## COMMERCE 4FE3:Options and Futures

Please show all your works and use 4 decimal points in your answers.

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

made.

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.

When your answers involve taking positions in securities or lending/borrowing, please

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