Question

Use the put-call parity relationship to demonstrate that a butterfly spread using calls should cost the same as a butterfly spread using puts, when the underlying asset, strike price and maturity dates of each spread are the same.

Answer 1

Answer

In this case it is actually needed to take into consideration some of the most significant example in order to demonstrate cost of butterfly spread of European put is same to that of the cost of a butterfly spread using European Call.

Here in this case c1, c2 and c3 are the three-call price. On the other hand, p1, p2 and p3 are the put price whereas k1, k2 and k3 is considered as the strike price. The expansion of the same is shown below:

c1+ k1e^-rt = p1 + S

c2+ k2e^-rt = p2 + S

c3+ k3e^-rt = p3 + S

Hence,

c1+c3-2c2 + (k1+k3 – 2k2) e^-rt = p1+p3 – 2p2

as k2-k1 = k3-k2, it follows that k1+ k3 – 2k2 = 0 and

c1 + c3 – 2c2 = p1 + p3 – 2p2.

This is the process by which butterfly spread using European calls cost the same as a butterfly spread using European puts, when the underlying asset, strike price and maturity dates of each spread are the same.