Bus 316 Final Exam, December 2011
Student Name ________________________
Student Number ________________________
Bus 316 Final Exam, December 2011
CAUTION In accordance with the Academic Honesty Policy (T10.02), academic dishonesty in any form will not be tolerated. Prohibited acts include, but are not limited to, the following: - Making use of any books, papers, electronic devices or memoranda, other than those authorized by the examiners. - Speaking or communicating with other students who are writing examinations. - Copying from the work of other candidates or purposely exposing written papers to the view of other candidates.
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Student Name ________________________
Student Number ________________________
Question 1: Multiple Choice (20 Total marks, 2 marks for each answer) Select the best answer for each of the following multiple choice questions. Enter your answers on the scantron bubble sheet. If more than one answer is given for any item, that item will not be marked. Incorrect answers will be marked as zero. Explanations provided beside the answers will not be considered. 1. A short forward contract on a non-dividend paying stock was entered into some time ago at a delivery price of $24. It currently has six months to maturity. The risk-free interest rate is 5% and the current stock price is $25. Find the current value of the forward contract. a) – $1.63 b) – $1.59 c) $1.59 d) $1.63 e) $24.00 2. On May 1, an investor takes a short position in ten December gold Futures contracts on the New York Commodity Exchange (COMEX). The current Futures price is $1,600, the contract size is 100 ounces, the initial margin is $2,000 per contract, and the maintenance margin is $1,500. If by the end of May 1 the Futures price has dropped to $1,598: a) the margin balance increases by $200. b) the margin balance increases by $2000. c) the margin balance decreases by $2000. d) the investor receives a margin call. e) the broker automatically closes out the position. 3. Consider a 1-year Futures contract to buy one corporate bond. The current spot price of the bond is $920.00, the continuous interest rate for 1-year is 10%. One coupon payment of $10.00 will be paid in 9-months. If the Futures price of the bond in the market is $1003, an arbitrage profit can be generated by: a) shorting the Futures contract and short selling one Bond. b) shorting the Futures contract and buying one Bond. c) taking a long position in the Futures contract 9-months from today. d) taking a long position in the Futures contract and buying one Bond. e) taking a long position in the Futures contract and short selling one Bond. 4. An investor has an asset that is currently worth $500, and the continuously compounded rate at all risk-free maturities is 3 percent. If the asset pays a continuous dividend of 3 percent, which of the following is the closest to the no arbitrage price of the 3-month forward contract? a) $496.26 b) $500.00 c) $502.00 d) $503.00 e) $503.76
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5. The three month interest rates in New Zealand and Canada are 6% and 4% per annum respectively, with continuous compounding. The spot price of one New Zealand Dollar is Canadian Dollars 0.80. (i.e. 1 NZD = 0.80 CAD). What is the theoretical value of the 3 month futures contract? a) 0.7960 b) 0.8000 c) 0.8040 d) 0.8080 e) 0.8162 6. A shareholder wants to use 1-year put options to hedge 50% of a portfolio containing 10,000 shares of USCorp, a US firm. We denote by ST the spot price in one year for one share of USCorp and by K the strike price of the options purchased by the shareholder. The total value of the hedged portfolio (neglect the option premiums paid upfront) in one year is: a) 10,000 × K b) 10,000 × max(K,ST). c) 10,000 × ST + 5,000 × max(K,ST). d) 10,000 × ST + 10,000 × max(K,ST). e) 5,000 × ST + 5,000 × max(K,ST). 7. Consider a European put option on a stock currently price at $50. The put option has an expiration of 6 months, a strike price of $40, and the risk-free rate is 5 percent. The price of the put option must be less than: a) b) c) d) e)
$0 $1.23 $10 $10.99 $39.01
8. Is it ever optimal to exercise an American call option on gold early? a) Yes. You should exercise the option when the price of gold goes above the strike price. b) Yes. You should exercise the option when the price of gold drops below the strike price. c) Yes. You should exercise the option when the option’s time value starts decreasing. d) No. You should not exercise the option since the later the strike price is paid out, the better. e) No. You cannot exercise the option early since only European options can be exercised early. 9. A three-month call with a strike price of $25 costs $2. A three-month put with a strike price of $20 costs $3. A trader uses these options to create a strangle. For which two values does the trader breakeven? a) II. and IV. b) I. and V. c) I. and IV. d) III. and IV. e) II. and V.
