremely low for this option with a long time to expiry.

With Figure 2, in general the gearing is reduced around the upper strike but increased at the lower strike. As the ratio tends towards 100:0:0 the gearing at the top  strike will be zero and at the lower strike that of the digital put option.

### Eachway Puts and Volatility

In this section the 100:50:0 and 100:25:0 eachway puts with 10 days to expiry are compared over the same range of volatilities.

The midpoint between the strikes is at (98.50 + 101.00)/2 = 99.75. In Figure 3a with ratio 100:50:0 the profiles pivot around the strike midpoint of 99.75. Here we have the situation where a volatility increase or decrease has no discernible effect on the eachway put value.

Figure 3b paints a different picture where, at 99.50, increases in volatility increase the value of the eachway put. At 99.50 this eachway put has a positive eachway put vega. In this instant an increase in the volatility increases the probability of both the payoff being both 100 or 0. The incremental increase in ‘vol’ increases the incremental expected return on the upside. This leads to all the profiles intersecting each other at a ‘breakeven’ point closer to the lower strike.

### Summary

Eachway calls and puts are as likely to be as popular as conventional ratio call and put spreads. 1×2, 1×3, 2×3 spreads are not uncommon strategies conventional options market-makers answer request for quotes for. A buyer of those strategies leaves themselves with an unlimited  loss profile. Eachway calls and puts always have a capped loss  profile.

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