1、 CHAPTER 29 Interest Rate Derivatives: The Standard Market Models Practice Questions Problem 29.1. A company caps three-month LIBOR at 10% per annum. The principal amount is $20 million. On a reset date, three-month LIBOR is 12% per annum. What payment would this lead to under the cap? When would th
2、e payment be made? An amount 2 0 0 0 0 0 0 0 0 0 2 0 2 5 1 0 0 0 0 0$ 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 certain time in the future with
3、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 face value at the beginning of the
4、 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 bond is $125, the strike price is
5、$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, 0 1 10 (1 2 5 1 0 ) 1 2 7 0 9Fe , 110K , 0 1 1(0 )P T e , 008B , and 10T . 2121l n ( 1
6、2 7 0 9 1 1 0 ) ( 0 0 8 2 ) 1 8 4 5 60 0 80 0 8 1 7 6 5 6ddd From equation (29.2) the value of the put option is 0 1 1 0 1 11 1 0 ( 1 7 6 5 6 ) 1 2 7 0 9 ( 1 8 4 5 6 ) 0 1 2e 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
7、five-year cap. When spot volatilities are used to value a cap, a different volatility is used to value each caplet. 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
8、are a function of the maturity of the cap. Problem 29.5. Calculate the price of an option that caps the three-month 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 (quo
9、ted with quarterly compounding), 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 , 012kF , 013KR , 0115r , 012k , 125kt , 1(0 ) 0 8416kPt . 250kL 212l n ( 0 1 2 0 1 3 ) 0 1 2 1 2 5 2
10、 0 5 2 9 50 1 2 1 2 50 5 2 9 5 0 1 2 1 2 5 0 6 6 3 7dd The value of the option is 2 5 0 0 8 4 1 6 0 1 2 ( 0 5 2 9 5 ) 0 1 3 ( 0 6 6 3 7 ) NN 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 mat
11、uring in 10 years is used to price a 9-year option on the bond. Would you 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 m
12、aturity. In the case of a five year option on a ten year bond, the bond has 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 consider
13、ably less than that of a five year bond price observed in five years. (See 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 bon
14、d price is $105, the cash price of a four-year bond with the same coupon is $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 t
15、he four year bond is 4 0 1100 67 032e . The present value of the coupons is, therefore, 1 0 2 6 7 0 3 2 3 4 9 6 8 . This means that the forward price of the five-year bond is 4 0 1(1 0 5 3 4 9 6 8 ) 1 0 4 4 7 5e The parameters in Blacks model are therefore 104 475BF , 100K , 01r , 4T , and 002B . 21
16、21l n 1 0 4 4 7 5 0 5 0 0 2 4 1 1 1 4 40 0 2 40 0 2 4 1 0 7 4 4ddd The price of the European call is 0 1 4 1 0 4 4 7 5 ( 1 1 1 4 4 ) 1 0 0 ( 1 0 7 4 4 ) 3 1 9e 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 is specified as 22%, how
17、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 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 being 4
18、 2 0 0 7 0 2 2 0 0 6 4 7 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 strike 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 s
19、trike 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 is the swap rate. Note that the swap is actually a forward swap that excludes the first exchange. (See Business Snapshot 29.1) Problem 2
20、9.10. Derive a putcall parity relationship for European bond options. There are two way of expressing the putcall parity relationship for bond options. The first is in terms of bond prices: 0RTc I K e p B where c is the price of a European call option, p is the price of the corresponding European pu
21、t 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, and R is the risk-free interest rate for a maturity equal to the life of the options. To prove this we can consider two portfolios. The
22、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 coupons 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
23、expressing the putcall parity relationship is R T R TBc K e p F e where BF is the forward bond price. This can also be proved by considering two portfolios. The first consists of a European put option plus a forward contract on the bond plus the present value of the forward price; the second consist
24、s of a European call option plus the present value of the strike price. Both can be seen to be worth the same at the maturity of the options. Problem 29.11. Derive a putcall parity relationship for European swap options. The putcall parity relationship for European swap options is c V p where c is t
25、he value of a call option to pay a fixed rate of Ks and receive floating, p is the value of a put option to receive a fixed rate of Ks and pay floating, and V is the value of the forward swap underlying the swap option where Ks is received and floating is paid. This can be proved by considering two
26、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 the maturity of the options is greater than Ks . The call will be exercised and the put will not be exercised. Both portfolios are then worth zero. Suppose next
27、 that the actual swap rate at the maturity of the options is less than Ks . The put option is exercised and the call option is not exercised. Both portfolios are equivalent to a swap where Ks is received and floating is paid. In all states of the world the two portfolios are worth the same at time T
28、 . 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 is different from that of a floor. Do the broker quotes in Table 29.1 present an arbitrage opportunity? Suppose that th
