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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 $20 million. On a reset date, three-month LIBOR is 12% per annum. What payment would this lead to under the cap? When would th
2、e payment be made? An amount 20 000 000 0 02 0 25100 000$ 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 a certain fi
3、xed 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 life of the
4、 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 $110, the on
5、e-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 1 0 (125 10)127 09Fe , 110K , 0 1 1 (0)PTe , 0 08 B , and 1 0T . 2 1 21 ln(127 09 110)(0 082) 1
6、 8456 0 08 0 081 7656 d dd From equation (29.2) the value of the put option is 0 1 10 1 1 110( 1 7656) 127 09( 1 8456)0 12eNeN or $0.12. Problem 29.4. Explain carefully how you would use (a) spot volatilities and (b) flat volatilities to value a five-year cap. When spot volatilities are used to valu
7、e 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 are a function of the maturity of the cap. Problem 29.
8、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 (quoted with quarterly compounding), the 18-month risk-fre
9、e interest rate (continuously compounded) is 11.5% per annum, and the volatility of the forward rate is 12% per annum. In this case 1000L , 0 25 k , 0 12 k F , 0 13 K R , 0 115r , 0 12 k , 1 25 k t , 1 (0)0 8416 k Pt . 250 k L 2 1 2 ln(0 12 0 13)0 121 25 2 0 5295 0 12 1 25 0 52950 12 1 250 6637 d d
10、The value of the option is 250 0 8416 0 12 ( 0 5295)0 13 ( 0 6637)NN 0 59 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 y
11、ou 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
12、 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 considerably less than that of a five year bond price observed in five years. (
13、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 bond price is $105, the cash price of a four-year bond with the same coupo
14、n 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 the four year bond is 4 0 1 10067 032e . The present value of the coupon
15、s is, therefore, 102 67 03234 968. This means that the forward price of the five-year bond is 4 0 1 (105 34 968)104 475e The parameters in Blacks model are therefore 104 475 B F , 100K , 0 1r , 4T , and 0 02 B . 2 1 21 ln1 044750 5 0 024 1 1144 0 02 4 0 02 41 0744 d dd The price of the European call is 0 1 4104 475 (1 1144) 100 (1 0744) 3 19eNN 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 should the option be valued? Assume that, based on today
