The strike (exercise) price on a European call option is $10. The volatility of the stock is 40% and the risk-less rate is 5%. Which option has the highest gamma?

• Stock = $13 and option expires in 3 months
• Stock = $7 and option expires in 3 months
• Stock = $10 and option expires in 3 months
• Stock = $10 and option expires in 6 months

(don't peek until you try!)

 

 

 

 

 

Answer:

We don't really need the volatility and riskless rate to make comparisons, although they are needed for the calculation of Gamma as shown below

Gamma tends to be highest for options that are near to expiration (less time to maturity) and at-the-money. (but see graphs below for nuances).

Gamma is the second partial derivative with respect to the stock (underlying asset) price. Put another way, gamma is the rate of change of delta with respect to stock (asset price); i.e., the first derivative of the the first derivative. If delta is "velocity," gamma is "acceleration:" the rate of change in delta.

In the EditGrid below, I calculated gamma in the green matrix: stock price (top row) versus term (first column). The graph on the left, for term = 0.25 (3 months), shows how gamma tends to peak when the option in at-the-money. The graph on the right shows gamma for in-, at- and out-of-the money.

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