Quiz Instructions: Term Structure Models II and Introduction to Credit Derivatives

Questions 1 and 2should be answered by building and calibrating a 10-period Black-Derman-Toy (BDT) 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:

Period1 2 3 4 5 6 7 8 9 10

Spot Rate3.0% 3.1% 3.2% 3.3% 3.4% 3.5% 3.55% 3.6% 3.65% 3.7%

The market-price of a zero-coupon bond that matures in period66isZ_0^6 = 100/(1+.035)^6 = 81.35Z06=100/(1+.035)6=81.35assuming a face value of 100.

Link: https://docs.google.com/spreadsheets/d/1Y7UPjESIZWOfFan9NWjn7byyhG6ck09_QepnR9vi0tM/edit?usp=sharing

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Questions 3-5refer to the material on defaultable bonds and credit-default swaps (CDS).

Question 1: Assumeb=0.05b=0.05is a constant for alliiin the BDT model. Calibrate thea_iaiparameters so that the model term-structure matches the market term-structure. Be sure that the final error returned by Solver is at most10^{-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 timet=3t=3with an option strike of00. You may assume the underlying swap has a fixed rate of3.9\%3.9%and that if the option is exercised then cash-flows take place at timest=4, \ldots , 10t=4,...,10. (The cash-flow at timet=it=iis 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 ofb = 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 an = 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=0.9andq=1-q=1/2q=1q=1/2. Assume that the 1-step hazard rate in node(i,j)(i,j)is given byh_{ij} = a b^{j-\frac{i}{2}}hij=abj2iwherea = 0.01a=0.01andb = 1.01b=1.01. Compute the price of a zero-coupon bond with face valueF = 100F=100and recoveryR = 20\%R=20%.

Submission Guideline:Give your answer rounded to two decimal places. For example, if you compute the answer to be 73.2367, submit 73.24.

Question 4: The true price of 5 different defaultable coupon paying bonds with non-zero recovery are specified in worksheet{\tt Calibration}Calibrationin the workbook{\tt Assignment5\_cds.xlsx}.Assignment5_cds.xlsx. Link:https://docs.google.com/spreadsheets/d/1Y7UPjESIZWOfFan9NWjn7byyhG6ck09_QepnR9vi0tM/edit?usp=sharing

The interest rate isr = 5\%r=5%per annum. Calibrate the six month hazard rates{\tt A6}A6to{\tt A16}A16to by minimizing the{\tt Sum \,Error}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{\tt A6} \leq {\tt A7}, \ldots, {\tt A15} \leq {\tt A16}A6A7,...,A15A16. Report the hazard rate at time00as a percentage.

Submission Guideline:Give your answer inpercentrounded to two decimal places. For example, if you compute the answer to be 73.2367%, submit 73.24.

Question 5: Modify the data on the{\tt CDS \,pricing}CDSpricingworksheet in the workbook{\tt bonds\_and\_cds.xlsx}bonds_and_cds.xlsxLink: https://docs.google.com/spreadsheets/d/1Y7UPjESIZWOfFan9NWjn7byyhG6ck09_QepnR9vi0tM/edit?usp=sharing

It is required compute a par spread in basis points for a 5yr CDS with notional principalN =10N=10million assuming that the expected recovery rateR = 25\%R=25%, the 3-month hazard rate is a flat1\%1%, and the interest rate is5\%5%per annum.

Submission Guideline:Give your answer inbasis pointsrounded to two decimal places (1 bps = 0.01%). For example, if you compute the answer to be 73.2367 bps, submit 73.24.