期权期货与其他衍生产品第九版课后习题与答案Chapter.docx

1、期权期货与其他衍生产品第九版课后习题与答案Chapter期权期货与其他衍生产品第九版课后习题与答案ChapterCHAPTER 29Interest Rate Derivatives: The Standard Market ModelsPractice QuestionsProblem 29.1.A company caps three-month LIBOR at 10% per annum. The principal amount is $20 million. On a reset date, three-month LIBOR is 12% per annum. What paym

2、ent would this lead to under the cap? When would the payment be made?An amount20000000002025100000$,?.?.=,would be paid out 3 months later.Problem 29.2.Explain why a swap option can be regarded as a type of bond option.A swap option (or swaption) is an option to enter into an interest rate swap at a

3、 certain time in the future with a certain fixed rate being used. An interest rate swap can be regarded as the exchange of a fixed-rate bond for a floating-rate bond. A swaption is therefore the option to exchange a fixed-rate bond for a floating-rate bond. The floating-rate bond will be worth its f

4、ace value at the beginning of the life of the swap. The swaption is therefore an option on a fixed-rate bond with the strike price equal to the face value of the bond.Problem 29.3.Use the Blacks model to value a one -year European put option on a 10-year bond. Assume that the current value of the bo

5、nd is $125, the strike price is $110, the one-year risk-free interest rate is 10% per annum, the bonds forward price volatility is 8% per annum, and the present value of the coupons to be paid during the life of the option is $10.In this case, 0110(12510)12709F e .?=-=., 110K =, 011(0)P T e -.?,=, 0

6、08B =., and 10T =. 2121ln(12709110)(0082)1845600800817656d d d ./+./=.=-.=. From equation (29.2) the value of the put option is011011110(17656)12709(18456)012e N e N -.?-.?-.-.-.=.or $0.12.Problem 29.4.Explain carefully how you would use (a) spot volatilities and (b) flat volatilities to value a fiv

7、e-year cap.When spot volatilities are used to value a cap, a different volatility is used to value eachcaplet. When flat volatilities are used, the same volatility is used to value each caplet within a given cap. Spot volatilities are a function of the maturity of the caplet. Flat volatilities are a

8、function of the maturity of the cap.Problem 29.5.Calculate the price of an option that caps the three-m onth rate, starting in 15 months time, at 13% (quoted with quarterly compounding) on a principal amount of $1,000. The forward interest rate for the period in question is 12% per annum (quoted wit

9、h quarterlycompounding), the 18-month risk-free interest rate (continuously compounded) is 11.5% per annum, and the volatility of the forward rate is 12% per annum.In this case 1000L =, 025k =., 012k F =., 013K R =., 0115r =., 012k =., 125k t =., 1(0)08416k P t +,=.250k L =212052*6637d d =-.=-.-.=-.

10、 The value of the option is25008416012(05295)013(06637)N N ?.?.-.-.-.059=. or $0.59.Problem 29.6.A bank uses Blacks model to price European bond options. Suppose that an implied price volatility for a 5-year option on a bond maturing in 10 years is used to price a 9-year option on the bond. Would yo

11、u expect the resultant price to be too high or too low? Explain.The implied volatility measures the standard deviation of the logarithm of the bond price at the maturity of the option divided by the square root of the time to maturity. In the case of a five year option on a ten year bond, the bond h

12、as five years left at option maturity. In the case of a nine year option on a ten year bond it has one year left. The standard deviation of a one year bond price observed in nine years can be normally be expected to be considerably less than that of a five year bond price observed in five years. (Se

13、e Figure 29.1.) We would therefore expect the price to be too high.Problem 29.7.Calculate the value of a four-year European call option on bond that will mature five years from today using Blacks model. The five -year cash bond price is $105, the cash price of a four-year bond with the same coupon i

14、s $102, the strike price is $100, the four-year risk-free interest rate is 10% per annum with continuous compounding, and the volatility for the bond price in four years is 2% per annum.The present value of the principal in the four year bond is 40110067032e -?.=. The present value of the coupons is

15、, therefore, 1026703234968-.=. This means that the forward price of the five-year bond is401(10534968)104475e ?.-.=. The parameters in Blacks model are therefore 104475B F =., 100K =, 01r =., 4T =,and 002B =.212111144010744d d d =.=-.=. The price of the European call is014104475(11144)100(10744)319e

