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.
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.
| European Digitals | Call Accumulator | Call Accumulator Gamma | Call Accumulator Theta | Call Accumulator Vega |
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.
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