The University of Hong Kong FACULTY OF ENGINEERING DEPARTMENT OF COMPUTER SCIENCE COMP 7802 Introduction to financial computing Date: December 9, 2017 Time: 2:30pm-4:30pm Time allowed Student l.D. 2 hours Only approved calculators as announced by the Examinations Secretary can be used in this examination. It is candidates' responsibility to ensure that their calculator operates satisfactorily, and candidates must record the name and type of the calculator used on the front page of the examination script. Brand and type of calculator Candidates are permitted to bring to the examination one piece of A 4-sized paper with printed or written notes on both sides. Answer ALL questions. Page I of7 COMP7802 Universitv Number: I. [15 points] Loan a. [5 points] On 13 Dec 2016, a customer entered into a loan of$100,000 with a bank. The annual interest rate is 5%. The repayment schedule is Pon 13 Dec 2017, 2P on 13 Dec 2018 and 3P on 13 Dec 2019. Calculate P. Assume ACT/365 for year basis. 1. [ 4 points] Show the equation for calculating P. ii. [1 point] Calculate P. b. [10 points] On 13 Dec 2016, a customer entered into a loan of $100,000 with a bank. The repayment schedule is $55,000 on 13 Dec 2017 and $55,000 on 13 Dec 2018. Estimate the annual interest rater. Again, assume ACT/365 for year basis. 1. [ 4 points] Show the equation for calculating r. ii. [ 6 points] Calculate r. 2. [15 points] Arbitrage Given the following market rates: 3 months Rates 5.05% - 5.10% 6 months Rates FRA 3x6 months 4.95% 5.00% 4.95% 5.00% Is there any opportunity for arbitrage? If yes, clearly indicate the transactions and the respective times you have to execute. You may use a notional of $1,000,000 for illustration. If you transact an FRA, you should also show the notional. Show a diagram of the cashflows (individual cashflows instead of netted should be shown). If no, clearly explain why. Page 2 of? COMP 7802 Universitv Number: 3. [15 points] An FRA market maker sells a EUR 100 million 3v6 FRA at a rate of7.52%. He is exposed to the risk that interest rates will have risen by the FRA settlement date in three months' time. Date (spot) 21 December 3v6 FRA rate 7.52% March futures price 92.50 Current 3M spot rate 6.85% The dealer needs to calculate a hedge ratio. Assume 30/360, that is, 90/360 for 3 months. a. [4 points] If interest rate for the FRA period rises by 1 bp (basis point), calculate the loss to the dealer, discounting back to the spot day-. b. [2 points] Should the dealer buy or sell future contracts to hedge the FRA? c. [4 points] Given the following specification for the Three Month Euro (Euribor) Future Notional €1,000,000 Tick Size 0.5 bp Calculate the hedge ratio. Show your expression. d. [ 5 points] Market data after 3 months Date (spot) 21 March March futures settlement 92.38 3M spot rate 7.625% Calculate the P/L (Profit and Loss). 4. [5 points] True or False a. If economic activity is expected to accelerate in the future, the yield curve tends to become flatter. b. Wide spread quotes indicates an instrument has a high liquidity risk. c. Investing in commercial paper usually has a higher return than treasury notes. d. At inception, the theoretical NPV of an IRS is zero. e. You have to pay a non-refundable premium for entering into a futures contract. Page 3 of7 COMP7802 Universi Number: 5. [15 points] Near market closing'on a given day T, below information is observed for stock and European stock option of Company ABC. Current Stock price= $76.5 per share Option Strike= $75 Option time to expiry= 90 days Day/ year convention= ACT/360 Annualized Volatility= 16.52% Continuous compounding risk-free interest rate= 0.68% per annum The Investor would like to use 2 periods binomial option pricing model (Cox, Ross & Rubinstein) to calculate the theoretical put and call option price of the above option. In his spreadsheet model, the following tree diagram is shown. Peri·od 0 Period 1 Pe..-iod 2 85.98 Sl.l-0 76.:5--0 1< ~I 76.5-0 . I 72.16 I" I 6.S.07 ~E~·~~P~:Sit.:01,1!i:Q.~:. Y:a.i µ~ Period 0 Period 1 Pe..-iod 2 I '\?I 1<1 1< I ,74 ,~6 '7'-l I \75 1( I "~3 a) [5 points] Please calculate below intermediary parameters of binomial option pricing model. • • • • • i'-t = period interval in each binomial nodes DF = discount factor in each binomial nodes . . u = up JUmp size d = down jump size p =risk neutral probability of up jump size Please show the calculation expression of each intermediary parameters (round the calculated results to the nearest 6 decimal points) Page4of7 COMP7802 Universi Number: b) [7 points] Please use the tree diagram shown above with given stock prices projection to calculate the European call option theoretical price. You are required to calculate the value ofVl to V6, then indicate the theoretical price of European call option clearly. Please show the calculation expression and round to 6 decimal points accuracy c) [3 points] The current call option market price on T is the same as the option theoretical price computed in b) above, please determine the following on T: • Intrinsic value of the call option • Time value of the call option • Moneyness of the call option Please show the calculation expression and round to 6 decimal points accuracy. 