SampleExam3 by kramkov

INTERPOLATION OF DATA CURVES

DISCOUNT CURVE FROM SWAP RATES BY LOG LINEAR INTERPOLATION

period between payments = 0.5
initial time = 1

swap rates:
[0] = 0.12
[1] = 0.125
[2] = 0.13
[3] = 0.135
[4] = 0.14
[5] = 0.145
[6] = 0.15
[7] = 0.155
[8] = 0.16
[9] = 0.165

VALUES VERSUS TIME:

    time           value
   1.001        0.999883
 1.45555        0.948296
 1.91009        0.895794
 2.36464        0.842739
 2.81918        0.789473
 3.27373        0.736314
 3.72827        0.683565
 4.18282        0.631506
 4.63736        0.580394
 5.09191        0.530465

LEAST SQUARE FITTING OF DATA CURVES

LEAST-SQUARES FIT OF FORWARD EXCHANGE CURVE IN SVENSSON MODEL

lambda 1 = 0.05
lambda 2 = 0.06
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      

Fitted coefficients and their covariance matrix:

       value                        covariance matrix
   -0.113977       2.60018e-06      -2.60048e-06      -1.01783e-05        6.3313e-06
   0.0579761      -2.60048e-06       2.60079e-06       1.01795e-05      -6.33205e-06
     0.66787      -1.01783e-05       1.01795e-05       3.98543e-05      -2.47938e-05
   -0.494173        6.3313e-06      -6.33205e-06      -2.47938e-05       1.54253e-05

chi^2 error = 1.63532e-13

Fitted forward exchange rates and their errors:

    time         value           err      
       1           100             0      
 1.66667       96.3552      6.51381e-06      
 2.33333       92.8873      8.94881e-06      
       3       89.5944      1.34783e-05      
 3.66667       86.4731      1.54128e-05      
 4.33333       83.5186      1.74397e-05      
       5       80.7253      2.27222e-05      
 5.66667       78.0869      2.67706e-05      
 6.33333       75.5966      2.85235e-05      
       7       73.2473      5.41307e-05      

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

STRIKE OF VARIANCE SWAP IN ASSET MODEL

maturity = 0.5
number of variance times = 12

RISK REPORT: 

price = 0.199891
delta = -0.000313838
one percent gamma = 5.69586e-06

OPTION VALUES VERSUS SPOT:

    spot            option
 90.4837          0.199927
 92.3116          0.199919
 94.1765          0.199911
 96.0789          0.199904
 98.0199          0.199898
     100          0.199891
  102.02          0.199885
 104.081          0.199879
 106.184          0.199873
 108.329          0.199868
 110.517          0.199864

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

CALLABLE CAPPED FLOATER IN INTEREST RATE MODEL

cap parameters:
notional = 1000
period between payments = 0.25
number of payments = 6
rate = 0.077

spread over LIBOR = 0.01

RISK REPORT: 

price = 998.351
delta = 578.48
one percent gamma = -477.101

OPTION VALUES VERSUS SHORT RATE:

 rate            option
 0.17           879.198
 0.15           903.709
 0.13           928.906
 0.11           954.604
 0.09           979.469
 0.07           998.351
 0.05            1002.4
 0.03           1002.48
 0.01           1002.49
-0.01           1002.51
-0.03           1002.52

