Question

6. The following table shows the premiums of European call and put options having the same underlying stock, the same time to expiration but different strike prices: StrikeCall Premium Put Premium $20 $23 $25 $3.59 $2.45 $1.89 $2.64 $4.36 $5.70 You use the above call and put options to construct an asymmetric butterfly spread with the following characteristics (i) The maximum payoff of 6 is attained when the stock price at expiration is 23 (ii) The payoff is strictly positive as long as the stock price at expiration is strictly between 20 and 25 Calculate your profit from the asymmetric butterfly spread if the stock price at expiration is 21.(Hint: You may need to use Put Call Parity for two different strike prices to find eT)

Answer 1

Butterfly spread is constructed by using options at 3 different expiry. It is used when the trader believes the price of the underlying asset will not deviate much from the current price. Hence, it is a neutral strategy.
A butterfly spread is constructed using 4 options, at 3 different strike prices. the middle strike price is called the body and the 2 strike prices at the end are called as winds.
If the trader believes that the price of the underlying will not move much, it can use a long butterfly strategy. If the trader believes that the price of the underlying will move a lot, it can use a short butterfly spread
The problem states that maximum payoff is attained when the stock price at expiration is 23 (body) and the payoff is positive between stock prices 20 and 25 (wings). Hence, the assumption is the trader wants to construct a long butterfly spread where he does not expect large price movements.

Long call butterfly-
Buy 1 call each at strike price 20 and 25 and sell 2 calls at strike price 23
Cash flow from premiums

=-3.58-1.89+2*2.45	-0.57
Stock price at expiry	23

Options payoff at expiry
strike price	State	Payoff
20	In the money	3
23	At the money	0
25	Worthless	0
Net cash flow
=3-0.47	2.53

Long put butterfly-
Buy 1 put each at strike price 20 and 25 and sell 2 puts at strike price 23

Cash flow from premiums
=-2.62-5.7+2*4.36	0.4
Stock price at expiry	23

Options payoff at expiry
strike price	State	Payoff
20	Worthless	0
23	At the money	0
25	In the money	2
Net cash flow
=2+0.4	2.4

Payoff table from options (ignoring the premiums)-
Strike price	Call butterfly	Call butterfly	Put butterfly	Put butterfly
20	=0+0+0	0	=0-2*3+5	-1
21	=1+0+0	1	=0-2*2+4	0
22	=2+0+0	2	=0-2*1+3	1
23	=3+0+0	3	=0-0+2	2
24	=4-1*2+0	2	=0-0+1	1
25	=5-2*2+0	1	=0+0+0	0
	Average payoff	1.5		0.5

It appears that long call butterfly is more profitable. Hence, we will use - Long call butterfly strategy. Since this is an asymmetric butterfly, the payoffs are different. At spot price of 21 at expiry, the payoff from options is $1. Including the option premiums, the net payoff becomes- 1-0.57= $0.43