Question

Show algebraically the payoff of a butterfly spread using calls in the following cases:

– ST < K1
– K1 < ST < K2
– K2 < ST < K3
– ST > K3

Assume K2 is the average of K1 and K3: K2 = .5*(K1 + K3) or 2K2 = K1 + K3 (i.e., the butterfly trade is symmetric). You can use this relationship to simplify the final expressions quite a bit.

Answer 1

Solution :----

The general formula for the payoff of the butterfly spread.

Payoff = max(ST – K1, 0) + max(ST – K3, 0) + 2(min(K2-ST, 0)),

where K1 and K3 (We can assume the K3>K1) are the strike prices for the two European long call, K2 is the strike price for the two European short calls, and ST is the stock price at time T

Then let us discuss:

1, ST < K1 Then payoff= 0+0+0 = 0

2, K1 < ST < ST K3 Then payoff= (ST – K1) + 0 + 0 = ST – K1

4, K2 < ST <K3

Then payoff=(ST – K1) + 0 + 2(K2 – ST)= 2K2 – K1 - ST = K3 – ST

5, ST > K3

Then payoff= (ST – K1) + (ST – K3) + 2(K2 – ST)= 2K2 – K1 – K3 = 0

From the discussion, we know

                               ST-K1     if K1 < S < K2

Payoff =                 K3-ST    if K2 < ST < K3

   0 , Otherwise

K2 = (K1 + K3)/2

K2 Will lie in between k1 and K3 in the payoff with relation to ST