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 10

Spot Rate 3.0% 3.1% 3.2% 3.3% 3.4% 3.5% 3.55% 3.6% 3.65% 3.7%

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 Z_0^6 =100/ (1+.035)^6= 81.35 assuming a face value of 100.

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 10-5.(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 O. 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.) 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 10-5.(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 O. 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.)