I. II. III. IV. V. 3
$15 $17 $22.50 $27 $30
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10. An increase in which of the following will increase the value of an American call? a) I. only b) II. only c) III. only d) I. and III. only e) I. and III. and IV. only
I. II. III. IV.
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The stock price The strike price The volatility The time remaining before expiration
Student Name ________________________
Student Number ________________________
Question 2: Option Pricing & Arbitrage (12 marks total) Consider a stock that is currently priced at $40. The stock is a non-dividend paying stock. The continuous riskfree rate of interest is 10% per annum.
(3 marks) (a)
If you own a 3-month European put with a strike price of $45, calculate the price of the put based on the “lower bound”.
Put = _______________ (3 decimals)
(3 marks) (b)
IF the current market price of a 3-month European call with a strike price of $45 was $0.50. Using the information at the very beginning of the question, calculate what the price for the 3month European put should be if there were no arbitrage opportunities.
Put = _______________ (3 decimals) 5
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(3 marks) (c)
IF the current market price of a 3-month European put with a strike price of $45 was $4.25, based on your answer in part (b), describe the transactions would you enter into to earn a riskless profit? [If you were unable to answer part (b), you may use the price of $4.35]
Transaction #1: _______________________________________________________________ Transaction #2: _______________________________________________________________ Transaction #3: _______________________________________________________________ Additional Transactions:
(3 marks) (d)
Now assume that the stock is expected to pay a dividend of $0.75 in two months. If the current market price of a 3-month European call with a strike price of $45 was $0.50. Using the information at the very beginning of the question, calculate what the price for the 3-month European put should be if there was no arbitrage opportunities (3 decimals)
Put = _______________ (3 decimals) 6
Student Name ________________________
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Question 3: Option Trading Strategies (12 marks total) Consider a portfolio of options that consists of: Sell 3-month put with a strike price of $35 that priced at $4.50 Long 3-month put with strike price of $43 that is priced at $7.00 Currently the price of the underlying stock is $40.
(2 marks) (a) Draw the PROFIT diagram based on the above portfolio. Clearly label the breakeven(s).
(3 marks) (b) Complete the below table to Show Payoffs and Profit/Losses for the different possible price ranges of ST at expiration. Let ST denote the Stock Price at Expiration. Stock Price
Total Portfolio Payoff
Total Portfolio Profit/Loss
ST ≤ 35 35 < ST < 43 ST ≥ 43
(1 mark) (c) Calculate the profit or loss IF the value of the underlying asset at expiration was equal to $41.00.
Profit/Loss when ST=41 = ______________ (2 decimals) 7
Student Name ________________________
Student Number ________________________
(2 marks) (d) Explain the logic or reasoning behind why a trader would invest in the portfolio described at the beginning of the question. (maximum of 4 sentences)
(4 marks) (e) Name the trading strategy you drew in part (a) above: ________________________________ Based on your answer provided in part (d), describe the specific transactions you would enter into to create an “alternative portfolio” of derivatives that would have the same PAYOFF diagram to that of the portfolio described at the beginning of the question. Transaction #1: _______________________________________________________________ Transaction #2: _______________________________________________________________ Transaction #3: _______________________________________________________________ Additional Transactions:
Draw the PAYOFF diagram of this alternative portfolio
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Student Name ________________________
Student Number ________________________
Question 4: Futures Pricing & Arbitrage (10 marks) A trader is currently monitoring prices for a number of different variables and notices that: The current spot rate (S0) for cattle is $1.00 per pound. o The costs to feed cattle are $0.01 monthly per pound which is paid at the beginning of the month. o There is a rental agreement to house cattle of $4000 annually payable every end of quarter, which can accommodate 20,000 pounds of cattle o Currently there is ONLY 80,000 pounds of cattle available for purchase or sale in the spot market The current continuous risk-free rate is 5% per annum Today is January 1st
(2 marks) (a) Calculate what a 4-month Futures price on cattle (in dollars per pound) should be (4 decimals)
Futures Price = ______________ (4 decimals)
(2 marks) (b) A trader also observes that Currently the actual market price for a 1-month Futures price on cattle is $1.15 per pound. A 1-month futures contract is for delivery of 40,000 pounds of cattle Based on the answer in part (a), explain exactly what transactions might a trader enter into? (maximum of 3 sentences) [If you were unable to answer part (a), you may use the value of F0 = 1.05] Transaction #1: _______________________________________________________________ Transaction #2: _______________________________________________________________ Transaction #3: _______________________________________________________________ Additional Transactions:
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Student Name ________________________
Student Number ________________________
(6 marks) (c) As result of entering into the transactions described in part (b), calculate the total cash flows when the transaction was initially entered into (t=0) and at every important point in time Please fill out the below table to answer part (c) of the question. Use 2 decimals for cash flows and as many time periods as you need. Time period
Cash flows from each transaction (-) means cash outflow (+) means cash inflow
Transaction(s)
t=0
t= _______
t= _______
t= _______
t= _______
t= _______
t= _______
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Student Name ________________________
Student Number ________________________
Question 5: Binomial Pricing Model (10 marks total) A stock price is currently $50. Over each of the next two three-month periods it is expected to go up by 5.9% or down by 5.6%. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a six-month European call with a strike price of $51?
