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 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%

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 period6

6isZ_0^6 = 100/(1+.035)^6 = 81.35

Z0

6

=100/(1+.035)6

=81.35assuming a face value of 100.

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

Question1: Assumeb=0.05

b=0.05is a constant for alli

iin the BDT model as we assumed in the video lectures. Calibrate thea_i

ai

parameters 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=3

t=3with an option strike of0

0. 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 , 10

t=4,...,10. (The cash-flow at timet=i

t=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 = 0.1

b=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.