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Derivatives (Comm 4202) Final Examination (A)
(8:00 AM, 21/APR/2020 - 8:00 AM, 22/APR/2020, Halifax time)
Instructions: This take-home examination comprises eight questions/problems. You have 24 hours to complete the examination starting at 8:00 am April 21, 2020, Halifax time. You are required to type your answers in a Microsoft Word document and
upload it at the course website under “Assignments”. Please print your names and your Student ID on the document before
uploading it. Answers will be graded for content and appropriate presentation.
1. A manager has just taken a short position in a one-year forward contract on a common stock. The stock is traded at
$30 a share. It is expected that the stock will pay a dividend of $0.5 in 8 months. The risk-free rate of interest is 5%
per annum with continuous compounding, and the term structure is flat. What is the stock forward price? Six months
later, if the price of the stock is $31, what will be the value of the forward to the manager?
(10 marks)
2. The 12-month LIBOR is 4% with continuous compounding, and the 12-month Eurodollar futures price is traded at
95.75. What should the 15-month spot rate be? If the Eurodollar futures price at maturity is 96.25, estimate the
3-month LIBOR in 12 months.
(10 marks)
3. It is April 21, 2020. Two Treasury bonds are traded as in the following table:
Bond
A
B
Coupon Rate (%)
4.5
6.5
Coupon Payment Dates
January 1 and July 1
April 15 and October 15
Maturity
July 1, 2037
April 15, 2038
Market Price
92.65
108.25
Calculate the conversion factor of each bond for the September 2020 Treasury bond futures. If the September 2020
Treasury bond futures price is 110-05, which bond is cheaper for delivery under the Treasury futures contract?
(10 marks)
4. A stock is traded at $50 a share, and there will be no dividend on the stock in 6 months. The 6-month European calls
and puts on the stock with same strike price of $45 are traded at $8 and $2, respectively. The six-month risk-free rate
is 5% with continuous compounding. Is there an arbitrage opportunity? If yes, construct an arbitrage portfolio and
clearly explain how arbitrage profits are created. If no, why?
(10 marks)
5. The 3-month European call option on a non-dividend paying stock with strike price of $50 is traded at $1.15, and the
3-month European put option with strike price of $40 is traded at $0.67. The current 3-month interest rate is 5% with
continuous compounding, and the stock is traded at $45 a share. A trader has just bought one hundred shares of the
stock and would like to hedge against stock price risk, what option positions should the investor take to guarantee the
stock worth between $40 and $50 a share in 3 months? What is the cost of the hedging portfolio?
(15 marks)
6. Suppose the term structure of interest rates is flat in the United States and Canada. The USD interest rate is 3% per
annum and the CAD interest rate is 3.5% per annum. The current value of one CAD is 0.73 USD. In a swap agreement,
a financial institution pays 6% per annum in CAD and receives 5.5% per annum in USD. The principals in the two
currencies are $10 million USD and $14 million CAD. Payments are exchanged every 6 months, with one exchange
having just taken place. The swap will last two more years. What is the value of the swap to the financial institution?
Assume all interest rates are continuously compounded.
(15 marks)
7. A stock is currently traded at $50 a share with a volatility of 30% and a dividend yield of 2%. The risk-free rate is
5% with continuous compounding. Using the Black-Scholes-Merton formula, calculate the price of a 6-month European
put option with a strike price of $55. Now, assuming the put option is American with same strike price and maturity,
use a two-step binomial model to price the put option. Will the American put be exercised earlier before expiration?
Why or why not?
(15 marks)
8. A swap dealer has just entered into a two-year swap contract to semiannually exchange a fixed-rate interest for sixmonth LIBOR interest on $100 million. Six-month and twelve-month LIBOR rates are 3% and 3.125%, respectively.
The 12 × 18 and 18 × 24 LIBOR forward rates are 3.5% and 3.75%, respectively. Estimate the two-year swap rate. All
rates are expressed with continuous compounding.
