Question 1

Quiz instructions

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.

1 point

Question 2

Quiz instructions

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.

Question 3

Quiz instructions

Construct an = 10

n=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=1.1

u=1.1,d=0.9

d=0.9andq=1-q=1/2

q=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 byh_{ij} = a b^{j-\frac{i}{2}}

hij

=abj2

i

wherea = 0.01

a=0.01andb = 1.01

b=1.01. Compute the price of a zero-coupon bond with face valueF = 100

F=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

Quiz instructions

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.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 time0

0as 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

Quiz instructions

Modify the data on the{\tt CDS \,pricing}

CDSpricingworksheet in the workbook{\tt bonds\_and\_cds.xlsx}

bonds_and_cds.xlsxto compute a par spread in basis points for a 5yr CDS with notional principalN =10

N=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.