SampleExam2 by kramkov

INTERPOLATION OF DATA CURVES

VOLATILITY CURVE BY LINEAR INTERPOLATION OF VARIANCE CURVE

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      

VALUES VERSUS TIME:

    time           value
       1       0.0373058
     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

LEAST SQUARE FITTING OF DATA CURVES

LEAST-SQUARES FIT OF DISCOUNT CURVE FOR HULL AND WHITE MODEL

lambda = 0.05
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
   -0.152746       2.65228e-05      -2.86514e-05
    0.220502      -2.86514e-05       3.10054e-05

chi^2 error = 5.60212e-06

Fitted discount factors and their errors:

    time         value           err      
       1             1             0      
 1.66667      0.958153      0.000252476      
 2.33333      0.922419      0.000398269      
       3      0.892098      0.000472774      
 3.66667      0.866608      0.000523116      
 4.33333      0.845465      0.000610699      
       5      0.828266      0.000787039      
 5.66667      0.814679      0.00106712      
 6.33333      0.804431      0.00144333      
       7      0.797299      0.00190645      

OPTIONS ON A SINGLE STOCK IN BLACK MODEL

PARAMETERS OF BLACK MODEL:

interest rate = 0.07
spot price = 100
convenience yield = 0.02
sigma = 0.2
lambda = 0.05
initial time = 0

quality = 200

BOOST OPTION IN ASSET MODEL

notional = 100
lower barrier = 95
upper barrier = 110

barrier times:
[0] = 0.0416667
[1] = 0.0833333
[2] = 0.125
[3] = 0.166667
[4] = 0.208333
[5] = 0.25
[6] = 0.291667
[7] = 0.333333
[8] = 0.375
[9] = 0.416667
[10] = 0.458333
[11] = 0.5

RISK REPORT: 

price = 35.2092
delta = 169.571
one percent gamma = -81.8368

OPTION VALUES VERSUS SPOT:

    spot            option
 90.4837           4.11003
 92.3116           8.66121
 94.1765            15.319
 96.0789           23.0988
 98.0199           30.2781
     100           35.2092
  102.02           36.9129
 104.081           35.1275
 106.184           30.1918
 108.329             23.09
 110.517           15.4035

INTEREST RATE OPTIONS IN HULL-WHITE MODEL

PARAMETERS OF HULL-WHITE MODEL:

interest rate = 0.07
sigma = 0.02
lambda = 0.05
initial time = 0

quality = 200

PUTABLE BOND WITH RESETTABLE COUPON IN INTEREST RATE MODEL

bond parameters:
notional = 1000
period between payments = 0.25
number of payments = 6
rate = 0.07

reset value for coupon rate = 0.05
redemption price (percent of notional) = 0.98

RISK REPORT: 

price = 981.548
delta = 973.111
one percent gamma = 98.2623

OPTION VALUES VERSUS SHORT RATE:

 rate            option
 0.17            869.51
 0.15           892.744
 0.13           916.228
 0.11           939.922
 0.09           962.389
 0.07           981.548
 0.05           1004.02
 0.03           1031.87
 0.01           1061.11
-0.01            1091.2
-0.03           1122.16

