# Call Accumulator Delta The 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: where P is the call accumulator value and S is the asset price.

The call accumulator delta is the gradient of the price profile of the call accumulator.

### Evaluating Call Accumulator Delta

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

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

where the terms are the digital call delta with strikes K1, K2, K3 & K4.

and where

K1 < K2 < K3 < K4

and

Payout1 = 0.1, Payout2 = 0.2, Payout3 = 0.3 and Payout4 = 0.4.

### 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, 0.10, 0.30, 0.60 and 1.00. Figure 1 – Call Accumulator Delta w.r.t. Time to Expiry

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 call accumulator 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 digital call deltas. Figure 2 – Call Accumulator Delta w.r.t. Volatility

As volatility increases the delta falls to reflect the fact that an increase in volatility flattens out the price profile of the underlying call accumulator.

### Summary

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.

By: Hamish Raw

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