Dear Tutor,

Could you explain how to solve the Table 4,5,7 in the second part of the document?

Thank you so much!

Homework 1 FINA 4522 Options and Derivatives (I) Professor Hengjie Ai Undergrad, Fall 2014 O ce: 3-247 Phone: 612-626-7348 Carlson School of Management Email: hengjie@umn.edu University of Minnesota Homework 1 is due on Sep 10, Wednesday, at the beginning of the class. Your session sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m number, team number, and the names of all team members must appear on the front page of your submission. Your submission must be typed and submitted in hard copy. Identical homework submissions will be considered as violation of the student conduct code. You do not need to type in question II in this homework. You can simply print out page 3-5 of the pdf ...le and ...ll in the blanks with a pen or pencil. I Continuously Reinvesting Dividends (10 Points) This exercise is designed to lead you through the steps of compounding interest/dividend payment at dierent frequencies. After this exercise, you should understand the rationale for using continuous compounding, and the reason why the dierence between discrete compounding and continuous compounding becomes small as the compounding frequency increases. Suppose an investor invested in a stock that pays a constant dividend yield at 10% per year. This means if dividends are paid semiannually, then each dividend payment is 5% of the share price; if the dividends are paid quarterly, then each dividend payment is 2:5% of the share price; if dividends are made monthly, then each payment is 10% 12 = 0:833% of the share price, etc. Note that the stock price may uctuate, but constant dividend yield means Th that the ratio of dividend payment to stock price remains constant. Assume that the investor invested in 100 shares of stock at the begining of period. He plans to use dividend payments to repurchase shares whenever possible and hold all shares until the end of the ...fth year. 1) Suppose the investor is only allowed to reinvest the dividend of the stock at the end of the ...fth year. How many shares of the stock will the investor hold at the end of the ...fth year? 1 https://www.coursehero.com/file/10091258/Homework-1/ 2) Suppose dividends are paid semi-annually and the investor is allowed to reinvest to purchase more shares after each dividend payment, how many shares of the stock will the investor hold at the end of the ...fth year? 3) Suppose dividends are paid monthly and the investor is allowed reinvest monly, how many shares of the stock will the investor hold at the end of the ...fth year? 4) Suppose dividends are paid daily, and the investor is allowed to reinvest daily, how many shares of the stock will the investor hold at the end of the ...fth year? (Assuming there sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m are 365 days in a year). 5) Using the formula of continuous compounding to calculate the number of shares that the investor hold at the end of the ...fth year. Compare with the answers in 1)-5), what do Th you conclude? 2 https://www.coursehero.com/file/10091258/Homework-1/ II Constructing Arbitrage Portfolio (10 Points) (Grading criteria: 10 points in total, 1 point for each entry in the blanks.) This exercise will lead you through the steps of constructing arbitrage portfolio strategies when a forward contract is mis-priced. Constructing arbitrage portfolio is the key to understanding all derivative pricing issues. All of the arbitrage strategies used in this class fall into one of the two categories summarized below. Make sure you understand them. The same logic will be used repeatedly in dierent contexts throughout this class. Let S0 denote the current price of one share of a non-dividend-paying stock, and ST sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m denote the date T price of the stock. Let F0;T denote the forward price of one share of the stock with expiration date T 1 . The date 0 and date T cash ow of a forward contract on one share of the stock is as follows: Portfolio A Strategy Table 1 Date 0 Cash Flow Long Forward 0 Date T Cash Flow ST F0;T In order to calculate the no-arbitrage price of a derivative contract, we ...rst need to know how the payo of the derivative contract can be replicated by assets with known prices. We now consider the following portfolio strategy: Long (buy) 1 share of the stock and short (borrow) e rt F 0;T rt F risk free debt (or short e 0;T T-Bill). The cash ow of the above strategy is: Table 2 Date 0 Cash Flow Strategy Long 1 share Stock Portfolio B Borrow e rt F Total e 0;T e Date T Cash Flow S0 rt F 0;T rt F 0;T ST F0;T S0 ST F0;T Th Because Portfolio A and Portfolio B have the same payo (at date T ), they must have the same cost (price) by Law of One Price (at date 0). Therefore we have F0;T = __________ (...ll in the blank). Suppose now there is a non-dividend-paying stock named XYZ with S0 = $100. The continuously compounded interest rate is r = 5% per annum. The arbitrage-free forward price of one share of stock XYZ with two years of maturity is ___________ (...ll in the blank). 1 That is, it is the price speci...ed on a forward contract with expiration date T . Note this is dierent from the price or value of the forward contract itself. The value of a forward contract at its initiation date is 0. 3 https://www.coursehero.com/file/10091258/Homework-1/ Suppose the above forward contract is traded at $105 per contract (That is, F0;T = 105). The ...rst method of constructing an arbitrage portfolio can be summarized as \"Buy High and Sell Low \". The key to this type of strategy is to construct two portfolios with the same price (at date 0), but dierent payo (at date T ), with one payo dominating the other. The arbitrage portfolio involves purchasing the portfolio with higher payo and selling the portfolio with lower payo . The cash ow of the above forward contract is summarized as: Table 3 Date 0 Cash Flow Strategy sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m Portfolio A0 Date T Cash Flow Long 1 Forward 0 ST 105 With the \"correct\" forward price you calculated above, the cash ow of the portfolio strategy B is as follows (...ll in the blank): Table 4 Date 0 Cash Flow Strategy Long 1 unit Stock Portfolio B0 Borrow e rt F Date T Cash Flow 100 ST 100 0;T Total 0 Note portfolio B 0 and portfolio A0 have the same price at date 0, yet portfolio A0 has a higher payo at date T . Therefore, we can long the portfolio with higher date-T payo and short the portfolio with lower date-T payo (Hence the name Buy High and Sell Low ), as follows: Strategy Table 5 Date 0 Cash Flow Long A0 0 B0 0 Th Short Total Date T Cash Flow ST 105 0 Verify this strategy is indeed an arbitrage by ...lling in the blanks in the above table. The second type of portfolio strategy is called \"Buy Low, Sell High\". The key to this type of portfolio strategy is to construct two portfolios with identical payos but with dierent prices. The arbitrage portfolio strategy involves purchasing the portfolio with lower price and selling the portfolio with higher price. We know that if we set F0;T = 105 in table 2, then the portfolio strategy B replicates the payo of the forward contract in table 3. The cash ow of this strategy is: 4 https://www.coursehero.com/file/10091258/Homework-1/ Strategy Table 6 Date 0 Cash Flow Long 1 unit stock Portfolio B 00 Date T Cash Flow 100 Borrow 95:007 ST 95:007 Total 105 4:993 ST 105 In this case the cost of portfolio B 00 is higher than the cost of the forward. Therefore the arbitrage portfolio can be constructed as: Table 7 Date 0 Cash Flow Date T Cash Flow sh is ar stu ed d vi y re aC s o ou urc rs e eH w er as o. co m Strategy Long A0 short B 00 ST ST + 105 Th Total 5 https://www.coursehero.com/file/10091258/Homework-1/ Powered by TCPDF (www.tcpdf.org) 105