(15 marks)
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Derivatives (Comm 4202) Final Review
Key Concepts:
1
Interest Rates
• Compounding frequencies.
• Spot rates (zero rates) and forward rates, and their relations.
• Forward Rate Agreement.
– Definition
– Settlement, Payoff
– Valuation.
• Bond pricing, coupon rate, face value, yield to maturity, and par yield.
• Bootstrapping of interest rates with bond prices.
2
Forward and Futures Contracts
• Definition.
• Arbitrage.
• Futures trading on margin.
• Determination of forward and futures prices.
– Without income.
– With income.
– Stock, currencies, and commodities.
– Treasury bond futures.
– Eurodollar futures.
• Stock futures without (with) dividends, index futures, currency futures (forward).
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• Payoff.
• Valuation.
• Hedging strategies with forward and/or futures contracts
• changing portfolio systematic risk.
3
Options
Unlike futures contracts, the initial value of an option is nonzero. The long position
has the right to exercise the option (an asset), while the short position does not (a
liability).
• Definition of options: Calls and Puts, American versus European.
• Profits/losses of a portfolio of options and the underlying asset.
• Margin requirements.
• Option price bounds, put-call parity for European options.
• Trading strategies involving options and the underlying asset.
• Valuation of options.
– Arbitrage and/or replication arguments.
– Risk neutral valuation.
– Binomial method for both European and American options.
– BSM method for European options.
– Parameter conversion between the two methods.
– Options with dividends.
– Options on stock indices, currencies, and futures.
– Hedging with option contracts.
2
4
Interest Rate Futures
• Treasury bond futures
– Day counts, accrued interests, and price quotations.
– Cash price and quoted price.
– Conversion factor of a bond.
– Cheapest to deliver bond.
– Treasury futures price.
– Cash received on delivery under the futures contract.
• Eurodollar futures
– Eurodollar.
– Contract Price.
– Reference asset in a Eurodollar futures contract – future three-month LIBOR rate.
– Alternative explanation of a Eurodollar futures contract.
– Eurodollar futures rates and its relation to LIBOR forward rates.
– Determination of Eurodollar futures price.
– Bootstrapping interest rates with Eurodollar futures rate.
5
Swaps
• Definition of swaps, asset/liability transformation, roles of financial institutions.
A swap contract obligates two parties to exchange/swap cash flows at specified
future dates.
A swap is a portfolio of forwards.
• Interest rate swaps:
An Example of a ‘Plain Vanilla’ Interest Rate Swap: I agree to pay you 8% of
$40 million each year for the next five years, and you agree to pay me whatever
1-year LIBOR is (times $40 million) for each of the next five years.
– Swap rates, libor/swap zero rates.
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– How swap rates are determined.
– Bootstrapping zero rates with swap rates.
– Valuation of interest rate swaps:
∗ Valuation in terms of bond prices (fix and floating coupon bonds).
∗ Valuation in terms of forward rate agreements (FRAs).
• Currency swaps:
– Definition – the reference “asset” is the exchange rate between the two
countries.
– Principals are swapped.
– Two methods of valuation: in terms of bond pricing and forward contracts.
– Discount rates are the country interest rates for bond pricing in the valuation mechanism.
Note that the valuation of swaps does not take credit risk into consideration.
4
Review Questions and Answers
1. A company has a portfolio of stocks worth $15 million with a beta of 0.8. The
S&P 500 index futures contract is traded at 1500 and on $250 times the index
level. The initial margin is $2000 and the maintenance margin is $1500 per
contract, respectively. The balance of the margin account is $1,000,000. The
company currently takes a full hedging position in the futures contracts on its
stock portfolio.
(a) What position in the futures contracts does the company take?
Answer:
To hedge portfolio exposure, the company should short
0.8 ×
15000000
1500 × 250
= 32
S&P 500 index futures contracts.
(b) What futures price level would lead to a margin call?
Answer:
The maintenance margin is $1500 per contract, so the total margin required
is $48000. Let x be the futures price level to trigger a margin call.
