# Call Accumulator

### Call Accumulator Evaluation

The Call Accumulator is calculated as:

Call Accumulator    =    R1 x Digital Call(K1) + R2 x Digital Call(K2) + R3 x Digital Call(K3) + R4 x Digital 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 Accy at Expiry

The calculation above shows the call accumulator to be a weighted average of the four individual digital 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 three settlement price digital call options have now grown to nine different possible settlement values although the prices are still constrained by the limits of 0.0 and 1.00.

### Call Accy 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 digital 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 digital 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 risk.

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.

If the 1.50% volatility profile was omitted then the range of volatility reveals four profiles with only marginal difference when 5 days to expiry.

### Summary

This is a low risk directional expiry which now decries the objections of molly-coddling regulators who may consider digital options as too ‘hit-or-miss’, too ‘all-or-nothing’. “Ah, but this is a contrived strategy constructed of a number of different digital call options!” Absolutely! Just as every financial instrument ever invented is.

Just suppose that instead of four strikes there was a strike midway between each integral price, i.e. a strike every 0.005 of a price increment. Now there would be (101.50 – 98.50) / 0.01 = 3/0.01 = 300 strikes. Now add to this feature that the payoff of each incremental strike was not 0.1,  0.2, 0.3 and 0.4 but all payoffs were 1/300. We have just created a conventional call spread or a limit up/limit down future.

By: Hamish Raw

You cannot copy content of this page