金融衍生工具测试题-(26)

1、金融衍生工具测试题-(26)2.3.a)b)c)d)e)1. A box spread is a comb in ati on of a bull spread composed of two call opti ons with strike prices X1 and X 2 and a bear spread composed of two put options with the same two strike prices.a) Describe the payoff from a box spread on the expiration date of the options.b)

2、 What would be a fair price for the box spread today? Define variables asn ecessary.c) Un der what circumsta nces might an in vestor choose to con struct a box spread?d) What sort of in vestor do you thi nk is most likely to in vest in such an opti on comb in ati on, i.e. a hedger, speculator or arb

3、itrageur? Expla in your an swer.Form a long butterfly spread using the three call options in the table below.C1X = $90 T = 180 daysC2X = $100 T = 180 daysC3X = $110 T = 180 daysPrice16.330010.3000 I6.0600DELTA0.78600.6151 0.4365GAMMA0.01380.0181：0.0187THETA-11.2054-12.26071-11.4208VEGA20.461926.8416

4、 27.6602RHO30.708525.2515：18.5394a) What does it cost to establish the butterfly spread?b) Calculate each of the Greek measures for this butterfly spread position and expla in how each can be in terpreted.c) How would you make this opti on portfolio delta n eutral?What would beachieved by doing so?d

5、) Suppose that tomorrow the price of C1 falls to $12.18 while the prices of C2 and C3rema in the same. Does this create an arbitrage opport uni ty?Expla in.Con sider a six month America n put opti on on in dex futures where the curre nt futures price is 450, the exercise price is 450, the risk-free

6、rate of interest is 7 percent per annum, the con ti nu ous divide nd yield of the in dex is 3 perce nt, and the volatility of the in dex is 30 perce nt per annum. The futures con tract un derly ing the opti on matures in seve n mon ths. Usi ng a three-step bino mial tree, calculatethe price of the A

7、merica n put opti on now,the delta of the opti on with respect to the futures price,the delta of the opti on with respect to the in dex level, andthe price of the corresp onding Europea n put opti on on in dex futures.Apply the control variate technique to improve your estimate of the American optio

8、n price and of the delta of the opti on with respect to the futures price.Note that the Black-Scholes price of the European put option is $36,704 and the delta with respect to the futures price give n by Black-Scholes is-0.442.4. A finan cial in stituti on trades swaps where 12 month LIBOR is excha

9、nged for a fixed rate of in terest. Payme nts are made once a year. The on e-year swap rate (i.e., the rate that would be excha nged for 12 month LIBOR in a new on e-year swap) is 6 perce nt. Similarly the two-year swap rate is 6.5 perce nt.a) Use this swap data to calculate the one and two year LIB

10、OR zero rates, express ing the rates with continu ous compo unding.b) What is the value of an existing swap with a notional principal of $10 million that has two years to go and is such that financial institution pays 7 percent and receives 12 month LIBOR? Payme nts are made once a year.c) What is t

11、he value of a forward rate agreement where a rate of 8 percent will be received on a pri ncipal of $1 milli on for the period betwee n one year and twoyears?Note: All rates give n in this questi on are expressed with annual compo unding.5.The term structure is flat at 5% per annum with continuous co

12、mpounding. Some time ago a financial institution entered into a 5-year swap with aprin cipal of $100 millio n in which every year it pays 12-m on th LIBOR and receives 6%. The swap now has two years eight mon ths to run. Four mon ths ago 12-m on th LIBOR was 4% (with annual compo undin g). What is t

13、he valueof the swap today? What is the financial institution s credit exposure on the swap?6. An America n put opti onto sell a Swiss franc for USD has a strike price of0.80 and a time to maturity of 1 year. The volatility of the Swiss franc is 10%, the USD in terest rate is 6%, and the Swiss franc

14、in terest rate is 3% (bothin terest rates con ti nu ously compo un ded). The curre nt excha nge rate is 0.81. Use a three time step tree to value the opti on.7. A European call option on a certain stock has a strike price of $30, a time to maturity of one year and an implied volatility of 30%. A put

15、 option on the same stock has a strike price of $30, a time to maturity of one year and an implied volatility of 33%. What is the arbitrage opport unity ope n to a trader. Does the opport un ity work only whe n the log no rmal assumpti on un derly ing Black-Scholes holds. Expla in the reas ons for y

16、our an swer carefully.8.A put option on the S&P 500 has an exercise price of 500 and a time to maturity of one year. The risk free rate is 7% and the divide nd yield on the index is 3%. The volatility of the index is 20% per annum and the current level of the index is 500. A financial institution ha

17、s a short position in the optio n.a) Calculate the delta, gamma, and vega of the positi on. Expla in how they can be in terpreted.b) How can the positi on be made delta n eutral?c) Suppose that one week later the in dex has in creased to 515. How can delta n eutrality be preserved?9. An interest rat

18、e swap with a principal of $100 million invoIves the exchange of 5% per annum (semia nnu ally compo un ded) for 6-m onth LIBOR. The remai ning life is 14 mon ths. In terest is excha nged every six mon ths. The 2 mon th, 8 month and 14 mon th rates are 4.5%, 5%, and 5.4% with con ti nu ous compo undi