32 ∗ 250 ∗ (x − 1500) = 1000000 − 48000
x = 1619
(c) What position in the futures contracts should the company take to increase
the stock portfolio beta to 1.2?
Answer:
(1.2 − 0.8) ×
15000000
250 × 1500
= 16
Thus, the company should take the long position of 16 futures contracts to
increase the portfolio beta to 1.2.
2. A company enters into a short futures contract to sell 5,000 bushels of wheat for
$7.50 per bushel. The initial margin is $3,000 and the maintenance margin is
$2,000. What price change would lead to a margin call? Under what circumstances could $1,500 be withdrawn from the margin account?
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Answer:
There is a margin call if $1000 is lost on the contract. Let x be the futures price
level such that the company will lose $1000. Then
5000 ∗ (x − 7.50) = 1000
x = 7.70
That is, the company will receive a margin call if the futures price rises to $7.70.
Similarly, $1500 can be withdrawn if the futures price falls by 30 cents to $7.20
per bushel.
Remark: Some brokers require the traders to maintain the initial margin requirement when funds are withdrawn from the margin account, and some may
allow traders to keep funds at the maintenance level.
3. Suppose 3-month futures contracts on S&P 500 index are used to hedge an equity portfolio over the next two months in the following situation
Value of portfolio:
Dividend yield:
Beta of portfolio:
Contract multiplier:
$50,000,000
3%
0.87
250
S&P 500 Index:
Futures price:
Risk free rate:
1250
1259
6%
(a) What position should the manager take to hedge all exposure over the next
two months?
Answer:
The number of contracts the fund manager should short is
0.87 ×
50, 000, 000
1259 × 250
= 138.20
Rounding to the nearest whole number, 138 contracts should be shorted.
(b) If the index in two months is 1,000, 1,100, 1,200, 1,300, or 1,400, calculate
the gains or losses of the manager’s total portfolio. Assume that the 1 month
futures price is 0.25% higher than the index level in two months.
Answer:
We need to calculate the profit/loss from the equity portfolio and futures
contracts. To illustrate the calculations, we assume the index will be 1000 in
two months. The futures price is then
1000 ∗ 1.0025 = 1002.5
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The gain on the short futures position is then
(1259 − 1002.5) ∗ 250 ∗ 138 = 8, 849, 250
The return on the index including dividends is
0.03 ∗ 2/12 − 250/1250 = −19.5%
The risk free rate is 6% × 2/12 = 1% per two months. From the Capital
Asset Pricing Model, we expect that the return on the portfolio is
1% + 0.87 ∗ (−19.5% − 1%) = −16.835%
The loss on the portfolio is
16.835% ∗ 50, 000, 000 = 8, 417, 500
The total gain is 8,849,250-8,417,500 = 431,750
For all possible 5 scenarios, the company’s net gains/losses can be calculated
similarly.
Index in two months
Gain/loss on the hedged portfolio
1000
431,750
1100
1200
453,125 474,500
1300
495,875
4. The 6-month, 12-month, 18-month, and 24-month zero rates are 4%, 4.5%, 4.75%,
and 5%, with semiannual compounding.
(a) What are the rates with continuous compounding?
Answer:
Denote Rc as continuous compounding rate and R2 the semiannual compounding rate. Then,
Rc = 2 ln(1 + R/2)
and
R2 = 2(eRc /2 − 1).
Thus, the corresponding rates to the variant terms are 3.96%, 4.45%, 4.69%,
and 4.94%, with continuous compounding.
(b) What is the forward rate for the 6-month period beginning in 18 months?
Answer:
The 6 month forward rate starting in 18 months is
RF = (0.0494 ∗ 2 − 0.0469 ∗ 1.5)/(2 − 1.5) = 5.69%
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1400
517,250
with continuous compounding. With semiannual compounding, it is
2 ∗ (e0.0569/2 − 1) = 5.77%.
