## COMMERCE 4FE3:Options and Futures

 All interest rates herein are expressed with continuous compounding.
 In case you need to make assumptions, please clearly state them. Only reasonable
assumptions are acceptable. Assumptions that violate finance principles are not to be
 For numerical questions, simply calculating the numbers out will not be sufficient.
Please also provide me with the reasons for your calculations. Correct thinking
processes are much more important than correct numbers. Therefore, regardless of
whether or not your numbers are correct, no marks will be given if (a) no explanation
of your calculations is provided; or (b) the explanation provided is incorrect. On the
other hand, even if your numbers are incorrect, partial marks will be given if your
thinking process is correct.
 Please note that marks can only be given based on what you write. Therefore, I request
that you attempt to convey your ideas to me as clearly as possible. I will not make a
guess as to what you intend to mean if you do not make it obvious and clear.
mention all the relevant details such as the timing of the transactions, the side of the
contracts (e.g., long or short forward), the length of the contracts (e.g., 6-month
futures), the exercise prices (in case of options) and the interest rates (in case of lending
or borrowing).

2
1) (3 points) Consider an index-linked note that has a maturity of 3 years. Suppose you
buy this note today (say, with a principal of \$100). After three years, you will get the
principal back plus interest at the rate equal to the percentage increase in the TSX 60
stock index over the three-year period up to 30%. If the index declines over the threeyear period, the interest will be zero.
As an example, suppose that the TSX 60 index today is 800. The percentage return on
the note will be as follows:
– If after 3 years, the index is ≤ 800, the percentage return on the note will be
zero (i.e., you will only get the principal back).
– If after 3 years, the index is between 800 and 1,040 (i.e., an increase of up
to 30% over the three-year period), the percentage return on the note will
be equal to the percentage increase in the index.
– If after 3 years, the index is > 1,040 (i.e., an increase of > 30% over the
three-year period), the percentage return on the note will be 30%.
Please explain how you can replicate the payoffs of this note.
2) (3 points) You want to short a 6-month forward contract on a stock. You contacted
your bank and were offered a forward price of \$39.85 [Note: This forward price is
available only to customers who want to take a short position. Customers who want to
take a long position will get a different quote.]. You also observe the following
information:
– Current price of the stock = \$40
– Expected dividend on the stock = \$0.50, payable 3 months from now
– Your spot 3-month lending rate = 2.50% p.a. [i.e., This is the rate that you will
get if you lend money for 3 months, starting now.]
– Your spot 3-month borrowing rate = 4.00% p.a. [i.e., This is the rate that you
will have to pay if you borrow money for 3 months, starting now.]
– Your transaction cost in buying one stock in the spot market = \$0.10, payable
at the time of the transaction.
– Your transaction cost in short selling one stock in the spot market = \$0.20,
payable at the time of the transaction (This already includes the transaction cost
for closing out the short-sale transaction).
What is the minimum 6-month spot lending rate that you need to get in order for you
to reject the bank’s offer?
3
3) (3 points) A friend of yours believes that Apple stock is now overvalued. As a result,
she expects that their stock price, now at \$140 per share, will not go up any further.
She is prepared to back up her conviction by offering you the following bet. If in
exactly one year from now Apple stock price is higher than \$140/share, she will give
you two hundred times the amount of the difference (e.g., suppose the price in one year
is \$145 per share, you will get \$1,000). On the other hand, if the price in one year ends
up below \$140/share, you will have to give her two hundred times the amount of the
difference (e.g., suppose the price in one year is \$137 per share, you will have to give
her \$600). Suppose that the expected return on Apple stock is 10% p.a. Suppose also
that the current risk-free rate is 2% p.a. and Apple does not plan to pay dividends in the
next year. What is the value of this bet to you?
Hint: Write out the payoff function of this bet. What does it look like to you?
4) (3 points) Your boss is offering you a choice between two bonus schemes. Under the
first scheme, you will receive a sum of \$20,000 per year for three years, with the first
payment starting one year from now. Under the second scheme, you will receive 100
shares of your company’s stocks per year for three years, with the first payment starting
one year from now.
You observe the following information:
Current spot price of your company’s stock: \$190/share
Spot interest rate (for both lending and borrowing): 4% p.a. for all terms
Continuous dividend yield on your company’s stock: 1% p.a.

Which bonus scheme will you choose?
Hint: The stock scheme is risky because you do not know