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Put-Call Parity, Options and Arbitrage

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.

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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