Problem Set We have now covered the basic properties of options. To conclude, he

Question

Problem Set
We have now covered the basic properties of options. To conclude, here are some exercises to
practice what you have just read. For the problems below refer to the following set of information. As
of July 26, the following European options, which expire on August 20, were written on XYZ
Corporation. Their prices are shown in Table C.
Table C
Exercise Price,| Calls,| Puts
$35,| $4 1/8,| $1 3/16
$40,| $1 5/16,| Not traded
$45|, $3/8,| Not traded
Source: Casewriters.
XYZ pays no cash dividends to its shareholders. A Treasury bill maturing on August 20 that pays
$35 at maturity costs $34.94 on July 26.
Problem 1 Could an American call option on a stock ever be worth less than a European call?
How does an increase in time to maturity affect the value of European calls? American calls?

Problem 2 Suppose the stock on a company is at $150, and the company pays no dividends.
Your friend has an American option on the stock with an exercise price of $100, but it is still a few
months to expiration. He has decided to lock in his profit and exercise the option. Can you think of a
way of arbitraging this situation?

Problem 3 Suppose that XYZ’s stock price on July 26 was $50. On that date, what is the very
least a $35 call could be worth? The most? What trade would you do if the call was $15?
Problem 4 What price would you expect the $45 put on XYZ above to trade at if the stock were
at $37.50? If it traded above this price, what arbitrage opportunity would exist?
Problem 5 Suppose a stock, which pays no dividends, sells for $10 today. Next period, it will
either move to $7 or $14. You do not know the probabilities of these two outcomes. Riskless zerocoupon
bonds, paying $1.10 in one period, cost $1.00 today.

1. What price would an at-the-money call sell for today?

2. If you wished to synthetically manufacture the at-the-money call option, how many bonds
would you buy? How many shares of stock?

3. If the call sold for $3.00, how would you capture arbitrage profits?

4. Now, consider what the stock might do in the second period. If it moves to $14 in the first
period, it can either move up to $18 or down to $11 in the second period. If it moves to $7 in
the first period, it can either move up to $11 or down to $4 in the second period. Assuming
that riskless zero-coupon bonds, paying $1.10 in the second period, cost $1.00 at the end of the
first period, what price would an at-the-money call sell for today?

Explanation / Answer

Problem 1

Could an American call option on a stock ever be worth less than a European call?

An American option can be exercised at any time, whereas a European option can only be exercised at the expiration date but American options carries all rights and privileges as a European option, it cannot be worth less than a European option. If it were you can make arbitrage profit by a selling European call and using part of proceeds to buy an equivalent American call option. Thus American call option on a stock ever be worth less than a European call.

How does an increase in time to maturity affect the value of European calls? American calls?

The increase in time to maturity of the option, the greater is the chance that at some time exchange rate will move above the strike price of a call and below the strike price of a put.

For American options, the longer the time to the exercise date more valuable the options. The holder of one year option can turn into six month option by simply exercising it early. The right to delay exercising option is worth something, so the option with later exercise date will be more valuable.

The longer the time to the exercise date more valuable the options cannot work for European options because one year option cannot turn into six month option.

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