Call accumulator gamma describes the change in the fair value of a call accumulator delta due to a change in the underlying price. The call accumulator gamma is the first derivative of the call accumulator delta with respect to a change in underlying price. It is depicted as:
where Δ is the call accumulator delta value and S is the asset price.
In effect the call accumulator gamma is the gradient of the call accumulator delta profile.
Evaluating Call Accumulator Gamma
Call Accumulator Gamma = Payout1 x Digital Call Gamma(K1) + Payout2 x Digital Call Gamma(K2)
+ Payout3 x Digital Call Gamma(K3) + Payout4 x Digital Call Gamma(K4)
where the terms are the digital call gamma with strikes K1, K2, K3 & K4 respectively.
Payout1 = 10%, Payout2 = 20%, Payout3 = 30% and Payout4 = 40%.
Call Accumulator Gamma Over Time
The call accumulator gamma is displayed against time to expiry in Figure 1. The 0.1-day profile shows the volatility of this metric with the profile on a switchback tide through the strikes.
The flatness of the call accumulator delta with 8 and 25 days to expiry leads to the flatness of the 8 and 25 day gamma. The gamma is positive at the lower asset prices for the 2, 8 and 25 day profiles. All profiles turn negative above the upper strike.
| European Digitals | Call Accumulator | Call Accumulator Delta | Call Accumulator Theta | Call Accumulator Vega |
Call Accumulator Gamma and Volatility
Figure 2 shows the gamma over a range of implied volatilities. Unfortunately the 1.5% volatility profile does not add much to this illustration apart from telling us that this metric is worthless at extremely low volatility.
The 4.5% and higher volatility profiles show clearly positive gamma at low asset prices and negative at higher asset prices. Yet, above 102.20, we can already see the 1.50% profile back to zero while the 4.50% profile, having turned upwards from the trough, is also heading to zero. The 7.50%, 10.50% and 13.50% will all, in turn, head back to zero and remain there as the asset price increases.
Summary
As more strikes are added to what was originally a digital call option, then an eachway call option, to the present call accumulator the Greeks become alot less volatile as the gamma becomes an average of more strikes’ gammas. We have travelled from the most highly geared instrument available to this now placid creature with its low geared gamma.