FNCE 403 v4: Assignment 1

Due Date:
Assignment #1 should be submitted after you have completed Unit 2 (Lesson 7)

Credit Weight:
This assignment is worth 15 percent of your final grade.

Instructions

· This assignment contains seven problems, with a total of 100 marks. The maximum mark for each problem is noted at the beginning of the problem. Read the requirements for each problem, and plan your responses carefully.

· You are encouraged to contact your academic expert if you encounter any assignment problems that you do not understand or know how to solve. However, You must try your best to work on the problems for yourself first, and approach your academic expert with specific questions, not—“I don’t know how to do Questions 1, 2, 3, 4, 5, 6, and 7 in Assignment 1. Can you help?”

· Most, if not all, of the problems in Assignment 1 are referenced either in the assigned textbook readings or the online lesson notes (Lessons 1–7). You should not have trouble finding similar problems in the course materials.

· Although your responses should be concise, ensure that you answer each of the required components completely. If supporting calculations are required, present them in good form.

· Show ALL WORK! Use formulas to derive your answers for all the questions in this assignment. Do not use the Excel files provided by the authors of the textbook unless you are asked to do so in the question.

· When you have completed Assignment 1, go to the
FNCE 403
course site, and upload your completed assignment file(s) in the Assignment 1 drop box. Ensure that you have filled in all the correct information and uploaded the correct file(s), and then click the Submit button to submit the assignment. Note: Once you have clicked the Submit button, you will not be able to change your solutions to the assignment.

· When you receive your graded assignment, carefully review the comments the evaluator has made. This review component is an important step in your learning process. If you have any questions or concerns about the evaluation, please contact an academic expert.

Question 1. Margin Account and Settlement (6 marks)

Suppose that you bought two one-year gold futures contracts when the one-year futures price of gold was US$1,340.30 per troy ounce. You then closed the position at the end of the sixth trading day. The initial margin requirement is US$5,940 per contract, and the maintenance margin requirement is US$5,400 per contract. One contract is for 100 troy ounces of gold. The daily prices on the intervening trading days are shown in the following table.

Assume that you deposit the initial margin and do not withdraw the excess on any given day. Whenever a margin call occurs on Day t, you would make a deposit to bring the balance up to meet the initial margin requirement at the start of trading on Day t+1, i.e., the next day.

a. What are the initial margin and maintenance margin on your margin account?

Take the number of contracts (which is 2) multiplied by the initial margin contract. So, 2*5940 =
$11880

Then for maintenance margin, number of contracts (2) and then times by maintenance margin (5400) so maintenance margin =
$108000

(1 mark)

b. Fill the appropriate numbers in the blank cells in the following table. (Hint: See solution to Q19 in Lesson 2 Learning Activity.)
(4 marks)

c. What is your total profit after you closed out your position? 11880+580+860=13380

(1 mark)

Question 2. Binomial Model and Option Pricing (14 marks)

The shares of XYZ Inc. are currently selling for $120 per share. The shares are expected to go up by 10 percent or down by 5 percent in each of the following two months (Month 1 and Month 2). XYZ Inc. is also expected to pay a dividend yield of 2 percent at the end of Month 1. The risk-free rate is 0.5 percent per month.

a. What is the value of an American call option on XYZ shares, with an exercise price of $125 and two months to expiration? Use the binomial model to obtain the answer.

(12 marks)

b. Draw a binomial tree diagram for this American call option, showing the share price, call price, and whether the call should be exercised at each state during the next two months.

(2 marks)

Question 3. Currency Option Pricing with Binomial Model (10marks)

On January 11, the spot exchange rate for the U.S. dollar is $0.70 per Canadian dollar. In one year’s time, the Canadian dollar is expected to appreciate by 20 percent or depreciate by 15 percent. We have a European put option on U.S. dollars expiring in one year, with an exercise price of 1.39 CND$/US$, that is currently selling for a price of $2.93. Each put option gives the holder the right to sell 10,000 U.S. dollars. The current one-year Canadian Treasury Bill rate is 2 percent, while the one-year U.S. Treasury Bill rate is 3percent, both compounded annually. Treat the Canadian dollar as the domestic currency.

a. What is the estimated value of this put option by using the binomial model?

