FNCE 403 v4: Assignment 1
假设您存入了初始保证金，并且在任何给定的日期都没有提取多余的保证金。每当在第t天发生追加保证金通知时，您都会在第t + 1天（即第二天）开始交易时进行存款以使余额达到初始保证金要求。
0 $ 1340.30
1 $ 1345.50
2 $ 1339.20
3 $ 1330.60
4 $ 1327.70
5 $ 1337.70
6 $ 1340.60
XYZ Inc.目前的股价为每股120美元。在接下来的两个月中（第1个月和第2个月），预计股价将分别上涨10％或下跌5％。 XYZ Inc.还有望在第1个月末支付2％的股息收益率。无风险利率为每月0.5％。
Question 3. Currency Option Pricing with Binomial Model (10 marks)
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 3 percent, both compounded annually. Treat the Canadian dollar as the domestic currency.
- What is the estimated value of this put option by using the binomial model? (5 marks)
- 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)
- 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.
- 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)
- 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? (5.5 marks)
Question 5. Volatility and Option Hedging (34 marks)
Today, is January 4, 2016. IBM common stock is selling at $135.95 per share. The stock has a dividend yield of 4% per year. The following table contains the monthly stock prices for IBM shares during the last 12 months.
|Month (2015)||IBM Share Price|
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.
- 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)
- 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)
- 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)
- 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)
- 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:
|Strike/Expiry||January 22, 2016||February 19, 2016||March 18, 2016|
|Strike/Expiry||January 22, 2016||February 19, 2016||March 18, 2016|
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.
- Write out the payoff and profit function. (4 marks)
- Use a table to show the payoffs and profits when the put option expires in-the-money and out-of-the-money. (2 marks)
- 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.) (2 marks)
- What is the breakeven share price at expiration for this protective put? (1 mark)
- What is the maximum profit and maximum loss on this protective put? (1 mark)
Question 7. Box Spread (9 marks)
Use the data on Suncor Inc. presented in Question 6 above to answer this question.
- Construct a box-spread using the March option contracts with exercise prices of 24 and 25. (2.5 marks)
- Construct a profitable riskless arbitrage opportunity using this box-spread, with the requirement of $0 investment today. Calculate the NPV of the riskless profit.(6.5 marks)
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