(1 mark) (a)
Please write the “p” that you used here _________________ (4 decimal places)
(2 marks) (b)
Complete the below Stock Price Tree (answer in dollars & cents to 2 decimal places)
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Student Name ________________________
Student Number ________________________
(6 marks) (c)
Complete the below Option Price Tree for the European call (answer in dollars & cents to 2 decimal places)
D B A
E C F
(1 mark) (d)
Were this call an American call, at which nodes in the above option price tree would you exercise? (circle all nodes where you would exercise) None A
B
C
D
E
F
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Student Name ________________________
Student Number ________________________
Question 6: Binomial Pricing Model (12 marks total) To receive partial credit, you should show all calculations. Please write your answers in the spaces provided. At each node: the upper number should be the stock price the lower number should be the option price A stock price is currently $95. The volatility of the stock is 12% per annum. The stock does not pay any dividends. The risk-free continuous compounding interest rate is 5% per annum.
(4 marks) (a) Using the Binominal Option Pricing model calculate the value of a six-month European call option with a strike price of $100 (final answer 2 decimals, intermediate calculations 4 decimals)
p = _____________ u = _____________ d = _____________ (all to 4 decimals)
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Student Name ________________________
Student Number ________________________
(4 marks) (b) Using the Binominal Option Pricing model calculate the value of a six-month European put option with a strike price of $100 (to 2 decimals)
(1 mark) (c) Verify that the European call and European put prices you calculated satisfy put-call parity.
(2 marks) (d) What is the delta of the European call?
Δc= _______ (1 mark) (e) Now assume that the put option is an American option. Without actually calculating a final price for the put option at node A, using the values calculated for the various nodes (other than node A) determine if it would be optimal to exercise the American put early. Would you exercise early?
Please circle your answer:
If Yes, at which node? (please circle) 14
A
Yes B
C
No D
E
F
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Student Number ________________________
Question 7: Black-Scholes Pricing (12 marks total) The price of the underlying asset of the option is $156. The underlying asset pays a continuous dividend yield of 4%. The volatility of the underlying asset is 30%, and the risk free interest rate is 6%.
(6 marks) (a) Using the Black Scholes model, calculate the value and delta of a six-month European call option with a strike price of $160 ( 4 decimals when calculating d1 and d2)
d1 = ____________ d2 = ____________
cvalue = __________ Δc = ____________ 15
Student Name ________________________
Student Number ________________________
(6 marks) (b) There is another underlying asset with the same volatility that is currently priced at $150.00 and pays a dividend in four months of $5.00 Using the Black Scholes model, calculate the value of a six-month American put option with a strike price of $160 (3 decimals – round to 2 decimals when calculating d1 and d2) [Hint: use Black’s Approximation]
d1 = ____________
d1 = ____________
d2 = ____________
d2 = ____________
p = _____________
p = _____________ 16
P = _____________
Student Name ________________________
Student Number ________________________
Question 8: The Greeks (12 marks total) An investor has created a portfolio consisting of a combination of calls and puts (on the same underlying stock) and the characteristics of the options are described in the following table
Option A Option B
Number of contracts -500 -300
Delta of option
Gamma of option
Vega of option
0.85 -0.25
2.20 1.25
2.5 0.60
(5 marks total, 1 mark each) (please circle your answers) (a)
Which option is a Call?
