1. Assume b=0.05 is a constant for all ii 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...

1. Assume b=0.05 is a constant for all ii in the BDT model as we assumed in the video lectures. Cali
ate the a_iai ​parameters so that the model term-structure matches the market term-structure. Be sure that the final e
or returned by Solver is at most 10-8. (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 cali
ated, compute the price of a payer swaption with notional $1M that expires at time t=3 with an option strike of 0. 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 a
ears.)
Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be 10,456.67, submit 10457.
    Value of
    0.05
     
    Nominal value
    1000000
    
    Expiry duration (in years)
    10
    
    Fixed rate of interest
    0.039
    
     
     
    
    Duration of expiration
    Rate of interest
    Valuation of bonds at different time intervals
    Market value of bond (face value 100)
    1
    0.03
    $48,543.69
    $485.44
    2
    0.031
    $47,038.41
    $470.38
    3
    0.032
    $45,491.57
    $454.92
    4
    0.033
    $43,910.53
    $439.11
    5
    0.034
    $42,302.62
    $423.03
    6
    0.035
    $40,675.03
    $406.75
    7
    0.0355
    $39,166.91
    $391.67
    8
    0.036
    $37,678.36
    $376.78
    9
    0.0365
    $36,211.48
    $362.11
    10
    0.037
    $34,768.22
    $347.68
     
     
    $4,15,786.82
     
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.
    Value of
    0.1
     
    Nominal value
    1000000
    
    Expiry duration (in years)
    10
    
    Fixed rate of interest
    0.039
    
     
     
    
    Duration of expiration
    Rate of interest
    Valuation of bonds at different time intervals
    Market value of bond (face value 100)
    1
    0.03
    $97,087.38
    $970.87
    2
    0.031
    $94,076.83
    $940.77
    3
    0.032
    $90,983.14
    $909.83
    4
    0.033
    $87,821.07
    $878.21
    5
    0.034
    $84,605.25
    $846.05
    6
    0.035
    $81,350.06
    $813.50
    7
    0.0355
    $78,333.81
    $783.34
    8
    0.036
    $75,356.71
    $753.57
    9
    0.0365
    $72,422.96
    $724.23
    10
    0.037
    $69,536.44
    $695.36
     
     
    $8,31,573.65
     
3. Construct a n =10-period binomial model for the short-rate, ri,j. The lattice parameters are: r0,0=5%, u=1.1, d=0.9 and q=1-q=1/2. This is the same lattice that you constructed in Assignment 5. Assume that the 1-step hazard rate in node (i,j) is given by hij=abj−1/2 where a =0.01 and b =1.01. Compute the price of a zero-coupon bond with face value F =100 and recovery 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.
Solution: The price of a zero coupon bond is calculated as:
Price = M/ (1 + r) n
Where M = Maturity value or face value of the bond
R = required rate of interest
N = number of years until...