(5 marks)

b. Calculate the estimated value of this put option for U.S. T-Bill rates of 0%, 1%, 2%, 4%, 5%, and 6%. Plot these values in a graph (by hand or using Excel), with put option values on the y-axis and U.S. T-bill rates on the x-axis. What can we conclude about the relationship between foreign interest rates and foreign currency put option values?
(2.5 marks)

c. Calculate the estimated value of this put option for Canadian T-Bill rates of 0%, 1%, 2%, 4%, 5%, and 6%. Plot these values in a graph (by hand or using Excel), with put option values on the y-axis and Canadian T-bill rates on the x-axis. What can we conclude about the relationship between domestic interest rates and foreign currency put option values?
(2.5 marks)

Question 4. Option Pricing with Black-Scholes-Merton Model (17 marks)

Today is January 12, 2017. The shares of XYZ Inc. are currently selling for $120 per share. The shares have an estimated volatility of 25%. XYZ Inc. is also expected to pay a dividend of $1.50 with an ex-dividend date of January 25, 2017. The risk-free rate is 6.17 percent per year with continuous compounding. Assume that one call option gives the holder the right to purchase one share.

a. Use the Black-Scholes-Merton model to estimate the fair value of a European call option on XYZ shares, with exercise price of $125 and expiration date of March 21, 2017. (Note that 2017 is not a leap year.)
(11.5 marks)

b. This European call option has a market price of $3.00. Is it correctly priced? If not, how can an investor use the put-call parity to take advantage of this arbitrage opportunity?

Fair value of European call = $3.65



The call option is undervalued

Explanation:

Part A

Calculate the fair value of European call

Current price S = $120

Strike price K = $125

Time to expiration t = 68 days or 68/365

Implied volatility σ = 25%

Risk-free rate r = 6.17%

d1 = (ln(S/K) + (r + σ^2 / 2) * t) / σ * sqrt(t)

= (ln($120/$125) + (6.17% + 25%^2 / 2) * 68/365) / (25% * sqrt(68/365))

= -0.217830



d2 = d1 - σ * sqrt(t)

= -0.217830 - 25% * sqrt(68/365)

= -0.325736



Applying Excel NORM.S.DIST

N(d1) = NORM.S.DIST(d1,cummulative)

= NORM.S.DIST(-0.217830,1)

= 0.4138



N(d2) = NORM.S.DIST(d2,cummulative)

= NORM.S.DIST(-0.325736,1)

= 0.3723



C = SN(d1) - N(d2)Ke^-rt

= $120*0.4138 - 0.3723*$125*2.71828^-(6.17%*68/365)

= $3.65



The fair value of European call is $3.65

Part B

Fair value = $3.65

Market price = $3.00



The call option is not correctly priced. It is undervalued. I will arbitrage and use the following strategy; I will long call option at $3.

(5.5 marks)

Question 5. Volatility and Option Hedging (34 marks)

c

A call option with a March 18, 2016 expiration date and an exercise price of $130 is currently trading at $6.50. Each option entitles the holder to purchase 100 IBM shares. The risk-free rate is 0.58%, compounded continuously. Shares and options can only be bought and sold in whole numbers. Note that 2016 is a leap year.

a. Compute the historical volatility in terms of annualized standard deviation on the IBM shares, using the 12-month price data in the table above. Note that the volatility should be calculated on the stock returns and not on the stock prices. Obtain your answer to four decimal places (or two decimal places in percentage).
(3 marks)

b. Based on the market price of $6.50, derive the implied volatility on the IBM shares. You may use the BlackScholesMertonImpliedVolatility10e.xlsm file provided by the textbook’s authors to derive the implied volatility. Take a screen shot of the answer provided in this Excel spreadsheet, and copy and paste it into your answer for this question. Obtain your answer to four decimal places (or two decimal places in percentage).
(2 marks)

c. Construct a delta-hedge position on January 4, 2016 involving the sale of 1,000 calls. Then rebalance the portfolio at the end of the next day, when the share price goes down to $135 per share. Assume the market call price is correct. That is, use the implied volatility as the correct volatility for the IBM shares. (You may calculate the deltas using the formula or the BlackScholesMertonBinomial10e.xlsm file provided by the textbook’s authors. If you use the latter, include a screen shot of the Excel spreadsheet in your answer.)