(c) What is the value of an FRA that promises to pay 6% interests with semiannual compounding on a principal of $1 million for the 6-month period starting in 18-months?
Answer:
The value of the FRA is
1 ∗ (0.0577 − 0.06) ∗ (1 − 0.5)e−0.0494∗2 = −0.0010million.
5. The prices of two bonds with a principal of $100 and coupons paid semiannually
are given below:
Time to maturity
6 months
12 months
Annual coupon rate
0%
3.5%
Bond price
$99
$101
(a) Estimate the 6-month and 12-month risk-free interest rates.
Answer:
Let R1 and R2 be the 6-month and 12-month zero rates.
100e−0.5R1 = 99.
3.5/2e−0.5R1 + (100 + 3.5/2)e−R2 = 101.
R1 = −2 ∗ ln(99/100) = 0.0201, and
R2 = − ln((101 − 1.75 ∗ exp(−0.0201 ∗ 0.5))/101.75) = 0.0247
with continuous compounding.
(b) Estimate the price and par yield of a one-year bond providing a 4% coupon
rate with semiannual payment.
Answer:
P = 2 ∗ exp(−0.0201 ∗ 0.5) + 102 ∗ exp(−0.0247 ∗ 1) = 101.49 and the
par yield is 4% with semiannual compounding.
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(c) Calculate the 6-month forward rate beginning in 6 months.
Answer:
The 6-month forward rate in 6 months is
RF =
0.0247 ∗ 1 − 0.0201 ∗ 0.5
1 − 0.5
= 0.0293,
with continuous compounding.
6. Six months ago, a one-year long forward contract on a non-dividend-paying stock
was entered into when the stock price was $30 and the one-year zero rate was
5% per annum with continuous compounding.
(a) What is the delivery price of the forward contract?
Answer:
(a) The delivery price of the forward contract was
F0 = S0 erT = 30e0.05∗1 = 31.5381.
(b) The stock is currently priced at $31 and the 6-month zero rate is 4.75%.
What are the forward price and the value of the forward contract started 6
months ago?
Answer:
The (6-month) forward price is
F1 = S1 er1 T1 = 31e0.0475∗0.5 = 31.7451.
The value of the forward contract started 6 months ago is
f = S1 − F0 e−r1 T1 = 31 − 31.5381 ∗ e−0.0475∗0.5 = 0.2021.
Alternatively, the value of the forward contract is the discounted value of the
difference of the delivery prices of the forward contract started today and the
forward contract started 6 months ago. That is, the value of the forward is
f = (F1 − F0 )e−r1 T1 = (31.7451 − 31.5381)e−0.0475∗0.5 = 0.2021.
7. The two-month interest rates in Switzerland and the United States are 2% and
5% per annum, respectively, with continuous compounding. The spot price of
the Swiss franc is $0.8000. The futures price for a contract deliverable in two
months is $0.8100. What arbitrage opportunities does this create?
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Answer:
The futures price should be
0.8e(0.05−0.02)∗2/12 = 0.8040
which is lower than the market price of 0.81. Thus, there is an arbitrage opportunity. We should take a short position in the futures contract to sell the Swiss
franc at $0.81 in two months. In the meantime, borrow $0.80 USD at 5% to buy
1 Swiss franc in the spot market and deposit the Swiss franc in Switzerland. In
two months, the cash flow in USD is
0.81 ∗ e0.02∗2/12 − 0.80 ∗ e0.05∗2/12 = 0.0060.
We make a positive profit with initial zero cost - an arbitrage opportunity.
8. A stock is expected to pay a dividend of $2 per share in two months and $3 in
five months. The stock price is $10, and the term structure for the risk-free rates
are given as
Term One month Two-month
Rate
0.75%
1.00%
Three-month
1.25%
Four-month
1.50%
Five-month
1.75%
Six-month
2.00%
An investor has just taken a short position in a six-month forward contract on
the stock.
(a) Calculate the forward price.
