The put accumulator gamma describes the change in value of a put accumulator delta due to a change in the underlying price. This gamma is the first derivative of the put accumulator delta with respect to a change in underlying price. It is depicted as:
where Δ is the put accumulator delta value and S is the asset price.
Evaluating Put Accumulator Gamma
Put Accumulator Gamma = R1 x Digital Put Gamma(K1) + R2 x Digital Put Gamma(K2)
+ R3 x Digital Put Gamma(K3) + R4 x Digital Put Gamma(K4)
where the terms are the individual digital put gamma with strikes K1, K2, K3 & K4 respectively.
Where:
K1 < K2 < K3 < K4
and where:
R1 + R2 + R3 + R4 = 1 and R1 > R2 > R3 > R4
The payouts in the below examples are:
R1 = 40%, R2 = 30%, R3 = 20% and R4 = 10%
Put Accy Gamma Over Time
The put ‘accy’ delta 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 | Put Accumulator | Put Accumulator Delta | Put Accumulator Theta | Put Accumulator Vega |
Put Accy Gamma and Volatility
Figure 2 shows the gamma over a range of implied volatilities. It’s fair to say that with this amount of time to expiry (25 days) the gamma will not inject any excitement into the trading.
It is not until the 2% profile travels don below the lowest strike that anything of note takes place. Then we see a nosedive to -0.4 as the delta recovers from its own low.