Questions 1 and 2 should be answered by building and calibrating a 10-period Black-Derman-Toy model for the short-rate, ri,j. You may assume that the term-structure of interest rates observed in the market place is: Period 1 2 3 4 5 6 7 8 9 10 Spot Rate 3.0% 3.1% 3.2% 3.3% 3.4% 3.5% 3.55% 3.6% 3.65% 3.7% As in the video modules, these interest rates assume per-period compounding so that, for example, the market-price of a zero-coupon bond that matures in period 6 is Z06=100/(1+.035)6=81.35 assuming a face value of 100. Quiz instructions Question 1 Assume b=0.05 is a constant for all i in the BDT model as we assumed in the video lectures. Calibrate the ai parameters so that the model term-structure matches the market term-structure. Be sure that the final error returned by Solver is at most 108. (This can be achieved by rerunning Solver multiple times if necessary, starting each time with the solution from the previous call to Solver. Once your model has been calibrated, compute the price of a payer swaption with notional $1M that expires at time t=3 with an option strike of 0. You may assume the underlying swap has a fixed rate of 3.9% and that if the option is exercised then cash-flows take place at times t=4,,10. (The cash-flow at time t=i is based on the short-rate that prevailed in the previous period, i.e. the payments of the underlying swap are made in arrears.) Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be 10,456.67, submit 10457. Question 2 Repeat the previous question but now assume a value of b = 0.1b=0.1. Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be 10,456.67, submit 10457. Question 3 construct a n = 10n=10-period binomial model for the short-rate, r_{i,j}ri,j. The lattice parameters are: r_{0,0}= 5\%r0,0=5%, u=1.1u=1.1, d=0.9d=0.9 and q=1-q=1/2q=1q=1/2. This is the same lattice that you constructed in Assignment 5. Assume that the 1-step hazard rate in node (i,j)(i,j) is given by h_{ij} = a b^{j-\frac{i}{2}}hij=abj2i where a = 0.01a=0.01 and b = 1.01b=1.01. Compute the price of a zero-coupon bond with face value F = 100F=100 and recovery R = 20\%R=20%. Question 4 The true price of 5 different defaultable coupon paying bonds with non-zero recovery are specified in worksheet Calibration in the workbook Assignment5_cds.xlsx. The interest rate is r = 5\%r=5% per annum. Calibrate the six month hazard rates A6 to A16 to by minimizing the SumErrorensuring that the term structure of hazard rates are non-decreasing. You can model the non-decreasing hazard rates by adding constraints of the form A6A7,,A15A16. Report the hazard rate at time 00 as a percentage. Question 5 Modify the data on the CDSpricing worksheet in the workbook bonds_and_cds.xlsx to compute a par spread in basis points for a 5yr CDS with notional principal N =10N=10 million assuming that the expected recovery rate R = 25\%R=25%, the 3-month hazard rate is a flat 1\%1%, and the interest rate is 5\%5% per annum. Questions 1 and 2 should be answered by building and calibrating a 10-period Black-Derman-Toy model for the short-rate, r_{i,j}ri,j. You may assume that the term-structure of interest rates observed in the market place is:Period 1 2 3 4 5 6 7 8 9 10Spot Rate 3.0% 3.1% 3.2% 3.3% 3.4% 3.5% 3.55% 3.6% 3.65% 3.7%As in the video modules, these interest rates assume per-period compounding so that, for example, the market-price of a zero-coupon bond that matures in period 66 is Z_0^6 = 100/(1+.035)^6 = 81.35Z06=100/(1+.035)6=81.35 assuming a face value of 100.

Assume b=0.05b=0.05 is a constant for all ii in the BDT model as we assumed in the video lectures. Calibrate the a_iai parameters so that the model term-structure matches the market term-structure. Be sure that the final error returned by Solver is at most 10^{-8}108. (This can be achieved by rerunning Solver multiple times if necessary, starting each time with the solution from the previous call to Solver.

Once your model has been calibrated, compute the price of a payer swaption with notional $1M that expires at time t=3t=3 with an option strike of 00. You may assume the underlying swap has a fixed rate of 3.9\%3.9% and that if the option is exercised then cash-flows take place at times t=4, \ldots , 10t=4,,10. (The cash-flow at time t=it=i is based on the short-rate that prevailed in the previous period, i.e. the payments of the underlying swap are made in arrears.)

Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be 10,456.67, submit 10457.