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Put-Call Parity, Options and Arbitrage

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Question 1
What is put-call parity and why does it hold? Please show your proof?

Question 2
The following prices are observed. How are you going to profit from the opportunity?

Stock A is selling for $95.00
Call options on stock A with exercise price of 90 and with April expiration are selling for $9 per share
Put options on stock A with exercise price of 90 and with April expiration are selling at $2.5 per share
At the current t-bill rate, $89 invested today will grow to 90 at the option's maturity date.

Question 3
The following prices are observed. Use an arbitrage strategy to get an arbitrage profit.

Stock A is selling for $95.
Call options on stock A with an exercise price of 85 is selling for $12 per share
Call options on stock A with an exercise price of 90 is selling for $10 per share
Put options on stock A with an exercise price of 85 is selling for $1.25 per share
Put options on stock A with an exercise price of 90 is selling for $1.75 per share
At the current t-bill rate, $89 invested today will grow to 90 at the option's maturity date and 84.06 invested today will grow to 85 at the option's maturity date.

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Solution Preview

Please see responses below.

Question 1
What is put-call parity and why does it hold? Please show your proof?

Put-call parity is a relationship between the prices of call options and put options, when both options have the same expiration date and the same strike price.

Assume a put, a call, the underlying stock, and a riskless bond are all available, and that the put, call, and bond all have the same maturity. Then at each time until maturity, the relationship is

C + KB = P + S,

where

C = call price
K = option strike price
B = price of riskless bond which pays 1 at maturity
P = put price
S = stock price.

Proof: Let F be the stock price at the maturity date. If you hold a call option and K bonds today, then at maturity your holdings are worth K if F <= K. This is because your K bonds will be worth 1 each (total of K), and your call option will be worth 0. If on the other hand F > K, your holdings will be worth F, because the bonds will be worth K and the call will be worth F - K.

If you ...

Solution Summary

The solution discusses the put-call parity, options and arbitrage.

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