6. [20 marks] Below are the current market price observation for Company ABC on a given day T. Current Stock price on T ($) 76.5 Continuous com pounding risk free rate {p.a.) 0.68% Charateristic Call Oetion Put Oetion Price{$) 3.20 1.78 Strike($) 7S 75 Time to expiry 90 days 90 days day count basis ACT/360 ACT/360 Option Style European European The Investor said that "the arbitrage opportunity exists based on the above option price quotation in the market". He would like to consult your advice on the appropriate investment strategy to exploit the optimal arbitrage profit in the market based on Put-Call Parity theory. (Please show the calculation expression with 2 decimal points accuracy in this question.) a) [2 points] Please apply the Put-Call Parity theory to prove that the statement advised by Investor is correct. b) [9 points] Based on Put-Call Parity Theory, the Investor is considering to use the following products with 1 unit each to exploit the arbitrage opportunity per observed market data, namely (i) bond at price of PV(X) on T, (ii) stock, (iii) call option and (iv) put option. He said his expectation is "all the arbitrage profit should be locked up on T and no additional profit or loss at option expiry date regardless the stock price at expiry is above or below the option strike." i) [ 4 points] Please advise the Investor the appropriate buy I sell action of each product on T which will meet the expectation required by the investor. Page 5 of7 COMP7802 Universit Number: ii) [5 points] Then, please list out the cost of investment of each product and calculate the arbitrage profit locked up on T. (Please indicate +ve and -ve sign as cash-inflow and cash- outflow respectively) c) (9 points] Based on your advice given in b above, please explain what you will do to settle the products involved at option expiry date in each scenario below in order to validate that there will have "no additional profit or loss at option expiry date regardless the stock price at expiry is above or below the option strike: i) stock price= $70 at option expiry ii) stock price= $75 at option expiry iii) stock price= $80 at option expiry In your solution, you are required to provide (i) each option value at expiry, (ii) each option exercise status at expiry and (iii) the respective product delivery actions to be taken at option expiry. 7. (15 marks] The Investor is working on the spreadsheet to estimate the stock price volatility. Below is the sequence of closing prices of the stock of Company ABC for the preceding month- end shown in the spreadsheet. However, the calculation is only partially completed . Month I x, (x,-X)2 . I 0 58.35 .. . 1 61.45 0.051764: 0.000859 2 62.25 0.012935. 0.000091 3 66.00I O.OS8496. 0.001299 62.55: -0.053689. 0.005799 ----------"" ·-----~-i- --"-"''""-·-·· 5 63.15: 0.009547. 0.000167 6 64.25 0.017269: 0.000027 T 67.90 0.055254: 0.001075 8 72.85 0.070366. 0.002295 9 78.40 0.073421 0.002597 --- -- ----- - --- - --- - - 10 76.00 -0.031091 0.002868 11 76.55 0.007211 0.00()233 • -----.---- 76.40 -0.001961 0.000596 where: St= observed stock price at time period t Xt = LN (Stl S1-1) x = Average(.)(,;) = (IXr ) I n n =no. of months over the year Page 6 of7 COMP7802 Universi Number: a) [3 points] Using the above data, please help him to complete the annual volatility calculation. Please show the calculation expression with 6 decimal points accuracy and express volatility result in percent with 2 decimal points. b) [12 points] At the end ofmonth-9, Investor executed the 1 unit of long straddle option strategy at the following option market prices of Company ABC Current Stock price on T ($) 76.5 Continuous compounding risk free rate (p.a.) 0.68% Charateristic Call O~tion Put O~tion Price($) 3.40 1.78 Strike($) 75 75 Ti me to expiry 3 months 3 months Option Style European European Please show the calculation expression with 2 decimal points accuracy in this question. i) [2 points] Please explain the long straddle option strategy in terms of (i) the option combination and (ii) the expectation in this strategy deployment. ii) [ 4 points] Please calculate the following of 1 unit long straddle option strategy taken by the Investor at option expiry date. • potential maximum loss • potential maximum gain • brealceven point (stock price) when the stock price goes up • breakeven point (stock price) when the stock price goes down iii) [6 points] After 3 months, the option expired. Please calculate the profit or loss of this 1 unit long straddle option strategy at expiry in the following 2 scenarios: • the option expired and settled at month-12 stock closing price • the option expired and settled at stock price of $100 per share -END OF PAPER- Page 7 of7
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