19、ng. Six-m onth LIBOR was 5.5% four mon ths ago. What is the value of the swap?10.11.12.13.14.The Deutschemark-Ca nadia n dollar excha nge rate is curre ntly 1.0000. At the end of 6 mon ths it will be either 1.1000 or 0.9000. What is the value of a 6 mon th opti on to sell one millio n Can adia n dol

20、lars for 1.05 millio n deutschemarks. Verify that the answer given by risk neutral valuation is the same as that given by no-arbitrage arguments. Is the option the same as one to buy 1.05 million deutschemarks for 1 million Canadian dollars? Assume that risk-free in terest rates in Can ada and Germa

21、 ny are 8% and 6% per annum respectively.An America n put futures opti on has a strike price of 0.55 and a time to maturity of 1 year. The current futures price is 0.60. The volatility of the futures price is 25% and the interest rate(continuously compounded) is 6% per annum. Use a four time step tr

22、ee to value the opti on.Is it ever optimal to exercise early an American call option on a) the spot price of gold, b) the spot price of copper, c) the futures price of gold, and d) the average price of gold measured betwee n time zero and the curre nt time. Explai n your an swers.The future probabil

23、ity distribution of a stock price has a fatter right tail and thinner left tail tha n the log no rmal distributio n. Describe the effect of this on the prices of in-the-m on ey a nd out-of-the-m oney calls and puts. What is the volatility smile that would be observed?A bank has just sold a call opti

24、 on on 500,000 shares of a stock. The strike pric is 40; the stock price is 40; the risk-free rate is 5%; the volatility is 30%; and the time to maturity is 3 mon ths.a) What position should the company take in the stock for delta neutrality?b) Suppose that the bank does set up a delta n eutral posi

25、ti on as soon as the optio has been sold and the stock price jumps to 42 within the first hour of trading. What trade is n ecessary to mai ntai n delta n eutrality? Expla in whether the bank has gained or lost money in this situatio n. (You do not n eed to calculate the exact amount gained or lost.)

26、c) Repeat part b) on the assumpti on that the stock jumps to 38 in stead of 4215. A bank has sold a product that offers investors the total return (excluding divide nds) on the Toronto 300 in dex over a one year period. The retur n is capped at 20%. If the index goes down the original investment of

27、the inv estor is returned.a) What opti on positi on is equivale nt to the productb) Write dow n the formulas you would use to value the product and expla in in detail how you would decide whether it is a good deal to the inv estorUse a three step tree to value a three month America n put optio n on

28、wheat futures. The curre nt futures price is is 380 cen ts, the strike price is 370 cents, the risk-free rate is 5% per annum, and the volatility is 25% per annum. Expla in carefully what happe ns if the inv estor exercises the opti on after two mon ths. Suppose that the futures price at the time of

29、 exercise is 362 and the most rece nt settleme nt price is 360.a) A bank s assets and liabilities both haveduration of 5 years. Is the bank hedged aga in st in terest rate moveme nts? Explai n carefully any limitati ons of the hedgi ng scheme it has chose n.b) Explain what is meant by basis risk in

30、the situation where a company knows it will be purchas ing a certa in asset in two mon ths and uses a three-m onth futures con tract to hedge its risk.18. a) Give an example of how a swap might be used by a portfolio manager.b) Explain the nature of the credit risks to a financial institution in a s

31、wap agreeme nt1101.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.ANSWERS(a) The box spread pays off X 2-X1 in all circumstances (b) It should be worth the present value of X 2-X1 today, c) and d) An arbitrageur might invest in a box spread if it is mispriced in the market today.(a) 1.79 b) Greek letters are

32、-0.0077, -0.0037, etc c) For delta neutrality we buy 0.0077 of the un derly ing asset. Small cha nges in the price of the underlying asset then have very little effect on the value of the whole portfolio d) Yes. We have a positive cash flow when we set up the butterfly spread today and a zero or pos

33、itive cash in 180 daysa) 40.13, b) -0.449, c) F0=S0e0.04*7/12 or S0=O.9769F。so that delta with respect to the in dex level is =0.439, d) 39.81, e) America n opti on price becomes 40.13+36.704-39.81=37.02. Delta becomes -0.449 +0.444-0.442=-0.447a) One year rate is 5.827%, two-year rate is 6.313%, b)

34、 -$91,239, c) $8,504 3.50. This is also the credit exposure.0.021Put is priced too high relative to call. Sell put and buy call. This works regardless of whether the assumpti ons un derly ing Black-Scholes hold a) 0.371, -0.0038, -1.85, b) Sell 0.371 of in dex for each option sold, c) Delta cha nges

35、 to 0.317 so 0.054 of in dex must be bought back$841,000 assu ming float ing is received.Assume that the excha nge rate is DM per $. p is the n 0.450 and the value of the opti on is about 80,000 DM. Yes the two optio ns are the same.0.036 a) no b)yes c)yes d)yesThis will lead to a smile where volatility in creases with strike price. this is the opposite of what is usually observed.a) Delta of long position in one option is 0.563. Ba nk should buy 281,500 shares b) Delta chan ges to 0.686. Bank should buy a further 61,500 shares. The bank has a n e