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Positive Gamma, Negative Gamma

Gamma is the first derivative of delta in the theoretical pricing model. The rate of change for delta with respect to the underlying security's price, gamma changes relative to underline security prices first, and second, the options contract time to expiration.

Option contracts strike: 35, underlying security price: 35, expiration: 30 days, interest rate: 4%, dividend yield: 2%, implied volatility: 46%. ATM option contracts gamma is higher in any time interval.

For example, an option contracts strike price: 35, underlying security price: 35, expiration: 175 days, interest rate: 4%, dividend yield: 2%, implied volatility: 46%. For the call option contract holder calculations, delta 58 (decimal 0.58), gamma 3.5 (in decimal 0.035). If the underlying security changes up a dollar from $35.00 to $36.00, the call option contract price will rise by $58.00. New delta is the old delta plus the gamma, or 58 +3.5 = 61.5, new gamma 3.33 (in decimal 0.0333).

Put option contract holder calculations: delta -41.9 (decimal -0.419), gamma 3.5 (in decimal 0.035). If the underlying security changes up a dollar from $35.00 to $36.00, the put option contract price will drop by $41.90. New delta is the old delta plus the gamma, or -41.9 + 3.5 = -38.4, new gamma 3.33 (in decimal 0.0333).

90 Days to Expiration
Call option contract holder calculations: delta 55.8 (decimal 0.558), gamma 4.93(in decimal 0.0493), If underlying security changes up a dollar from $35.00 to $36.00 the call option contract price will rise by $55.80. New delta is the old delta plus the gamma, or 55.88 + 4.93 = 60.81, new gamma 4.67 (in decimal 0.0467).

Put option contract holder calculations: delta -44.2 (decimal -0.442), gamma 4.9 (in decimal 0.049).  If the underlying security changes up a dollar from $35.00 to $36.00, the put option contract price will drop by $44.20. New delta is the old delta plus the gamma, or -44.2 + 4.9 = -39.3, new gamma 4.67 (in decimal 0.0467).

30 Days to Expiration
Call option contract holder calculations: delta 53.4 (decimal 0.534), gamma 8.6 (in decimal 0.086). If the underlying security changes up a dollar from $35.00 to $36.00, the call option contract price will rise by $55.80. New delta is the old delta plus the gamma, or 53.4 + 8.6 = 62, new gamma 8 (in decimal 0.08).

Put option contract holder calculations: Delta -46.6 (decimal -0.466), gamma 8.61 (in decimal 0.0861). If the underlying security changes up a dollar from $35.00 to $36.00, the put option contract price will drop by $46.60. New delta is the old delta plus the gamma, or -46.6 + 8.61 = -37.99, new gamma 8 (in decimal 0.08).

The gamma accelerates when it reaches maturity (expiration date).

Two Days to Maturity
Call option contract holder calculations: Delta 50.8 (decimal 0.508), gamma 33.5 (in decimal 0.335).  If the underlying security changes up a dollar from $35.00 to $36.00, the call option contract price will rise by $55.80. New delta is the old delta plus the gamma, or 50.8 + 33.5 = 84.3, new gamma 22.7 (in decimal 0.227).

Put option contract holder calculations: Delta -49.1 (decimal -0.491), gamma 33.5 (in decimal 0.335). If the underlying security changes up a dollar from $35.00 to $36.00, the put option contract price will drop by $49.10. New delta is the old delta plus the gamma, or -49.1 + 33.5 = -15.6, new gamma 22.7 (in decimal 0.227).