Option A
Option B
Both
Can’t Tell
(b)
Which option is deeper in-the-money?
Option A
Option B
Neither
Can’t Tell
(c)
Which option is more sensitive to volatility changes? Option A
Option B
Neither
Can’t Tell
(d)
Which option’s price will go down if the underlying stock price goes up? If the underlying goes up by one, which option will change in value more?
Option A
Option B
Neither
Can’t Tell
Option A
Option B
Neither
Can’t Tell
(e)
(2 marks) (f)
Define the delta of a portfolio of options which is based on a single stock, and explain how the delta of a portfolio is used in hedging. (4 sentences maximum)
(g)
Based on the information given in the table, calculate the Delta of the portfolio of options, the Gamma of the portfolio of options and the Vega of the portfolio of options (1 decimal)
(3 marks)
Portfolio Delta = ___________ Portfolio Gamma = ___________ Portfolio Vega = ___________ 17
Student Name ________________________
Student Number ________________________
(2 marks) (h)
Based on your answer in part (b) for the portfolio of options, clearly describe how you could modify your portfolio of options to be JUST delta neutral (show calculations to receive full marks 7 circle your final answer) [If you were unable to answer part (e) you may use the value of -500 for the Delta of the portfolio of options]
THE LAST PAGE OF THE EXAMINATION
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Student Name ________________________
Student Number ________________________
Formula sheet
Optimal hedge ratio: h
S F
N*
Optimal number of contracts:
h * QA QF N
Optimal number of contract for equity hedging:
Capital Asset Pricing Model (CAPM) : Changing Beta:
( initial new )
VA VF
E(Rt) = rf + β [ E(Rmt) – rf]
VA VF
Conversion formulas for interest rates:
R Rc m ln 1 m m Rc / m Rm m e 1
Forward – Spot relationship: No income, no storage cost:
F = SerT
Income, no storage cost:
F0 = (S0 – I )erT or F0 = S0 e(r-q)T
Storage cost:
F0 = (S0+U) erT or F0 = S0 e(r+u )T
Cost of Carry:
F0 = S0 e(c–y)T
Forwards on Currencies: Value of a forward contract:
F0 S 0 e
( r r f )T
ƒ = (F0 – K)e-rT or ƒ = (K – F0 )e–rT
Options: Put – Call Parity:
c + Ke-rT = p + S0
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Student Name ________________________
Student Number ________________________
Binomial Option Pricing Model
p
e rT d ud
p
e ( r q ) dt d ud
u e
t
d e
t
1/ u
Black-Scholes Formulas For underlying assets: -
with no dividends:
c S 0 N (d1 ) K e rT N (d 2 ) p K e rT N (d 2 ) S 0 N (d1 ) where d1 d2
-
ln( S 0 / K ) (r 2 / 2)T
T
ln( S 0 / K ) (r 2 / 2)T
T
with known dividend yield:
d1 T
c S 0 e qT N (d1 ) Ke rT N (d 2 ) p Ke rT N (d 2 ) S 0 e qT N (d1 ) where, d1 d2
currency:
c S0e
rf T
N (d1 ) Ke rT N (d 2 )
p Ke rT N (d 2 ) S 0 e where, d1
rf T
ln( S 0 / K ) (r q 2 / 2)T
T ln( S 0 / K ) (r q 2 / 2)T
T
N (d1 )
ln( S 0 / K ) (r r 2 / 2)T f
d2
T ln( S 0 / K ) (r r 2 / 2)T f
T
Greeks Gamma:
c p
Theta:
2
S 2 2t Se .5( d1 ) c rKe rt N (d 2 ) 2 2t
p Rho:
e .5( d1 )
Se .5( d1 ) 2 2t
2
1 rS0 2 S 02 r 2
rKe rt N (d 2 )
c Kte rt N (d 2 ) p Kte rt N (d 2 )
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Student Name ________________________
Student Number ________________________
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Student Name ________________________
Student Number ________________________
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Student Number ________________________
Bus 316 Final Exam, December 2011
CAUTION In accordance with the Academic Honesty Policy (T10.02), academic dishonesty in any form will not be tolerated. Prohibited acts include, but are not limited to, the following: - Making use of any books, papers, electronic devices or memoranda, other than those authorized by the examiners. - Speaking or communicating with other students who are writing examinations. - Copying from the work of other candidates or purposely exposing written papers to the view of other candidates.
