Homework 2 by kramkov

INTERPOLATION OF DATA CURVES

FORWARD EXCHANGE CURVE BY LINEAR INTERPOLATION OF COST-OF-CARRY RATES

spot FX rate = 100
initial time = 1

Input discount factors:

      time      domestic DF      foreign DF      
       1.5        0.978976        0.952052      
         2        0.959773        0.907937      
       2.5        0.942207         0.86728      
         3        0.926118         0.82975      
       3.5        0.911365        0.795053      
         4         0.89782        0.762926      
       4.5        0.885373        0.733137      
         5        0.873923        0.705478      
       5.5        0.863381         0.67976      
         6        0.853665        0.655817      
       6.5        0.844705        0.633498      
         7        0.836436        0.612667      

VALUES VERSUS TIME:

    time           value
       1             100
    1.05         99.7215
     1.1         99.4438
    1.15         99.1669
     1.2         98.8907
    1.25         98.6153
     1.3         98.3407
    1.35         98.0668
     1.4         97.7937
    1.45         97.5214

LOG INTERPOLATION OF DISCOUNT CURVE

initial time = 1

Input discount factors:

      time           value      
       1.5          0.9674      
         2        0.939098      
       2.5        0.914449      
         3        0.892917      
       3.5        0.874059      
         4        0.857504      
       4.5         0.84294      
         5        0.830103      
       5.5        0.818769      
         6        0.808748      
       6.5        0.799874      
         7        0.792008      

log interpolation with Steffen method:

VALUES VERSUS TIME:

    time           value
       1               1
     1.6        0.961412
     2.2         0.92883
     2.8        0.901188
     3.4        0.877639
       4        0.857504
     4.6        0.840239
     5.2        0.825399
     5.8        0.812611
     6.4        0.801566

LEAST SQUARE FITTING OF DATA CURVES

LEAST-SQUARES FIT OF VOLATILITY CURVE FOR BLACK MODEL

lambda = 0.05
initial time = 1

Input volatilities:

      time      volatility      
       1.5       0.0373058      
         2       0.0398669      
       2.5       0.0427134      
         3       0.0458791      
       3.5       0.0494018      
         4       0.0533233      
       4.5       0.0576907      
         5       0.0625563      
       5.5       0.0679787      
         6       0.0740231      

Fitted coefficients and their covariance matrix:

       value      covariance matrix
   0.0562152       2.30437e-05

chi^2 error = 0.00181903

Fitted volatilities and their errors:

    time         value           err      
       1      0.0562152      0.00480039      
 1.55556      0.0554434      0.00473448      
 2.11111      0.0546892      0.00467008      
 2.66667      0.0539523      0.00460715      
 3.22222      0.053232      0.00454564      
 3.77778      0.0525282      0.00448554      
 4.33333      0.0518403      0.0044268      
 4.88889      0.0511679      0.00436938      
 5.44444      0.0505107      0.00431326      
       6      0.0498683      0.0042584      

LEAST-SQUARES FIT OF DISCOUNT CURVE IN SVENSSON MODEL

lambda1 = 0.02
lambda2 = 0.1
initial time = 1

Input discount factors:

      time           value      
       1.5          0.9674      
         2        0.939098      
       2.5        0.914449      
         3        0.892917      
       3.5        0.874059      
         4        0.857504      
       4.5         0.84294      
         5        0.830103      
       5.5        0.818769      
         6        0.808748      
       6.5        0.799874      
         7        0.792008      

Fitted coefficients and their covariance matrix:

       value                        covariance matrix
    -2.10627        0.00295885       -0.00295897       -0.00345526       0.000103238
     2.17626       -0.00295897        0.00295909        0.00345541      -0.000103242
     2.51282       -0.00345526        0.00345541        0.00403509      -0.000120593
   -0.220775       0.000103238      -0.000103242      -0.000120593        3.6119e-06

chi^2 error = 2.32637e-11

Fitted discount factors and their errors:

    time         value           err      
       1             1             0      
 1.66667      0.957523      7.72067e-07      
 2.33333      0.922292      1.06039e-06      
       3      0.892917      1.60243e-06      
 3.66667      0.868304      1.84503e-06      
 4.33333      0.847595      2.1108e-06      
       5      0.830104      2.78696e-06      
 5.66667      0.815287      3.3322e-06      
 6.33333      0.802706      3.61126e-06      
       7      0.792016      6.97974e-06      

