金融衍生工具測(cè)試題

1、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) What would be a fair price

2、for the box spread today? Define variables as n 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 arbitrageur? Expla in your an

3、swer.2. 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 16.0600DELTA0.78600.6151 '0.4365GAMMA0.01380.0181：0.0187THETA-11.2054-12.2607 I-11.4208VEGA20.461926.8416：27.6602RHO30.7085

4、25.2515：18.5394a) What does it cost to establish the butterfly spread?b) Calculate each of the Greek measures for this butterfly spread position and explain how each can be in terpreted.c) How would you make this opti on portfolio delta n eutral? What would be achieved by doing so?d) Suppose that to

5、morrow the price of C1 falls to $12.18 while the prices of C2 and C3 rema in the same. Does this create an arbitrage opport un ity? Expla in.3. 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 rate of inte

6、rest 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. Using a three-step bino mial tree, calculatea) the price of the American pu

7、t option now,b) the delta of the option with respect to the futures price,c) the delta of the option with respect to the index level, andd) the price of the corresponding European put option on index futures.e) Apply the control variate technique to improve your estimate of the American option price

8、 and of the delta of the option 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 is0.442.4. A financial institution trades swaps where 12 month LIBOR is exchanged for a fi

9、xed rate of in terest. Payme nts are made once a year. The on e-year s (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 s is 6.5 perce nt.a) Use this s to calculate the one and two year LIBOR zero rates, expressing the rates w

10、ith continuous compounding.b) What is the value of an existing s 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?Payments are made once a year.c) What is the value of a forward rate agreement where a rate

11、of 8 percent will be received on a principal of $1 million for the period between one year and two years?Note: All rates given in this question are expressed with annual compounding.5. The term structure is flat at 5% per annum with continuous compounding. Sometime ago a financial institution entere

12、d into a 5-year s a principal of $100 million in which every year it pays 12-month LIBOR and receives 6%. The s has two years eight months to run. Four months ago 12-month LIBOR was 4% (with annual compounding). What is the value of the s? What is the financial institution ' s credit exposure on

13、 the swap?6. An American put option to 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 interest rate is 6%, and the Swiss franc interest rate is 3% (both interest rates continuously compounded). The current exch

14、ange rate is 0.81. Use a three time step tree to value the option.7. A European call option on a certain stock has a strike price of $30, a time tomaturity of one year and an implied volatility of 30%. A put option on the same stock has a strike price of $30, a time to maturity of one year and an im

15、plied volatility of 33%. What is the arbitrage opportunity open to a trader. Does the opportunity work only when the lognormal assumption underlying Black-Scholes holds. Explain the reasons for your answer carefully.8. A put option on the S&P 500 has an exercise price of 500 and a time to maturi

16、tyof one year. The risk free rate is 7% and the dividend 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 has a short position in the option.a) Calculate the delta, gamma, and vega of the position. Explain how t

17、hey can be interpreted.b) How can the position be made delta neutral?c) Suppose that one week later the index has increased to 515. How can delta neutrality be preserved?9. An interest rate s a principal of $100 million involves the exchange of 5% perannum (semiannually compounded) for 6-month LIBOR

18、. The remaining life is 14 months. Interest is exchanged every six months. The 2 month, 8 month and 14 month rates are 4.5%, 5%, and 5.4% with continuous compounding. Six-month LIBOR was 5.5% four months ago. What is the value of the swap?10. The Deutschemark-Canadian dollar exchange rate is current

19、ly 1.0000. At the end of 6 months it will be either 1.1000 or 0.9000. What is the value of a 6 month option to sell one million Canadian dollars for 1.05 million deutschemarks. Verify that the answer given by risk neutral valuation is the same as that given by noarbitrage arguments. Is the option th

20、e same as one to buy 1.05 million deutschemarks for 1 million Canadian dollars? Assume that risk-free interest rates in Canada and Germany are 8% and 6% per annum respectively.11. An American put futures option has a strike price of 0.55 and a time to maturity of 1 year. The current futures price is