1
Student Name ________________________
Student Number ________________________
Question 1: Multiple Choice (20 Total marks, 2 marks for each answer) Select the best answer for each of the following multiple choice questions. Enter your answers on the scantron bubble sheet. If more than one answer is given for any item, that item will not be marked. Incorrect answers will be marked as zero. Explanations provided beside the answers will not be considered. 1. A short forward contract on a non-dividend paying stock was entered into some time ago at a delivery price of $24. It currently has six months to maturity. The risk-free interest rate is 5% and the current stock price is $25. Find the current value of the forward contract. a) – $1.63 b) – $1.59 c) $1.59 d) $1.63 e) $24.00 2. On May 1, an investor takes a short position in ten December gold Futures contracts on the New York Commodity Exchange (COMEX). The current Futures price is $1,600, the contract size is 100 ounces, the initial margin is $2,000 per contract, and the maintenance margin is $1,500. If by the end of May 1 the Futures price has dropped to $1,598: a) the margin balance increases by $200. b) the margin balance increases by $2000. c) the margin balance decreases by $2000. d) the investor receives a margin call. e) the broker automatically closes out the position. 3. Consider a 1-year Futures contract to buy one corporate bond. The current spot price of the bond is $920.00, the continuous interest rate for 1-year is 10%. One coupon payment of $10.00 will be paid in 9-months. If the Futures price of the bond in the market is $1003, an arbitrage profit can be generated by: a) shorting the Futures contract and short selling one Bond. b) shorting the Futures contract and buying one Bond. c) taking a long position in the Futures contract 9-months from today. d) taking a long position in the Futures contract and buying one Bond. e) taking a long position in the Futures contract and short selling one Bond. 4. An investor has an asset that is currently worth $500, and the continuously compounded rate at all risk-free maturities is 3 percent. If the asset pays a continuous dividend of 3 percent, which of the following is the closest to the no arbitrage price of the 3-month forward contract? a) $496.26 b) $500.00 c) $502.00 d) $503.00 e) $503.76
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Student Name ________________________
Student Number ________________________
5. The three month interest rates in New Zealand and Canada are 6% and 4% per annum respectively, with continuous compounding. The spot price of one New Zealand Dollar is Canadian Dollars 0.80. (i.e. 1 NZD = 0.80 CAD). What is the theoretical value of the 3 month futures contract? a) 0.7960 b) 0.8000 c) 0.8040 d) 0.8080 e) 0.8162 6. A shareholder wants to use 1-year put options to hedge 50% of a portfolio containing 10,000 shares of USCorp, a US firm. We denote by ST the spot price in one year for one share of USCorp and by K the strike price of the options purchased by the shareholder. The total value of the hedged portfolio (neglect the option premiums paid upfront) in one year is: a) 10,000 × K b) 10,000 × max(K,ST). c) 10,000 × ST + 5,000 × max(K,ST). d) 10,000 × ST + 10,000 × max(K,ST). e) 5,000 × ST + 5,000 × max(K,ST). 7. Consider a European put option on a stock currently price at $50. The put option has an expiration of 6 months, a strike price of $40, and the risk-free rate is 5 percent. The price of the put option must be less than: a) b) c) d) e)
$0 $1.23 $10 $10.99 $39.01
8. Is it ever optimal to exercise an American call option on gold early? a) Yes. You should exercise the option when the price of gold goes above the strike price. b) Yes. You should exercise the option when the price of gold drops below the strike price. c) Yes. You should exercise the option when the option’s time value starts decreasing. d) No. You should not exercise the option since the later the strike price is paid out, the better. e) No. You cannot exercise the option early since only European options can be exercised early. 9. A three-month call with a strike price of $25 costs $2. A three-month put with a strike price of $20 costs $3. A trader uses these options to create a strangle. For which two values does the trader breakeven? a) II. and IV. b) I. and V. c) I. and IV. d) III. and IV. e) II. and V.
