Homework 1 by kramkov

DATA CURVES FOR FINANCIAL MODELS

VOLATILITY CURVE FROM VARIANCE CURVE

initial time = 0.75

The result:

    time         value           err      
    0.75          0.25      1.03675e-08      
0.861111      0.277778             0      
0.972222      0.305556             0      
 1.08333      0.333333             0      
 1.19444      0.361111             0      
 1.30556      0.388889             0      
 1.41667      0.416667             0      
 1.52778      0.444444             0      
 1.63889      0.472222             0      
    1.75           0.5             0      

COST-OF-CARRY RATE IN BLACK MODEL

theta = 0.03
lambda = 0.05
sigma = 0.2
initial time = 0.75

VALUES VERSUS TIME:

    time           value
    0.75            0.05
    0.85       0.0498255
    0.95       0.0496518
    1.05       0.0494791
    1.15       0.0493073
    1.25       0.0491363
    1.35       0.0489663
    1.45       0.0487971
    1.55       0.0486288
    1.65       0.0484614

SVENSSON YIELD CURVE

c0 = 0.02
c1 = 0.04
c2 = 0.06
c3 = 0.03
lambda 1 = 0.05
lambda 2 = 0.07
initial time = 1.5

VALUES VERSUS TIME:

    time           value
     1.5            0.06
       2       0.0607547
     2.5       0.0614696
       3       0.0621464
     3.5       0.0627863
       4       0.0633906
     4.5       0.0639607
       5       0.0644979
     5.5       0.0650033
       6       0.0654781

FORWARD PRICES FOR A COUPON BOND

interest rate = 0.07
initial time = 1

bond parameters:
notional = 1
period between payments = 0.25
number of payments = 6
rate = 0.07

clean prices:

VALUES VERSUS TIME:

    time           value
       1         0.99913
 1.13636        0.999168
 1.27273        0.999269
 1.40909        0.999323
 1.54545        0.999413
 1.68182        0.999484
 1.81818        0.999563
 1.95455         0.99965
 2.09091        0.999718
 2.22727        0.999822

dirty prices:

VALUES VERSUS TIME:

    time           value
       1         0.99913
 1.13636         1.00871
 1.27273         1.00086
 1.40909         1.01046
 1.54545          1.0026
 1.68182         1.01221
 1.81818         1.00434
 1.95455         1.01397
 2.09091         1.00608
 2.22727         1.01573

