SampleExam1 by kramkov

DATA CURVES FOR FINANCIAL MODELS

YIELD CURVE IN VASICEK MODEL

theta = 0.02
lambda = 0.05
sigma = 0.01
r_0 = 0.04
initial time = 1.5

VALUES VERSUS TIME:

    time           value
     1.5            0.04
       2       0.0444668
     2.5       0.0488679
       3       0.0532042
     3.5       0.0574766
       4        0.061686
     4.5       0.0658334
       5       0.0699196
     5.5       0.0739455
       6        0.077912

INTERPOLATION OF DATA CURVES

FORWARD EXCHANGE CURVE BY LOG-LINEAR INTERPOLATION

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

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

UP-RANGE-OUT PUT OPTION IN ASSET MODEL

strike = 100
maturity = 0.5
upper barrier = 102
number of barrier events needed to cancel put = 4

barrier times:
[0] = 0.04
[1] = 0.08
[2] = 0.12
[3] = 0.16
[4] = 0.2
[5] = 0.24
[6] = 0.28
[7] = 0.32
[8] = 0.36
[9] = 0.4

RISK REPORT: 

price = 3.74647
delta = -45.9803
one percent gamma = 2.11803

OPTION VALUES VERSUS SPOT:

    spot            option
 90.4837           9.14938
 92.3116           7.95535
 94.1765           6.81547
 96.0789           5.73147
 98.0199            4.7057
     100           3.74647
  102.02           2.87251
 104.081           2.11013
 106.184           1.48173
 108.329          0.994489
 110.517          0.638265

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

FUTURES PRICE OF CHEAPEST TO DELIVER COUPON BOND IN INTEREST RATE MODEL

maturity of futures contract = 0.5
number of futures times = 20

Bond 0:
notional = 1000
coupon rate = 0.07
period = 0.25
number of payments = 4

Bond 1:
notional = 1000
coupon rate = 0.07
period = 0.25
number of payments = 5

Bond 2:
notional = 1000
coupon rate = 0.07
period = 0.25
number of payments = 6

Bond 3:
notional = 1000
coupon rate = 0.07
period = 0.25
number of payments = 7

Bond 4:
notional = 1000
coupon rate = 0.07
period = 0.25
number of payments = 8

RISK REPORT: 

price = 994.35
delta = 1344.52
one percent gamma = -198.128

OPTION VALUES VERSUS SHORT RATE:

 rate            option
 0.17             838.9
 0.15           868.602
 0.13           899.398
 0.11           931.324
 0.09           964.061
 0.07            994.35
 0.05           1017.57
 0.03           1037.13
 0.01           1056.57
-0.01           1076.37
-0.03           1096.54

