1. Distinguish between hedgers, speculators, and arbitrageurs. 2. ATX is a European call option on stock A and BED is a European call option on stock B with parameters as given in the following chart. Assume that both stock A and B pay no dividends.
Call
T
X
S
r
Option price
A
0.25
105
100
0.05
$15
B
0.25
105
100
0.05
$12
Which stock, A or B, has the higher volatility? Explain.
(5 marks) 3. The market price of a one-year European call option on a non-dividend-stock is $4. The strike price of the option is $42, the stock price is $40, and the risk-free rate is 8%.
a. What is the price of an otherwise identical put option?
(6 marks) b. What is the price of an otherwise identical American call option?
(4 marks) 4. A three-month European call on a non-dividend-paying stock is currently selling for $1. The strike price of the option is $80. The underlying stock trades for $80. The risk-free interest rate is 6%. Does there exist any arbitrage opportunity? If yes, what type of transaction should you execute to make an arbitrage profit?
(15 marks) 5. A stock price is currently $60. Over each of the next two six-month periods, it is expected to rise by 10% or drop by 10%. The risk-free rate is 4% for each period. Using the two-period binomial model, calculate the option price in the four situations that follow.
a. Calculate the price of a one-year European call option on the stock with a strike price of $62.
(6 marks) b. Calculate the price of a one-year European put option with a strike price of $62. Is your result consistent with the put-call parity?
(6 marks) c. Calculate the value of a one-year American call option with a strike price of $62. What is the early exercise premium?
(4 marks) d. Calculate the value of a one-year American put option with a strike price of $62. What is the early exercise premium?
(4 marks) 6. A non-dividend-paying stock sells for $20 per share. The continuously compounded risk-free interest rate is 8% per annum, and the volatility of the stock price is 20% per annum. Using the Black-Scholes model, determine the price of a six-month European call on the stock with a strike price of $20.
(14 marks) 7. Consider three six-month European put options on a stock with the strike prices of $100, $110, and $120. The market price of the stock is $105, and the market prices of the puts are $11.50, $6.50, and $3.30, respectively.
a. Construct a butterfly spread using those three put options.
(4 marks) b. Use formula, table, and diagram to show the payoff and profit pattern of the butterfly.
(6 marks) c. For what range of stock prices would the butterfly spread lead to a profit?
(3 marks) d. What are the maximum potential profit and loss?
(2 marks) 8. The following is information regarding three one-year options on stock SBY. Assume the market price of stock SBY is $30.
Option
Call 1
Put 1
Call 2
Strike price
30
30
35
Option price
3.0
1.8
1.1
a. Construct a straddle using Call 1 and Put 1. Draw a diagram showing the profit from the strategy. Indicate the breakeven points and the maximum loss.
(8 marks) b. A strangle is similar to a straddle except that the call and put have different strike prices. Construct a strangle using Call 2 and Put 1. Draw a diagram showing the profit from the strategy. Indicate the breakeven points and the maximum loss. (8 marks)
Part (b)
1. Suppose you bought four five-year gold futures contracts when the five-year futures price of gold was $395 per ounce, and closed the position at the end of the sixth trading day. The initial margin requirement is $2,000 per contract, and the maintenance margin requirement is $1,500 per contract. One contract is for 100 ounces of gold. The daily prices on the intervening trading days are shown in the following table.
Day
Settlement Price
Mark-to-Market ($)
Other
Entries
Account Balance ($)
Explanation
395.00
--
--
1
397.00
2
394.00
3
389.00
4
391.00
5
393.00
6
396.00
Closed
--
--
--
--
Assume that you deposit the initial margin and do not withdraw the excess on any given day. Whenever a margin call occurs, you would make a deposit to bring the balance up to meet the initial margin requirement. Fill the appropriate numbers in the blank cells. (Hint: Refer to Table 8.2 in the textbook on p. 287.)
