# Call Accumulator Delta

Binary options call accumulator delta describes the change in the fair value of a call accumulator due to a change in the underlying price. The call accumulator delta is the first derivative of the call accumulator fair value with respect to a change in underlying price. It is depicted as:

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

where P is the binary call value and S is the asset price.

In effect the binary call delta is the gradient of the price profile of the binary call option.

### Evaluating Call Accumulator Delta

Call Accumulator Delta =  Payout1 x Binary Call Delta(K1) + Payout2 x Binary Call Delta(K2)

+ Payout3 x Binary Call Delta(K3) + Payout4 x Binary Call Delta(K4)

where the terms are the binary call options delta with strikes K1, K2, K3 & K4 respectively. The payouts in the above example are:

Payout1 = 10%, Payout2 = 20%, Payout3 = 30% and Payout4 = 40%.

### Call Accumulator Delta Over Time

The call accumulator delta is displayed against time to expiry in Figure 1. It is not until time to expiry falls to 1-day that the humps of the deltas of the individual strikes is evident. The 0.1-day profile clearly shows the at-the-money delta increasing as the underlying rises which reflects the increasing payoff, in this case, 10, 30, 60 and 100.

The 25-day call accumulator is clearly a most benign animal with delta of roughly 0.1 indicating that if $10,000 was spent on this eachway call only$1,000 of the underlying need to be shorted to negate directional exposure.

### Call Accumulator Delta and Volatility

Figure 2 shows the delta over a range of implied volatilities. Only the 2% ‘vol’ indicates that the call accumulator delta is actually made up of four independent binary call deltas.

What is also apparent from both Figs 1 & 2 is how low the deltas are for the call accumulator. From the market-maker’s perspective this is a very low risk strategy to assume either a long or short position in so one would expect a keenly priced market, which in turn makes it very attractive to the end-user.

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