# Put Accumulator Gamma 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. Figure 1 – Put Accumulator Gamma w.r.t. Time to Expiry

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.

### 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. Figure 2 – Put Accumulator Gamma w.r.t. Volatility

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.

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