Obtain the value of this delta-hedge portfolio after it has been rebalanced. Compare this value to the target value of the portfolio should its initial value be invested at the risk-free rate. Explain the difference.
(12 marks)

d. There is another call option on IBM shares with an exercise price of $125 and the same expiration date (March 18, 2016). Construct a delta- and gamma-hedge portfolio on January 4, 2016 involving the sale of 1,000 of the 130-call option. Then rebalance the portfolio at the end of the next day, when the share price goes down to $135 per share. Again, use the implied volatility as the correct volatility for the IBM shares. (You may calculate the deltas and gammas using the formula or the BlackScholesMertonBinomial10e.xlsm file provided by the textbook’s authors. If you use the latter, include a screen shot of the Excel spreadsheet in your answer.)

Obtain the value of this delta-and-gamma-hedged portfolio after it has been rebalanced. Compare this value to the target value of the portfolio should its initial value be invested at the risk-free rate. Explain the difference.
(16 marks)

e. Explain the difference between the delta-hedged portfolio value in part (c) and the delta-and-gamma-hedged portfolio value in part (d).
(1 mark)

Question 6. Protective Put (10 marks)

Suncor Energy Inc. (SU) shares are listed on the New York Stock Exchange. At 9:30 a.m. on January 14, 2016, these shares sold for $21.85 per share. The volatility on the returns of Suncor shares is approximately 24%. The following call and put option contracts were available for the months of January, February, and March:

Each option contract involves 100 shares. The risk-free rates for these three expiration dates are 0.6%, 1%, and 1.2%. All three rates are continuously compounded.

Given the information on Suncor shares and options above, construct a protective put using the 23-put with February expiration. Hold the protective put position until expiration.

a. Write out the payoff and profit function.
(4 marks)

b. Use a table to show the payoffs and profits when the put option expires in-the-money and out-of-the-money.
(2 marks)

c. Calculate the potential profits for this protective put, using share prices ranging from 0 to 26. Plot a graph of these potential profits, with share prices on the x-axis, and profits on the y-axis. (Hint: It may be easier to do this in an Excel spreadsheet.)
(2marks)

d. What is the breakeven share price at expiration for this protective put?
(1 mark)

e. What is the maximum profit and maximum loss on this protective put?
(1 mark)

Protective put is an option startegy used to hedge risk by holding a long position in the underlying asset and buying a put option with strike price equal to or closer to the current price of the asset.

Therefore, protective put is constructed by buying 100 shares of Suncor Energy Inc. (SU) at current price of $ 21.85 per share and buying a put option contract ( 100 put options) at strike $ 23.

a) Payoff of protective put at time t = S_t+ Max(K-S_t,0)

where S_t- Price of share at time t

K- Strike Price

Profit from Protective Put = Number of Shares or options * (( S_t- S₀) + Max(K - S_t) - P_t)

where S_t- Price of share at time t

S₀- Price of Share at time 0

K - Strike of put option

P_t- Price of put option

b) Put option price is $2.01, if the put option is in-the-money, then share price at time t is less than share price at time 0.

Assuming the share price at time t is $20,

Payoff from protective put = 20 + Max(23-20,0)

= 20+3 = 23.

Profit = 100 * (( 20 - 21.85) + (23-20) - 2.01) = 100 * - 0.86

= - $86

There is a loss of $86 if the option expires in-the-money.

If the option expires out-of-the-money, the share price at time t will be preater than at time 0.

Assuming share price at time t = $24

Payoff = 24 + max(23-24,0)

= 24.

Profit = 100 * ((24-21.85) + Max(23-24,0) - 2.01)

= 100 * (0.14)

= $14.

c)

d) Breakeven Share price = $23.86

e)Maximum profit is unlimited

Maximum loss = -$86

Question 7. Box Spread (9 marks)

Use the data on Suncor Inc. presented in Question 6 above to answer this question.

a. Construct a box-spread using the March option contracts with exercise prices of 24 and 25.
(2.5 marks)

b. Construct a profitable riskless arbitrage opportunity using this box-spread, with the requirement of $0 investment today. Calculate the NPV of the riskless profit.