I. II. III. IV. V. 3
$15 $17 $22.50 $27 $30
Student Name ________________________
Student Number ________________________
10. An increase in which of the following will increase the value of an American call? a) I. only b) II. only c) III. only d) I. and III. only e) I. and III. and IV. only
I. II. III. IV.
4
The stock price The strike price The volatility The time remaining before expiration
Student Name ________________________
Student Number ________________________
Question 2: Option Pricing & Arbitrage (12 marks total) Consider a stock that is currently priced at $40. The stock is a non-dividend paying stock. The continuous riskfree rate of interest is 10% per annum.
(3 marks) (a)
If you own a 3-month European put with a strike price of $45, calculate the price of the put based on the “lower bound”.
Put = _______________ (3 decimals)
(3 marks) (b)
IF the current market price of a 3-month European call with a strike price of $45 was $0.50. Using the information at the very beginning of the question, calculate what the price for the 3month European put should be if there were no arbitrage opportunities.
Put = _______________ (3 decimals) 5
Student Name ________________________
Student Number ________________________
(3 marks) (c)
IF the current market price of a 3-month European put with a strike price of $45 was $4.25, based on your answer in part (b), describe the transactions would you enter into to earn a riskless profit? [If you were unable to answer part (b), you may use the price of $4.35]
Transaction #1: _______________________________________________________________ Transaction #2: _______________________________________________________________ Transaction #3: _______________________________________________________________ Additional Transactions:
(3 marks) (d)
Now assume that the stock is expected to pay a dividend of $0.75 in two months. If the current market price of a 3-month European call with a strike price of $45 was $0.50. Using the information at the very beginning of the question, calculate what the price for the 3-month European put should be if there was no arbitrage opportunities (3 decimals)
Put = _______________ (3 decimals) 6
Student Name ________________________
Student Number ________________________
Question 3: Option Trading Strategies (12 marks total) Consider a portfolio of options that consists of: Sell 3-month put with a strike price of $35 that priced at $4.50 Long 3-month put with strike price of $43 that is priced at $7.00 Currently the price of the underlying stock is $40.
(2 marks) (a) Draw the PROFIT diagram based on the above portfolio. Clearly label the breakeven(s).
(3 marks) (b) Complete the below table to Show Payoffs and Profit/Losses for the different possible price ranges of ST at expiration. Let ST denote the Stock Price at Expiration. Stock Price
Total Portfolio Payoff
Total Portfolio Profit/Loss
ST ≤ 35 35 < ST < 43 ST ≥ 43
(1 mark) (c) Calculate the profit or loss IF the value of the underlying asset at expiration was equal to $41.00.
Profit/Loss when ST=41 = ______________ (2 decimals) 7
Student Name ________________________
Student Number ________________________
(2 marks) (d) Explain the logic or reasoning behind why a trader would invest in the portfolio described at the beginning of the question. (maximum of 4 sentences)
(4 marks) (e) Name the trading strategy you drew in part (a) above: ________________________________ Based on your answer provided in part (d), describe the specific transactions you would enter into to create an “alternative portfolio” of derivatives that would have the same PAYOFF diagram to that of the portfolio described at the beginning of the question. Transaction #1: _______________________________________________________________ Transaction #2: _______________________________________________________________ Transaction #3: _______________________________________________________________ Additional Transactions:
Draw the PAYOFF diagram of this alternative portfolio
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Student Name ________________________
Student Number ________________________
Question 4: Futures Pricing & Arbitrage (10 marks) A trader is currently monitoring prices for a number of different variables and notices that: The current spot rate (S0) for cattle is $1.00 per pound. o The costs to feed cattle are $0.01 monthly per pound which is paid at the beginning of the month. o There is a rental agreement to house cattle of $4000 annually payable every end of quarter, which can accommodate 20,000 pounds of cattle o Currently there is ONLY 80,000 pounds of cattle available for purchase or sale in the spot market The current continuous risk-free rate is 5% per annum Today is January 1st
(2 marks) (a) Calculate what a 4-month Futures price on cattle (in dollars per pound) should be (4 decimals)
Futures Price = ______________ (4 decimals)
(2 marks) (b) A trader also observes that Currently the actual market price for a 1-month Futures price on cattle is $1.15 per pound. A 1-month futures contract is for delivery of 40,000 pounds of cattle Based on the answer in part (a), explain exactly what transactions might a trader enter into? (maximum of 3 sentences) [If you were unable to answer part (a), you may use the value of F0 = 1.05] Transaction #1: _______________________________________________________________ Transaction #2: _______________________________________________________________ Transaction #3: _______________________________________________________________ Additional Transactions:
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Student Name ________________________
Student Number ________________________
(6 marks) (c) As result of entering into the transactions described in part (b), calculate the total cash flows when the transaction was initially entered into (t=0) and at every important point in time Please fill out the below table to answer part (c) of the question. Use 2 decimals for cash flows and as many time periods as you need. Time period
Cash flows from each transaction (-) means cash outflow (+) means cash inflow
Transaction(s)
t=0
t= _______
t= _______
t= _______
t= _______
t= _______
t= _______
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Student Name ________________________
Student Number ________________________
Question 5: Binomial Pricing Model (10 marks total) A stock price is currently $50. Over each of the next two three-month periods it is expected to go up by 5.9% or down by 5.6%. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a six-month European call with a strike price of $51?