(10 marks) 2. Suppose that you enter into a long futures contract to buy May gasoline for $0.80 per gallon on the New York Commodity Exchange. The size of the contract is 42,000 gallons. The initial margin requirement is $2,000, and the maintenance margin is $1,500. What change in the futures price will lead to a margin call?
(10 marks) 3. Consider the three-month futures on stock BBM. Currently, the price of stock BBM is $100. The stock is expected to pay a dividend of $2.00 in two months. Assume that the simple interest rate is 6% per year. Determine the price of the futures assuming no arbitrage opportunites are present.
(10 marks) 4. Suppose that the market price of gold is $370 per ounce. The risk-free interest rate is 10% per year, compounded monthly, for all terms. Gold can be stored for $0.05 per month per ounce, paid at the end of the month. According to the cost of carry model, what should be the price of a gold futures contract, if expiration is six months away? What is the cost of carry?
(10 marks) 5. Suppose you entered a long position in a forward contract to buy 100 ounces of silver for $5.80 per ounce in December. Currently, the spot price of silver is $5.50, the December forward price of silver is $6.20, and the price of a T-bill expired in December with a face value of $100,000 is $96,500. What is the value of the forward contract you hold?
(6 marks) 6. Suppose the spot price and the six-month futures price of soybeans are $4.80 and $5.20 per bushel, respectively. One futures contract is for buying 5,000 bushels. The six-month risk-free interest is 6% per year. The storage cost for soybeans is $0.05 per bushel for six months, paid in advance. Could you make an arbitrage profit in the soybean market? How?
(15 marks) 7. It is March 15, 2005. Suppose you are a dealer in canola holding 80 tonnes of canola worth $380 per tonne. You consider using September 2005 canola futures to hedge. The price of the futures contract is $397 per tonne. Each contract is for 20 tonnes.
a. Determine the original basis.
(3 marks) b. How many contracts will you use? Long or short?
(3 marks) c. If you close your position on July 10, 2005, when the basis is -12, what would be the profit from a hedge?
(4 marks) 8. A coffee trader holds a current inventory of coffee worth $1 million at the present price of $1,250 per ton. The standard deviation of the value for the inventory is 0.27. She is considering a minimum-variance hedge of her inventory using the six-month coffee futures contract. The contract size is 10 tons. The volatility (i.e., standard deviation) of the futures is 0.33. For the particular grade of coffee in her inventory, the correlation between the futures and spot coffee is 0.85.
a. Compute the minimum-variance hedge ratio.
(7 marks) b. How many contracts she should trade? Long or short?
(7 marks) 9. You manage a $4 million portfolio, currently all invested in Canadian equities. The beta of the portfolio is 1.25. You believe that the market is on the verge of a big but short-lived downturn. You would move your portfolio temporarily into T-bills, but you do not want to incur the transaction costs of liquidating and reestablishing your equity position. Instead, you decide to hedge your portfolio with three-month S&P/TSX 60 index futures contracts for one month. Currently, the level of the S&P/TSX 60 index is 425, the three-month futures price of the S&P/TSX 60 is 428, and one contract is for $200 times the index. The annual risk-free rate is 3%.
a. How many futures contracts should you use? Long or short?
(5 marks) b. Suppose the return on the S&P/TSX 60 index is -5% in one month, and the S&P/TSX index futures price falls to 405 in one month. Calculate your gain or loss. (10 marks)
Part (c)
1. Suppose the spot exchange rate between the US dollar and Canadian dollar is US$0.75/C$. The Canadian dollar and US dollar risk-free rates are 2.5% and 1.5% per annum, respectively, compounded annually. The price of a two-year European call option with an exercise price of US$0.76/C$ is US$0.03. What is the price of the option in US dollars if it is a put?
(6 marks) 2. In a quote, the spot US dollar/Canadian dollar exchange rate is C$1.30/US$, and the six-month forward exchange is C$1.32/US$. The interest rates of the Canadian dollar and US dollar are 2.5% and 1.5% per year, respectively. Assume that you can lend or borrow at those rates in both currencies. Is there any "free lunch" in this market?