(6.5marks)

FNCE 403 v4: Assignment 1

Due Date:
Assignment #1 should be submitted after you have completed Unit 2 (Lesson 7)

Credit Weight:
This assignment is worth 15 percent of your final grade.

Instructions

· This assignment contains seven problems, with a total of 100 marks. The maximum mark for each problem is noted at the beginning of the problem. Read the requirements for each problem, and plan your responses carefully.

· You are encouraged to contact your academic expert if you encounter any assignment problems that you do not understand or know how to solve. However, You must try your best to work on the problems for yourself first, and approach your academic expert with specific questions, not—“I don’t know how to do Questions 1, 2, 3, 4, 5, 6, and 7 in Assignment 1. Can you help?”

· Most, if not all, of the problems in Assignment 1 are referenced either in the assigned textbook readings or the online lesson notes (Lessons 1–7). You should not have trouble finding similar problems in the course materials.

· Although your responses should be concise, ensure that you answer each of the required components completely. If supporting calculations are required, present them in good form.

· Show ALL WORK! Use formulas to derive your answers for all the questions in this assignment. Do not use the Excel files provided by the authors of the textbook unless you are asked to do so in the question.

· When you have completed Assignment 1, go to the
FNCE 403
course site, and upload your completed assignment file(s) in the Assignment 1 drop box. Ensure that you have filled in all the correct information and uploaded the correct file(s), and then click the Submit button to submit the assignment. Note: Once you have clicked the Submit button, you will not be able to change your solutions to the assignment.

· When you receive your graded assignment, carefully review the comments the evaluator has made. This review component is an important step in your learning process. If you have any questions or concerns about the evaluation, please contact an academic expert.

Question 1. Margin Account and Settlement (6 marks)

Suppose that you bought two one-year gold futures contracts when the one-year futures price of gold was US$1,340.30 per troy ounce. You then closed the position at the end of the sixth trading day. The initial margin requirement is US$5,940 per contract, and the maintenance margin requirement is US$5,400 per contract. One contract is for 100 troy ounces of gold. The daily prices on the intervening trading days are shown in the following table.

Assume that you deposit the initial margin and do not withdraw the excess on any given day. Whenever a margin call occurs on Day t, you would make a deposit to bring the balance up to meet the initial margin requirement at the start of trading on Day t+1, i.e., the next day.

a. What are the initial margin and maintenance margin on your margin account?

Take the number of contracts (which is 2) multiplied by the initial margin contract. So, 2*5940 =
$11880

Then for maintenance margin, number of contracts (2) and then times by maintenance margin (5400) so maintenance margin =
$108000

(1 mark)

b. Fill the appropriate numbers in the blank cells in the following table. (Hint: See solution to Q19 in Lesson 2 Learning Activity.)
(4 marks)

c. What is your total profit after you closed out your position? 11880+580+860=13380

(1 mark)

Question 2. Binomial Model and Option Pricing (14 marks)

The shares of XYZ Inc. are currently selling for $120 per share. The shares are expected to go up by 10 percent or down by 5 percent in each of the following two months (Month 1 and Month 2). XYZ Inc. is also expected to pay a dividend yield of 2 percent at the end of Month 1. The risk-free rate is 0.5 percent per month.

a. What is the value of an American call option on XYZ shares, with an exercise price of $125 and two months to expiration? Use the binomial model to obtain the answer.

(12 marks)

b. Draw a binomial tree diagram for this American call option, showing the share price, call price, and whether the call should be exercised at each state during the next two months.

(2 marks)

Question 3. Currency Option Pricing with Binomial Model (10marks)

On January 11, the spot exchange rate for the U.S. dollar is $0.70 per Canadian dollar. In one year’s time, the Canadian dollar is expected to appreciate by 20 percent or depreciate by 15 percent. We have a European put option on U.S. dollars expiring in one year, with an exercise price of 1.39 CND$/US$, that is currently selling for a price of $2.93. Each put option gives the holder the right to sell 10,000 U.S. dollars. The current one-year Canadian Treasury Bill rate is 2 percent, while the one-year U.S. Treasury Bill rate is 3percent, both compounded annually. Treat the Canadian dollar as the domestic currency.

a. What is the estimated value of this put option by using the binomial model?

