Delta/Gamma movements with Options

CFA Level II
#1

Can someone please explain why the following holds?

Long Call = +D, +G

Short Call = -D, - G

Long Put = -D, +G

Short Put = +D, -G

Thank you.

#2

Delta = Change in option price / Change in stock price

Long call: As stock price increases, value of call option increases and therefore value of position in long call increases. [+]

Short call: As stock price increases, value of call option increases and therefore value of position in short call decreases. [-]

Long put: As stock price increases, value of put option decreases and therefore value of position in long put decreases. [-]

Short put: As stock price increases, value of put option decreases and therefore value of position in short put increases. [+]

As for Gamma, am not sure about the above relationships.

From my understanding, if option is at-the-money and near expiration, Gamma is high and thus difficult to predict Delta.

#3

Thanks Kevin, that helps solidify some grey areas.

Can anyone else contribute to the positive and negative movements of Gamma with respect to my original post?

Thanks

#4

If you’re trying to memorize, your chart shows gamma positive for long positions, negative for short positions.

If you’ve had any calculus, think of gamma as the second derivative. The long positions will form a concave up curve and the short positions will form a concave down curve. A concave up (down) curve has a positive (negative) second derivative. And I’m sure someone will point out it is technically a second partial derivative, but I’m prolly reaching with the calculus analogy.

#5

gamma is the rate of change of delta so it measures the curvature of the slope whether it is increasing at an increasing/decreasing rate. Assume you long a call of EURUSD at 1.05 3 months ago and EURUSD is at 1.1 with 3 days to expiry. Also assume that Delta then when you long the EURUSD is 0.2 or any number. When close to the expiry date, delta will be close to 1 especially for such an in the money call option. So, during the 3 months when spot is moving higher to 1.1, you can imagine that the delta will also be rapidly moving from 0.2 to 1 at an increasing pace too. Hope that explains

#6

Close to expiry, delta will be close to 1 for almost any in-the-money call and close to zero for almost any out-of-the-money call.

#7

Thanks Spytheman. Helps a bit.

S2000 do you have another explanation for the gamma question I had? Thanks