16、 N N -.?.-.=.or $3.19.Problem 29.8.If the yield volatility for a five-year put option on a bond maturing in 10 years time isspecified as 22%, how should the option be valued? Assume that, based on todays interest rates the modified duration of the bond at the maturity of the option will be 4.2 years

17、 and the forward yield on the bond is 7%.The option should be valued using Blacks model in equation (29.2) with the bond price volatility being4200702200647.?.?.=. or 6.47%.Problem 29.9.What other instrument is the same as a five-year zero-cost collar where the strike price of the cap equals the str

18、ike price of the floor? What does the common strike price equal?A 5-year zero-cost collar where the strike price of the cap equals the strike price of the floor is the same as an interest rate swap agreement to receive floating and pay a fixed rate equal to the strike price. The common strike price

19、is the swap rate. Note that the swap is actually a forward swap that excludes the first exchange. (See Business Snapshot 29.1)Problem 29.10.Derive a put call parity relationship for European bond options.There are two way of expressing the put call parity relationship for bond options. The first is

20、in terms of bond prices:0RT c I Ke p B -+=+where c is the price of a European call option, p is the price of the corresponding European put option, I is the present value of the bond coupon payments during the life of the option, K is the strike price, T is the time to maturity, 0B is the bond price

21、, and Ris the risk-free interest rate for a maturity equal to the life of the options. To prove this we can consider two portfolios. The first consists of a European put option plus the bond; the second consists of the European call option, and an amount of cash equal to the present value of the cou

22、pons plus the present value of the strike price. Both can be seen to be worth the same at the maturity of the options.The second way of expressing the put call parity relationship isRT RT B c Ke p F e -+=+where B F is the forward bond price. This can also be proved by considering two portfolios. The

23、 first consists of a European put option plus a forward contract on the bond plus the present value of the forward price; the second consists of a European call option plus thepresent value of the strike price. Both can be seen to be worth the same at the maturity of the options.Problem 29.11.Derive

24、 a putcall parity relationship for European swap options.The putcall parity relationship for European swap options is+=c V pwhere c is the value of a call option to pay a fixed rate ofs and receive floating, p isKthe value of a put option to receive a fixed rate ofs and pay floating, and V is the va

25、lueKof the forward swap underlying the swap option wheres is received and floating is paid.KThis can be proved by considering two portfolios. The first consists of the put option; the second consists of the call option and the swap. Suppose that the actual swap rate at thes. The call will be exercis

26、ed and the put will not be maturity of the options is greater thanKexercised. Both portfolios are then worth zero. Suppose next that the actual swap rate at thes. The put option is exercised and the call option is not maturity of the options is less thanKs is received and floating is paid. exercised

27、. Both portfolios are equivalent to a swap whereKIn all states of the world the two portfolios are worth the same at time T. They must therefore be worth the same today. This proves the result.Problem 29.12.Explain why there is an arbitrage opportunity if the implied Black (flat) volatility of a cap

28、 is different from that of a floor. Do the broker quotes in Table 29.1 present an arbitrage opportunity?Suppose that the cap and floor have the same strike price and the same time to maturity. The following putcall parity relationship must hold:+=cap swap floorwhere the swap is an agreement to recei

29、ve the cap rate and pay floating over the whole life of the cap/floor. If the implied Black volatilities for the cap equal those for the floor, the Black formulas show that this relationship holds. In other circumstances it does not hold and there is an arbitrage opportunity. The broker quotes in Ta

30、ble 29.1 do not present an arbitrage opportunity because the cap offer is always higher than the floor bid and the floor offer is always higher than the cap bid.Problem 29.13.When a bonds price is lognormal can the bonds yield be negative? Explain your answer.Yes. If a zero-coupon bond price at some

31、 future time is lognormal, there is some chance that the price will be above par. This in turn implies that the yield to maturity on the bond is negative.Problem 29.14.What is the value of a European swap option that gives the holder the right to enter into a3-year annual-pay swap in four years wher

32、e a fixed rate of 5% is paid and LIBOR is received? The swap principal is $10 million. Assume that the LIBOR/swap yield curve is used for discounting and is flat at 5% per annum with annual compounding and the volatility of the swap rate is 20%. Compare your answer to that given by DerivaGem.Now suppose that allswap rates are 5% and all OIS rates are 4.7%. Use DerivaGem to calculate the LIBOR zero curve and the swap option value?