(5 marks)

b. Calculate the estimated value of this put option for U.S. T-Bill rates of 0%, 1%, 2%, 4%, 5%, and 6%. Plot these values in a graph (by hand or using Excel), with put option values on the y-axis and U.S. T-bill rates on the x-axis. What can we conclude about the relationship between foreign interest rates and foreign currency put option values?
(2.5 marks)

c. Calculate the estimated value of this put option for Canadian T-Bill rates of 0%, 1%, 2%, 4%, 5%, and 6%. Plot these values in a graph (by hand or using Excel), with put option values on the y-axis and Canadian T-bill rates on the x-axis. What can we conclude about the relationship between domestic interest rates and foreign currency put option values?
(2.5 marks)

Question 4. Option Pricing with Black-Scholes-Merton Model (17 marks)

Today is January 12, 2017. The shares of XYZ Inc. are currently selling for $120 per share. The shares have an estimated volatility of 25%. XYZ Inc. is also expected to pay a dividend of $1.50 with an ex-dividend date of January 25, 2017. The risk-free rate is 6.17 percent per year with continuous compounding. Assume that one call option gives the holder the right to purchase one share.

a. Use the Black-Scholes-Merton model to estimate the fair value of a European call option on XYZ shares, with exercise price of $125 and expiration date of March 21, 2017. (Note that 2017 is not a leap year.)
(11.5 marks)

b. This European call option has a market price of $3.00. Is it correctly priced? If not, how can an investor use the put-call parity to take advantage of this arbitrage opportunity?

Fair value of European call = $3.65



The call option is undervalued

Explanation:

Part A

Calculate the fair value of European call

Current price S = $120

Strike price K = $125

Time to expiration t = 68 days or 68/365

Implied volatility σ = 25%

Risk-free rate r = 6.17%

d1 = (ln(S/K) + (r + σ^2 / 2) * t) / σ * sqrt(t)

= (ln($120/$125) + (6.17% + 25%^2 / 2) * 68/365) / (25% * sqrt(68/365))

= -0.217830



d2 = d1 - σ * sqrt(t)

= -0.217830 - 25% * sqrt(68/365)

= -0.325736



Applying Excel NORM.S.DIST

N(d1) = NORM.S.DIST(d1,cummulative)

= NORM.S.DIST(-0.217830,1)

= 0.4138



N(d2) = NORM.S.DIST(d2,cummulative)

= NORM.S.DIST(-0.325736,1)

= 0.3723



C = SN(d1) - N(d2)Ke^-rt

= $120*0.4138 - 0.3723*$125*2.71828^-(6.17%*68/365)

= $3.65



The fair value of European call is $3.65

Part B

Fair value = $3.65

Market price = $3.00



The call option is not correctly priced. It is undervalued. I will arbitrage and use the following strategy; I will long call option at $3.

(5.5 marks)

Question 5. Volatility and Option Hedging (34 marks)

c

A call option with a March 18, 2016 expiration date and an exercise price of $130 is currently trading at $6.50. Each option entitles the holder to purchase 100 IBM shares. The risk-free rate is 0.58%, compounded continuously. Shares and options can only be bought and sold in whole numbers. Note that 2016 is a leap year.

a. Compute the historical volatility in terms of annualized standard deviation on the IBM shares, using the 12-month price data in the table above. Note that the volatility should be calculated on the stock returns and not on the stock prices. Obtain your answer to four decimal places (or two decimal places in percentage).
(3 marks)

b. Based on the market price of $6.50, derive the implied volatility on the IBM shares. You may use the BlackScholesMertonImpliedVolatility10e.xlsm file provided by the textbook’s authors to derive the implied volatility. Take a screen shot of the answer provided in this Excel spreadsheet, and copy and paste it into your answer for this question. Obtain your answer to four decimal places (or two decimal places in percentage).
(2 marks)

c. Construct a delta-hedge position on January 4, 2016 involving the sale of 1,000 calls. Then rebalance the portfolio at the end of the next day, when the share price goes down to $135 per share. Assume the market call price is correct. That is, use the implied volatility as the correct volatility for the IBM shares. (You may calculate the deltas using the formula or the BlackScholesMertonBinomial10e.xlsm file provided by the textbook’s authors. If you use the latter, include a screen shot of the Excel spreadsheet in your answer.)

