6. (23 points) The prices of zero-coupon bonds (each with a face value of $100) are...

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6. (23 points) The prices of zero-coupon bonds (each with a face value of $100) are given in the table on the right. a. (4 points) Fit the simple Black-Derman-Toy model to the zero-coupon bond prices. The length of a time period in the model shall be A = 0.25 years, and the volatility of the logarithm of the short rate is o = 10%. What are the 00, 0₁, and 0₂ values? Your answer should Maturity (years) Price (dollars) 98.43 96.81 95.16 93.51 0.25 0.50 0.75 1.00 consist of the three values, and a short (max. four sentences) description of how you obtained them. b. (4 points) Based on the model which you fitted in question a., calculate the price of a cap with maturity at t = 0.5 (i.e., the last cash flow of which is at t = 0.5), which has a face value of $100, a strike rate of 5.5% (with quarterly compounding), and which has quarterly cash flow payments. c. (5 points) You observe that the price of a cap, which has the same features as the cap in question b. except that this new cap has a strike rate of 6.5% (with quarterly compounding), is $0.1048. In this question, you are asked to fit the simple Black- Derman-Toy model so that it prices the zero-coupon bonds and also this new cap correctly. That is: find the 00, 0₁, 02, and o values so that the model-implied prices of the zero-coupon bonds and this new cap will all be equal to their observed prices. (Note: your values will not be the same as in question a. Also note: if you want to minimize several squared differences at the same time, you can do that by minimizing the sum of these squared differences.) Your answer should consist of the 00, 0₁, 02, and o values, and a short (max. four sentences) description of how you obtained them. d. (10 points) In this question, you are asked to build the recombinging full Black- Derman-Toy model. The zero-coupon bond prices are the same as in the above table. The price of a cap with maturity at t = 0.5, which has a face value of $100, a strike rate of 6.5% (with quarterly compounding), and which has quarterly cash flow payments, is $0.15. The price of the same cap but with maturity at t = 0.75 is $0.3. Furthermore, the price of the same cap but with maturity at t = 1 is $0.5. Find the values of 01, 02, 03, Z1,0, Z2,0, and Z3,0 such that the model prices correctly the four zero-coupon bonds and the three caps. (Note: Zi,o is the logarithm of the short rate ri,o.) Your answer should consist of the values 01, 02, 03, Z1,0, Z2,0, and Z3,0, and a short description (max. six sentences) of how you obtained them. 6. (23 points) The prices of zero-coupon bonds (each with a face value of $100) are given in the table on the right. a. (4 points) Fit the simple Black-Derman-Toy model to the zero-coupon bond prices. The length of a time period in the model shall be A = 0.25 years, and the volatility of the logarithm of the short rate is o = 10%. What are the 00, 0₁, and 0₂ values? Your answer should Maturity (years) Price (dollars) 98.43 96.81 95.16 93.51 0.25 0.50 0.75 1.00 consist of the three values, and a short (max. four sentences) description of how you obtained them. b. (4 points) Based on the model which you fitted in question a., calculate the price of a cap with maturity at t = 0.5 (i.e., the last cash flow of which is at t = 0.5), which has a face value of $100, a strike rate of 5.5% (with quarterly compounding), and which has quarterly cash flow payments. c. (5 points) You observe that the price of a cap, which has the same features as the cap in question b. except that this new cap has a strike rate of 6.5% (with quarterly compounding), is $0.1048. In this question, you are asked to fit the simple Black- Derman-Toy model so that it prices the zero-coupon bonds and also this new cap correctly. That is: find the 00, 0₁, 02, and o values so that the model-implied prices of the zero-coupon bonds and this new cap will all be equal to their observed prices. (Note: your values will not be the same as in question a. Also note: if you want to minimize several squared differences at the same time, you can do that by minimizing the sum of these squared differences.) Your answer should consist of the 00, 0₁, 02, and o values, and a short (max. four sentences) description of how you obtained them. d. (10 points) In this question, you are asked to build the recombinging full Black- Derman-Toy model. The zero-coupon bond prices are the same as in the above table. The price of a cap with maturity at t = 0.5, which has a face value of $100, a strike rate of 6.5% (with quarterly compounding), and which has quarterly cash flow payments, is $0.15. The price of the same cap but with maturity at t = 0.75 is $0.3. Furthermore, the price of the same cap but with maturity at t = 1 is $0.5. Find the values of 01, 02, 03, Z1,0, Z2,0, and Z3,0 such that the model prices correctly the four zero-coupon bonds and the three caps. (Note: Zi,o is the logarithm of the short rate ri,o.) Your answer should consist of the values 01, 02, 03, Z1,0, Z2,0, and Z3,0, and a short description (max. six sentences) of how you obtained them.

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Here are the answers a 00 00248 01 00049 02 00001 I fitted the simple BlackDermanToy m... View the full answer