(1 mark) (a)
Please write the “p” that you used here _________________ (4 decimal places)
(2 marks) (b)
Complete the below Stock Price Tree (answer in dollars & cents to 2 decimal places)
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Student Name ________________________
Student Number ________________________
(6 marks) (c)
Complete the below Option Price Tree for the European call (answer in dollars & cents to 2 decimal places)
D B A
E C F
(1 mark) (d)
Were this call an American call, at which nodes in the above option price tree would you exercise? (circle all nodes where you would exercise) None A
B
C
D
E
F
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Student Name ________________________
Student Number ________________________
Question 6: Binomial Pricing Model (12 marks total) To receive partial credit, you should show all calculations. Please write your answers in the spaces provided. At each node: the upper number should be the stock price the lower number should be the option price A stock price is currently $95. The volatility of the stock is 12% per annum. The stock does not pay any dividends. The risk-free continuous compounding interest rate is 5% per annum.
(4 marks) (a) Using the Binominal Option Pricing model calculate the value of a six-month European call option with a strike price of $100 (final answer 2 decimals, intermediate calculations 4 decimals)
p = _____________ u = _____________ d = _____________ (all to 4 decimals)
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Student Name ________________________
Student Number ________________________
(4 marks) (b) Using the Binominal Option Pricing model calculate the value of a six-month European put option with a strike price of $100 (to 2 decimals)
(1 mark) (c) Verify that the European call and European put prices you calculated satisfy put-call parity.
(2 marks) (d) What is the delta of the European call?
Δc= _______ (1 mark) (e) Now assume that the put option is an American option. Without actually calculating a final price for the put option at node A, using the values calculated for the various nodes (other than node A) determine if it would be optimal to exercise the American put early. Would you exercise early?
Please circle your answer:
If Yes, at which node? (please circle) 14
A
Yes B
C
No D
E
F
Student Name ________________________
Student Number ________________________
Question 7: Black-Scholes Pricing (12 marks total) The price of the underlying asset of the option is $156. The underlying asset pays a continuous dividend yield of 4%. The volatility of the underlying asset is 30%, and the risk free interest rate is 6%.
(6 marks) (a) Using the Black Scholes model, calculate the value and delta of a six-month European call option with a strike price of $160 ( 4 decimals when calculating d1 and d2)
d1 = ____________ d2 = ____________
cvalue = __________ Δc = ____________ 15
Student Name ________________________
Student Number ________________________
(6 marks) (b) There is another underlying asset with the same volatility that is currently priced at $150.00 and pays a dividend in four months of $5.00 Using the Black Scholes model, calculate the value of a six-month American put option with a strike price of $160 (3 decimals – round to 2 decimals when calculating d1 and d2) [Hint: use Black’s Approximation]
d1 = ____________
d1 = ____________
d2 = ____________
d2 = ____________
p = _____________
p = _____________ 16
P = _____________
Student Name ________________________
Student Number ________________________
Question 8: The Greeks (12 marks total) An investor has created a portfolio consisting of a combination of calls and puts (on the same underlying stock) and the characteristics of the options are described in the following table
Option A Option B
Number of contracts -500 -300
Delta of option
Gamma of option
Vega of option
0.85 -0.25
2.20 1.25
2.5 0.60
(5 marks total, 1 mark each) (please circle your answers) (a)
Which option is a Call?
