Question 1

Assumeb=0.05 is a constant for alli in the BDT model as we assumed in the video lectures. Calibrate thea_i 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. (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 with an option strike of0. You may assume the underlying swap has a fixed rate of3.9%and that if the option is exercised then cash-flows take place at timest=4,...,10. (The cash-flow at timet=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 of 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

Construct a n=10-period binomial model for the short-rate,r_i,j. The lattice parameters are:r_0,0=5%,u=1.1,d=0.9andq=1q=1/2.

Assume that the 1-step hazard rate in node(i,j) is given byh_ij=ab^j-1/2 where a=0.01 and b=1.01. Compute the price of a zero-coupon bond with face value F=100 and recoveryR=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.