Gamma--why is this true? price, positive, exercise

IAmNeil

"Call and put options on the same underlying asset with the same exercise price and time to maturity will have equal gammas. Long positions in calls and puts have positive gammas."

How are those conclusions drawn?

yodacaia

imagine you bought a call ands sold a put. all parameter are same. it is synthetic forward.

delta of forward = 1

say delta of long call = 0.5
delta of short put = 0.5
long call delta + short put delta = 1

spot goes up

forward delta no change stays = 1

long call delta goes to say = 0.7
short put delta must go to = 0.3
because we have synthetic forward and we know that total delta must be 1

both deltas changed by 0.2 - same gamma

tarunajwani

What this says (which is a bit strange but I'm not too sure of that) is that if the stock goes up by $1, then both the call and the put will change by the same amount (not percentage either)! That's what gamma says: change in delta/change in stock.

If stock price is $10, strike =$20, call price = $0.10, put price =$10....are they saying that if stock goes up to $11, both the call and the put will change by same amount? The put has just dropped $1...so that can't be true.

> Long positions in calls and puts have positive gammas."

This is fine because the deltas move in the same direction as the underlying price.

President1988

If you look at put/call parity you will see that only options are the ones with any gamma. They will have same vega too, delta will be different.

Houjichasan

still not following.

are they basically just saying that the rate of change of delta with a given change in stock price is the same for both calls and puts (with puts being the opposite direction)?

i guess this makes sense since the graphs of the two (pg 84 schweser book 5) look the same.