29、e cap and floor have the same strike price and the same time to maturity. The following putcall parity relationship must hold: cap swap floor where the swap is an agreement to receive the cap rate and pay floating over the whole life of the cap/floor. If the implied Black volatilities for the cap eq
30、ual 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 Table 29.1 do not present an arbitrage opportunity because the cap offer is always higher than the floor bid and the floo
31、r 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 future time is lognormal, there is some chance that the price will be above par. This in turn implies that the yiel
32、d 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 a 3-year annual-pay swap in four years where a fixed rate of 5% is paid and LIBOR is received? The swap principal is $10 million. Assume that the LIBOR/swap
33、 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 all swap rates are 5% and all OIS rates are 4.7%. Use DerivaGem to calculate the LIBOR zero curve and
34、the swap option value? In equation (29.10), 10 000 000L , 005Ks , 0 005s , 1 0 2 4 2 0 2d , 2.02 d , and 5 6 71 1 1 2 2 4 0 41 0 5 1 0 5 1 0 5A The value of the swap option (in millions of dollars) is 1 0 2 2 4 0 4 0 0 5 ( 0 2 ) 0 0 5 ( 0 2 ) 0 1 7 8NN This is the same as the answer given by DerivaG
35、em. (For the purposes of using the DerivaGem software, note that the interest rate is 4.879% with continuous compounding for all maturities.) When OIS discounting is used the LIBOR zero curve is unaffected because LIBOR swap rates are the same for all maturities. (This can be verified with the Zero
36、Curve worksheet in DerivaGem). The only difference is that 2790.2047.1 1047.1 1047.1 1 765 A so that the value is changed to 0.181. This is also the value given by DerivaGem. (Note that the OIS rate is 4.593% with annual compounding.) Problem 29.15. Suppose that the yield, R , on a zero-coupon bond
37、follows the process dR dt dz where and are functions of R and t , and dz is a Wiener process. Use Itos lemma to show that the volatility of the zero-coupon bond price declines to zero as it approaches maturity. The price of the bond at time t is ()RT te where T is the time when the bond matures. Usi
38、ng Its lemma the volatility of the bond price is ( ) ( )()R T t R T te T t eR This tends to zero as t approaches T . Problem 29.16. Carry out a manual calculation to verify the option prices in Example 29.2. The cash price of the bond is 0 0 5 0 5 0 0 0 5 1 0 0 0 0 5 1 0 0 0 5 1 04 4 4 1 0 0 1 2 2 8
39、 2ee 卐 e As there is no accrued interest this is also the quoted price of the bond. The interest paid during the life of the option has a present value of 0 0 5 0 5 0 0 5 1 0 0 5 1 5 0 0 5 24 4 4 4 1 5 0 4e e e e The forward price of the bond is therefore 0 0 5 2 2 5(1 2 2 8 2 1 5 0 4 ) 1 2 0 6 1e T
40、he yield with semiannual compounding is 5.0630%. The duration of the bond at option maturity is 0 0 5 0 2 5 0 0 5 7 7 5 0 0 5 7 7 50 0 5 0 2 5 0 0 5 0 7 5 0 0 5 7 7 5 0 0 5 7 7 50 2 5 4 7 7 5 4 7 7 5 1 0 04 4 4 1 0 0e 卐 eee 卐 e or 5.994. The modified duration is 5.994/1.025315=5.846. The bond price
41、volatility is therefore 5 8 4 6 0 0 5 0 6 3 0 0 2 0 0 5 9 2 . We can therefore value the bond option using Blacks model with 12061BF , 0 0 5 2 2 5( 0 2 2 5 ) 0 8 9 3 6Pe , 592B % , and 225T . When the strike price is the cash price 115K and the value of the option is 1.74. When the strike price is t
42、he quoted price 117K and the value of the option is 2.36. This is in agreement with DerivaGem. Problem 29.17. Suppose that the 1-year, 2-year, 3-year, 4-year and 5-year LIBOR-for-fixed swap rates for swaps with semiannual payments are 6%, 6.4%, 6.7%, 6.9%, and 7%. The price of a 5-year semiannual ca
43、p with a principal of $100 at a cap rate of 8% is $3. Use DerivaGem (the zero rate and Cap_and_swap_opt worksheets) to determine (a) The 5-year flat volatility for caps and floors with LIBOR discounting (b) The floor rate in a zero-cost 5-year collar when the cap rate is 8% and LIBOR discounting is
44、used (c) Answer (a) and (b) if OIS discounting is used and OIS swap rates are 100 basis points below LIBOR swap rates. (a) First we calculate the LIBOR zero curve using the zero curve worksheet of DerivaGem. The 1-, 2-, 3-, 4-, and 5_year zero rates with continuous compounding are 5.9118%, 6.3140%,
45、6.6213%, 6.8297%, and 6.9328%, respectively. We then transfer these to the choose the Caps and Swap Options worksheet and choose Cap/Floor as the Underlying Type. We enter Semiannual for the Settlement Frequency, 100 for the Principal, 0 for the Start (Years), 5 for the End (Years), 8% for the Cap/F
46、loor Rate, and $3 for the Price. We select Black-European as the Pricing Model and choose the Cap button. We check the Imply Volatility box and Calculate. The implied volatility is 25.4%. (b) We then uncheck Implied Volatility, select Floor, check Imply Breakeven Rate. The floor rate that is calcula
47、ted is 6.71%. This is the floor rate for which the floor is worth $3.A collar when the floor rate is 6.61% and the cap rate is 8% has zero cost. (c) The zero curve worksheet now shows that LIBOR zero rates for 1-, 2-, 3-, 4-, 5-year maturities are 5.9118%, 6.3117%, 6.6166%, 6.8227%, and 6.9249%. The OIS zero rates are 4.9385%, 5.3404%, 5.6468%, 5.8539%, and 5.9566%, respectively. When these are transferred to the cap and swaption worksheet and the Use OIS Discounting box is checked, the answer to a) becomes 24.81% and the answer to b) becomes 6.60%. Problem 29.18. Show that 12V f V
