Call Accumulator Gamma

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Binary options 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:

\Gamma = \frac{dΔ}{dS}

where Δ is the call accumulator delta value and S is the asset price.

In effect the binary call gamma is the gradient of the call accumulator delta profile of the binary call option.

Evaluating Call Accumulator Gamma

Call Accumulator Gamma =  Payout1 x Binary Call Gamma(K1) + Payout2 x Binary Call Gamma(K2)

+ Payout3 x Binary Call Gamma(K3) + Payout4 x Binary Call Gamma(K4)

where the terms are the binary call options gamma with strikes K1, K2, K3 & K4 respectively. The payouts in the above example are:

Payout1 = 10%, Payout2 = 20%, Payout3 = 30% and Payout4 = 40%.

Call Accumulator Gamma Over Time

The call accumulator 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.

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

Call Accumulator Gamma and Volatility

Figure 2 shows the gamma over a range of implied volatilities. Unfortunately the 2% volatility profile does not add much to this illustration as it is not easily interpreted.

Call Accumulator Gamma w.r.t. Volatility
Figure 2 – Call Accumulator Gamma w.r.t. Volatility

The 6% and above profiles show clearly positive gamma at low asset prices and negative at higher asset prices.

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