Answer:
The present value of the dividends before the delivery date is
I = 2 ∗ e−0.01∗2/12 + 3 ∗ e−0.0175∗2/12 = 4.9749.
Hence, the forward price of the 6 month forward contract is
F = (S − I)erT = (10 − 4.9749)e0.02∗6/12 = 5.0756.
(b) Three months later, the price of the stock is $8. What are the forward price
and the value of the short position in the forward contract started three
months ago? Assume the term structure is unchanged.
Answer:
Three months later, the present value of $3 dividend is
I1 = 3e−0.01∗2/12 = 2.9950.
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The forward price of the forward contract with the same maturity date is
F1 = (S1 − I1 )er1 T1 = (8 − 2.9950)e0.0125∗3/12 = 5.0207.
The value of the short position in the forward contract started 3 month ago
is
f = −(F1 − F )e−r1 T1 = −(5.0207 − 5.0756)e−0.0125∗3/12 = 0.0547.
Alternatively,
f = −(S1 − I1 − F e−r1 T1 ) = −(8 − 2.9950 − 5.0756e−0.0125∗3/12 ) = 0.0548
(rounding-off error)
9. A four-month European call option on a dividend-paying stock is currently selling for $5. The stock price is $64, the strike price is $60, and a dividend of $0.80
is expected in one month. The risk-free interest rate is 12% per annum for all
maturities. What opportunities are there for an arbitrageur?
Answer:
The lower bound of the option price is
max{S0 − D − K ∗ e−RT , 0} = 64 − 0.8e−0.12/12 − 60e−0.12∗4/12 = 5.5606
Thus, the option price is too low. The trader can make an arbitrage free profit by
shorting the stock, buying a call option, and depositing 0.8e−0.12/12 = 0.792 for
one month and depositing 60e−0.12∗4/12 = 57.6474 for 3 months. The net cost
(cash outflow) is
−64 + 5 + 0.792 + 57.6474 = −0.5606
The net cash flow in 4 months is zero and the dividend is financed by the investment in the risk free asset. Thus, there is an arbitrage free profits of $0.5606.
10. A European call with a strike price of $60 costs $6. A European put with a strike
price of $40 and the same expiration date costs $4. Construct a table that shows
the profit from a short strangle. For what range of stock prices would the short
strangle lead to a loss?
Answer:
A short strangle consists a short put with a lower strike price and short call with
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a high strike price. Let ST be the underlying stock price at tiem T . The payoff
from a short strangle using the put and the call is given as
payof f = − max{40 − ST , 0} − max{ST − 60, 0}
and the the net cost is −$4 − $6 = −$10. Hence, the profit is
payof f − (−10)
and a profit table can be constructed as
ST
Profit
10
-20
20 30 40 50 60 70
-10
0 10 10 10
0
80
-10
The break-even price points are ST = 30 or ST = 70. Thus, if the the stock
price at the expiration were either less than $30 or gretaer than $70, the short
strangle would lead to a loss.
11. The price of a stock is $40. The price of a one-year European put option on
the stock with a strike price of $30 is quoted as $7 and the price of a one-year
European call option on the stock with a strike price of $50 is quoted as $5.
Suppose that an investor buys one share, shorts one call option with strike 50,
and buys 1 put option with strike 30. Draw a diagram illustrating how the
investor’s profit or loss varies with the stock price over the next year. What is
the break-even price?
Answer:
Cost of the portfolio is
40 − 5 + 7 = 42.
Let S be the price of the stock next year. Payoff of the portfolio is
50, if S ≥ 50,
S − max(S − 50, 0) + max(30 − S, 0) =
S, if 30 < S < 50,
30, if S ≤ 30,
which is similar to a bull spread. The break-even stock price is such that S −
42 = 0. i.e., S = 42. Note, the payoff of stock+long put =long call + K, where
K is the strike price of the put.
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12. A stock price is currently $50. It is known that at the end of two months it
will be either $53 or $48. The risk-free interest rate is 10% per annum with
continuous compounding. What is the value of a two-month European call op ...