Three European put options on a stock have the same expiration date and strike p

Question

Three European put options on a stock have the same expiration date and strike prices of $55, $60, and $65. Their market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be created. What is the cost of the spread? Find the payoff (which ignores the cost) and profit (which includes the cost) from the following stock price ranges S65 60S<65 55S<60 S<55 For what range of stock prices would the butterfly spread lead to a loss? Consider three European call options on the same stock with the same expiration date and the same three trike prices of $55, $60, and $65. Their market prices are c1, c2, and c3, respectively. Use put–call parity to show that the cost of the butterfly spread created from these options is identical to that in Part (A). This is a general result which holds for any butterfly spreads.

Explanation / Answer

Butterfly spread: Buy 1 65 Put and 1 55 Put and sell 2 60 Put

Cost=8+3-2*5=1

Payoff

Payoff=Max(65-S,0)+Max(55-S,0)-2*Max(60-S,0)

S<=55: 65-S+55-S-2*(60-S)=0

55<=S<=60: 65-S-2*(60-S)=-55+S

60<=S<=65: 65-S

S>=65: 0

Profit

S<=55: 0-1=-1

55<=S<=60: -55+S-1=-56+S

60<=S<=65: 65-S-1=64-S

S>=65: 0-1=-1

Cost of butter fly spread using calls from put call parity:

C1=P1+S-55e^(-rt)

C2=P2+S-60e^(-rt)

C3=P3+S-65e^(-rt)

C1+C3-2*C2=P1+P3-2P2+S+S-2*2-(55+65-2*60)*e^(-rt)=P1+P3-2P2

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.