Obtain the value of this delta-hedge portfolio after it has been rebalanced. Compare this value to the target value of the portfolio should its initial value be invested at the risk-free rate. Explain the difference.
(12 marks)

d. There is another call option on IBM shares with an exercise price of $125 and the same expiration date (March 18, 2016). Construct a delta- and gamma-hedge portfolio on January 4, 2016 involving the sale of 1,000 of the 130-call option. Then rebalance the portfolio at the end of the next day, when the share price goes down to $135 per share. Again, use the implied volatility as the correct volatility for the IBM shares. (You may calculate the deltas and gammas using the formula or the BlackScholesMertonBinomial10e.xlsm file provided by the textbook’s authors. If you use the latter, include a screen shot of the Excel spreadsheet in your answer.)

Obtain the value of this delta-and-gamma-hedged portfolio after it has been rebalanced. Compare this value to the target value of the portfolio should its initial value be invested at the risk-free rate. Explain the difference.
(16 marks)

e. Explain the difference between the delta-hedged portfolio value in part (c) and the delta-and-gamma-hedged portfolio value in part (d).
(1 mark)

Question 6. Protective Put (10 marks)

Suncor Energy Inc. (SU) shares are listed on the New York Stock Exchange. At 9:30 a.m. on January 14, 2016, these shares sold for $21.85 per share. The volatility on the returns of Suncor shares is approximately 24%. The following call and put option contracts were available for the months of January, February, and March:

Each option contract involves 100 shares. The risk-free rates for these three expiration dates are 0.6%, 1%, and 1.2%. All three rates are continuously compounded.

Given the information on Suncor shares and options above, construct a protective put using the 23-put with February expiration. Hold the protective put position until expiration.

a. Write out the payoff and profit function.
(4 marks)

b. Use a table to show the payoffs and profits when the put option expires in-the-money and out-of-the-money.
(2 marks)

c. Calculate the potential profits for this protective put, using share prices ranging from 0 to 26. Plot a graph of these potential profits, with share prices on the x-axis, and profits on the y-axis. (Hint: It may be easier to do this in an Excel spreadsheet.)
(2marks)

d. What is the breakeven share price at expiration for this protective put?
(1 mark)

e. What is the maximum profit and maximum loss on this protective put?
(1 mark)

Protective put is an option startegy used to hedge risk by holding a long position in the underlying asset and buying a put option with strike price equal to or closer to the current price of the asset.

Therefore, protective put is constructed by buying 100 shares of Suncor Energy Inc. (SU) at current price of $ 21.85 per share and buying a put option contract ( 100 put options) at strike $ 23.

a) Payoff of protective put at time t = S_t+ Max(K-S_t,0)

where S_t- Price of share at time t

K- Strike Price

Profit from Protective Put = Number of Shares or options * (( S_t- S₀) + Max(K - S_t) - P_t)

where S_t- Price of share at time t

S₀- Price of Share at time 0

K - Strike of put option

P_t- Price of put option

b) Put option price is $2.01, if the put option is in-the-money, then share price at time t is less than share price at time 0.

Assuming the share price at time t is $20,

Payoff from protective put = 20 + Max(23-20,0)

= 20+3 = 23.

Profit = 100 * (( 20 - 21.85) + (23-20) - 2.01) = 100 * - 0.86

= - $86

There is a loss of $86 if the option expires in-the-money.

If the option expires out-of-the-money, the share price at time t will be preater than at time 0.

Assuming share price at time t = $24

Payoff = 24 + max(23-24,0)

= 24.

Profit = 100 * ((24-21.85) + Max(23-24,0) - 2.01)

= 100 * (0.14)

= $14.

c)

d) Breakeven Share price = $23.86

e)Maximum profit is unlimited

Maximum loss = -$86

Question 7. Box Spread (9 marks)

Use the data on Suncor Inc. presented in Question 6 above to answer this question.

a. Construct a box-spread using the March option contracts with exercise prices of 24 and 25.
(2.5 marks)

b. Construct a profitable riskless arbitrage opportunity using this box-spread, with the requirement of $0 investment today. Calculate the NPV of the riskless profit.

(6.5marks)