Option A
Option B
Both
Can’t Tell
(b)
Which option is deeper in-the-money?
Option A
Option B
Neither
Can’t Tell
(c)
Which option is more sensitive to volatility changes? Option A
Option B
Neither
Can’t Tell
(d)
Which option’s price will go down if the underlying stock price goes up? If the underlying goes up by one, which option will change in value more?
Option A
Option B
Neither
Can’t Tell
Option A
Option B
Neither
Can’t Tell
(e)
(2 marks) (f)
Define the delta of a portfolio of options which is based on a single stock, and explain how the delta of a portfolio is used in hedging. (4 sentences maximum)
(g)
Based on the information given in the table, calculate the Delta of the portfolio of options, the Gamma of the portfolio of options and the Vega of the portfolio of options (1 decimal)
(3 marks)
Portfolio Delta = ___________ Portfolio Gamma = ___________ Portfolio Vega = ___________ 17
Student Name ________________________
Student Number ________________________
(2 marks) (h)
Based on your answer in part (b) for the portfolio of options, clearly describe how you could modify your portfolio of options to be JUST delta neutral (show calculations to receive full marks 7 circle your final answer) [If you were unable to answer part (e) you may use the value of -500 for the Delta of the portfolio of options]
THE LAST PAGE OF THE EXAMINATION
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Student Name ________________________
Student Number ________________________
Formula sheet
Optimal hedge ratio: h
S F
N*
Optimal number of contracts:
h * QA QF N
Optimal number of contract for equity hedging:
Capital Asset Pricing Model (CAPM) : Changing Beta:
( initial new )
VA VF
E(Rt) = rf + β [ E(Rmt) – rf]
VA VF
Conversion formulas for interest rates:
R Rc m ln 1 m m Rc / m Rm m e 1
Forward – Spot relationship: No income, no storage cost:
F = SerT
Income, no storage cost:
F0 = (S0 – I )erT or F0 = S0 e(r-q)T
Storage cost:
F0 = (S0+U) erT or F0 = S0 e(r+u )T
Cost of Carry:
F0 = S0 e(c–y)T
Forwards on Currencies: Value of a forward contract:
F0 S 0 e
( r r f )T
ƒ = (F0 – K)e-rT or ƒ = (K – F0 )e–rT
Options: Put – Call Parity:
c + Ke-rT = p + S0
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Student Name ________________________
Student Number ________________________
Binomial Option Pricing Model
p
e rT d ud
p
e ( r q ) dt d ud
u e
t
d e
t
1/ u
Black-Scholes Formulas For underlying assets: -
with no dividends:
c S 0 N (d1 ) K e rT N (d 2 ) p K e rT N (d 2 ) S 0 N (d1 ) where d1 d2
-
ln( S 0 / K ) (r 2 / 2)T
T
ln( S 0 / K ) (r 2 / 2)T
T
with known dividend yield:
d1 T
c S 0 e qT N (d1 ) Ke rT N (d 2 ) p Ke rT N (d 2 ) S 0 e qT N (d1 ) where, d1 d2
currency:
c S0e
rf T
N (d1 ) Ke rT N (d 2 )
p Ke rT N (d 2 ) S 0 e where, d1
rf T
ln( S 0 / K ) (r q 2 / 2)T
T ln( S 0 / K ) (r q 2 / 2)T
T
N (d1 )
ln( S 0 / K ) (r r 2 / 2)T f
d2
T ln( S 0 / K ) (r r 2 / 2)T f
T
Greeks Gamma:
c p
Theta:
2
S 2 2t Se .5( d1 ) c rKe rt N (d 2 ) 2 2t
p Rho:
e .5( d1 )
Se .5( d1 ) 2 2t
2
1 rS0 2 S 02 r 2
rKe rt N (d 2 )
c Kte rt N (d 2 ) p Kte rt N (d 2 )
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Student Name ________________________
Student Number ________________________
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Student Name ________________________
Student Number ________________________
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