21、 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 tree to value the option.12. 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 p

22、rice of gold, and d) the average price of gold measured between time zero and the current time. Explain your answers.13. The future probability distribution of a stock price has a fatter right tail and thinner left tail than the lognormal distribution. Describe the effect of this on the prices of in

23、-the-money and out-of-the-money calls and puts. What is the volatility smile that would be observed?14. A bank has just sold a call option on 500,000 shares of a stock. The strike price is 40; the stock price is 40; the risk-free rate is 5%; the volatility is 30%; and the time to maturity is 3 month

24、s.a) What position should the company take in the stock for delta neutrality?b) Suppose that the bank does set up a delta neutral position as soon as the option has been sold and the stock price jumps to 42 within the first hour of trading.What trade is necessary to maintain delta neutrality? Explai

25、n whether the bank has gained or lost money in this situation. (You do not need to calculate the exact amount gained or lost.)c) Repeat part b) on the assumption that the stock jumps to 38 instead of 4215. A bank has sold a product that offers investors the total return (excluding dividends) on the

26、Toronto 300 index over a one year period. The return is capped at 20%. If the index goes down the original investment of the investor is returned.a) What option position is equivalent to the productb) Write down the formulas you would use to value the product and explain in detail how you would deci

27、de whether it is a good deal to the investor16. Use a three step tree to value a three month American put option on wheat futures. The current futures price is is 380 cents, the strike price is 370 cents, the risk-free rate is 5% per annum, and the volatility is 25% per annum. Explain carefully what

28、 happens if the investor exercises the option after two months. Suppose that the futures price at the time of exercise is 362 and the most recent settlement price is 360.17. a) A bank ' s assets and liabilities both have a duration of 5 years. Is the bank hedged against interest rate movements?

29、Explain carefully any limitations of the hedging scheme it has chosen.b) Explain what is meant by basis risk in the situation where a company knows it will be purchasing a certain asset in two months and uses a three-month futures contract to hedge its risk.18. a) Give an example of how a s be used

30、by a portfolio manager.b) Explain the nature of the credit risks to a financial institution in a sANSWERS1. (a) The box spread pays off X2-X1 in all circumstances (b) It should be worth the present value of X2-X1 today, c) and d) An arbitrageur might invest in a box spread if it is mispriced in the

31、market today.2. (a) 1.79 b) Greek letters are -0.0077, -0.0037, etc c) For delta neutrality we buy 0.0077 of the underlying asset. Small changes 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 t

32、he butterfly spread today and a zero or positive cash in 180 days3. a) 40.13, b) -0.449, c) F0=S0e0.04*7/12 or S0=0.9769F0 so that delta with respect to the index level is =0.439, d) 39.81, e) American option price becomes 40.13+36.704-39.81=37.02. Delta becomes -0.449 +0.444-0.442=-0.4474. a) One y

33、ear rate is 5.827%, two-year rate is 6.313%, b) -$91,239, c) $8,5045. 3.50. This is also the credit exposure.6. 0.0217. Put is priced too high relative to call. Sell put and buy call. This works regardless of whether the assumptions underlying Black-Scholes hold8. a) 0.371, -0.0038, -1.85, b) Sell 0

34、.371 of index for each option sold, c) Delta changes to 0.317 so 0.054 of index must be bought back9. $841,000 assuming floating is received.10. Assume that the exchange rate is DM per $. p is then 0.450 and the value of the option is about 80,000 DM. Yes the two options are the same.11. 0.03612. a)

35、 no b)yes c)yes d)yes13. This will lead to a smile where volatility increases with strike price. this is the opposite of what is usually observed.14. a) Delta of long position in one option is 0.563. Bank should buy 281,500 shares b) Delta changes to 0.686. Bank should buy a further 61,500 shares. The bank has a