# Call Accumulator

The call accumulator is an extension of the eachway call. There are four strike prices with the settlement prices at 0, 10, 30, 60 and 100 as outlined in Figure 1. This is a strategy which, when bought out-of-the-money, can provide extremely high gearing, and increasingly rewards the speculator for an increasingly accurate forecast.

The Call Accumulator is calculated as:

Call Accumulator    =    R1 x Binary Call(K1) + R2 x Binary Call(K2) + R3 x Binary Call(K3) + R4 x Binary Call(K4)

where:

K1 < K2 < K3 < K4

and:

R1 < R2 < R3 < R4 and R1 + R2 + R3 + R4 = 1

In the below example R1 = 10%, R2 = 20%, R3 = 30% and R4 = 40%

### Call Accumulator at Expiry

The calculation above shows the call accumulator to be a weighted average of the four individual binary call option’s values. The sum of the ratios must come to 1, e.g. 0.1 + 0.2 + 0.3 + 0.4 =  1. Clearly there are many combinations that add to 1 but the incremental increase of 0.1 as illustrated in Figure 1 lends itself to the betting terminology of ‘accumulator’ or ‘accy’ for short.

Our ‘binary’ options have now grown to nine different possible settlement values although the prices are still constrained by the limits of 0 and 1 (or 100, as in this pricing context).

### Call Accumulator Over Time To Expiry

The profiles of Figure 2 show how the call accumulator is affected over the passage of time. Since the strategy is a weighted average of the individual binary call options the profiles closely map each other until there is only one day or less to expiry. As with the eachway call option this strategy can be made more or less aggressive in terms of delta by adjusting the gaps between the strikes.

The binary options call accumulator provides a smooth price profile for the strategy with even only 0.5 days to expiry. It is only with 0.1-days to expiry that the profile looks remotely like the expiry profile. Subsequently, with 25 days to expiry this strategy is a great deal less risky for the market-maker as well as the speculator who buys it out-of-the-money. Why?

• the profile reflects a low delta which means directional risk at the trade’s inception is low.
• the vega risk is low as witnessed by Figure 3, and
• at expiry the maximum incremental change is just 30 (between the two highest strikes) which vastly reduces ‘pin’ risk1.

With these risks taken out of the equation the market-maker is likely to produce highly competitive, tight bid/ask quotes.

### Call Accumulator and Volatility

Figure 3 shows the effect of changes in implied volatility on the call accumulator. Even though the range of implied volatilities is wide the price profiles map each other closely reflecting a very low vega.