Session 2 by kramkov

INTERPOLATION OF DATA CURVES

LOG LINEAR INTERPOLATION OF FORWARD CURVE

spot = 100
initial time = 1

Input forward prices:

      time           value      
       1.5          103.37      
         2         106.485      
       2.5         109.355      
         3         111.992      
       3.5         114.409      
         4         116.617      
       4.5         118.632      
         5         120.467      
       5.5         122.135      
         6         123.648      

VALUES VERSUS TIME:

    time           value
       1             100
    1.05         100.332
     1.1         100.665
    1.15         100.999
     1.2         101.335
    1.25         101.671
     1.3         102.009
    1.35         102.347
     1.4         102.687
    1.45         103.028

DISCOUNT CURVE BY INTERPOLATION OF YIELDS

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      

initial short-term rate = 0.0673981

interpolation with Akima method:

VALUES VERSUS TIME:

    time           value
       1               1
    1.05        0.996636
     1.1        0.993283
    1.15        0.989945
     1.2        0.986625
    1.25        0.983327
     1.3        0.980057
    1.35        0.976821
     1.4        0.973626
    1.45        0.970483

LEAST-SQUARES FITTING OF DATA CURVES

FORWARD CURVE BY LEAST-SQUARES FIT OF COST-OF-CARRY RATES

spot = 100
initial time = 1

Input forward prices:

      time           value      
       1.5          103.37      
         2         106.485      
       2.5         109.355      
         3         111.992      
       3.5         114.409      
         4         116.617      
       4.5         118.632      
         5         120.467      
       5.5         122.135      
         6         123.648      

We fit with constant cost-of-carry rate.

Fitted coefficients and their covariance matrix:

       value      covariance matrix
   0.0532717       6.41176e-06

chi^2 error = 0.000577059

Fitted forward prices and their errors:

    time         value           err      
       1           100             0      
 1.05556       100.296      0.0141092      
 1.11111       100.594      0.028302      
 1.16667       100.892      0.0425788      
 1.22222       101.191       0.05694      
 1.27778       101.491      0.071386      
 1.33333       101.792      0.0859171      
 1.38889       102.093      0.100534      
 1.44444       102.396      0.115236      
     1.5       102.699      0.130025      

LEAST-SQUARES FIT OF FORWARD CURVE IN BLACK MODEL

lambda = 0.05
sigma = 0.2
spot = 100
initial time = 0

Input forward prices:

      time           value      
       0.5          103.37      
         1         106.485      
       1.5         109.355      
         2         111.992      
       2.5         114.409      
         3         116.617      
       3.5         118.632      
         4         120.467      
       4.5         122.135      
         5         123.648      

Fitted coefficients and their covariance matrix:

       value      covariance matrix
   0.0759665       5.06297e-06

chi^2 error = 0.000398907

Fitted forward prices and their errors:

    time         value           err      
       0           100             0      
0.555556       103.129      0.127144      
 1.11111       106.298      0.258509      
 1.66667       109.502      0.394008      
 2.22222       112.741       0.53354      
 2.77778       116.011      0.676999      
 3.33333        119.31      0.824267      
 3.88889       122.635       0.97522      
 4.44444       125.984       1.12972      
       5       129.354       1.28764      

