You hold two types of calls and two types of puts on a given stock. The deltas and gammas of the respective types are (+0.40, +0.03), (+0.55, +0.036), (−0.63, +0.028), and (−0.40, +0.032). You have a long position in 1,000 of the first type of call, a short position in 500 of the second type of call, a long position in 1,000 of the first type of put, and a short position in 500 of the second type of put. 

(a) What is the aggregate delta of your portfolio? The aggregate gamma? 

(b) Suppose you decide to gamma hedge your portfolio using only the first type of call. What is the resulting delta of the new portfolio? What position in the underlying is now required to create a delta-neutral gamma-neutral portfolio?