(8 marks) 3. Suppose that the US dollar interest rate is 2% per annum and Canadian dollar interest rate is 3% per annum for all maturities, annually compounded. The current exchange rate is US$0.75/ C$. Under the terms of a swap agreement, a bank receives 4% per annum in Canadian dollars and pays 5% per annum in US dollars. Payments are exchanged every year, with one exchange having just taken place. The principal amounts are C$14 million and US$10 million. The swap will last two more years.
a. Determine the cash flows to the bank.
(4 marks) b. What is the value of the swap to the bank in terms of Canadian dollars?
(6 marks) 4. CP Financial owns a $10 million 10-year maturity, non-callable corporate bond with a 6.8% coupon paid annually. The company pays annual LIBOR minus 1% on its three-year term time deposits.
VG Corporation owns an annual pay LIBOR floater and wants to swap this variable rate for a fixed rate. (http://books.google.com.pk/books?id=0C6K1N6ryskC&pg=PA353&lpg=PA353&dq=%22owns+an+annual+pay+%22&source=bl&ots=W-Q_qkLyPm&sig=RAsensvd8QSuHssf6FZrzO4VfIE&hl=en&ei=qfj1TKLsO4jprQeVw6mQBw&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCQQ6AEwAg#v=onepage&q=%22owns%20an%20annual%20pay%20%22&f=false)
CP and VG enter into a $10-million plain vanilla interest rate for three years in which CP receives annual LIBOR.
The spot term structure of LIBOR is as follows:
Years to maturity
LIBOR rate (c.c.)
1
5.0%
2
5.2%
3
5.5%
a. Determine the swap rate.
(8 marks) b. Diagram the cash flows between CP, VG, CP's depositors, and CP's corporate bond. Label the following items:
CP, VG, CP's depositors, and VG's corporate bond
the applicable interest rate at each line, specifying whether it is floating or fixed
the direction of each of the cash flows.
(6 marks) c. Calculate the first net swap payment between CP and VG, and indicate the direction of the net payment amount.
(4 marks) 5. Linda longs a FRA on three-month LIBOR with a fixed rate of 8.75% and a notional principal of $20 million. If the market LIBOR rate is 9% at expiration, what would be the payoff of the FRA to Linda?
(5 marks) 6. Are a long cap and a short floor with the exercise price set at the swap rate equivalent to a swap? Explain.
(10 marks) 7. A portfolio consists of 100,000 shares of stock A and 500,000 of stock B. Currently, the price of stock A is $80, and the price of stock B is 40. Stock A has an expected return of 15% per annum and a volatility of 20% per annum; stock B has an expected return of 20% per annum and volatility of 30% per annum. The correlation coefficient between the two stocks is 0.6. What is the daily 95% VAR?
(12 marks) 8. Consider three calls-Call 1, Call 2, and Call 3-all written on the same underlying stock, with the following information:
Current price of the underlying stock: S = $80
Volatility of the stock:
Risk-free interest rate: r = 0.07
Option
Strike price
Days to maturity
Option price
Delta
Gamma
Call 1
70
90
11.40
0.94
0.015
Call 2
75
90
7.16
0.81
0.034
Call 3
80
120
4.60
0.60
0.042
a. If the stock price rises to $80.10, what is your "best" estimate of what the Call 1 price will be?
(5 marks) b. Using Call 1 and Call 2, create a delta-neutral portfolio assuming that the position is long one Call 1.
(5 marks) c. Use all three calls to form a portfolio that is delta-neutral and gamma-neutral, assuming the portfolio is long one Call 1.
(5 marks) 9. A portfolio consists of 400 shares of stock and 200 calls on that stock. If the delta for the call is 0.6, what would be the dollar change in the value of the portfolio in response to a $1 decline in the stock price?
(6 marks) 10. Explain the difference between centralized and enterprise risk management. (10 marks)
