Homework 4 by kramkov

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

COLLAR IN INTEREST RATE MODEL

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

floor rate = 0.063

RISK REPORT: 

price = 0.487137
delta = -834.995
one percent gamma = 34.2297

OPTION VALUES VERSUS SHORT RATE:

 rate            option
 0.17           120.802
 0.15           96.2911
 0.13           71.0932
 0.11           45.3827
 0.09           20.3206
 0.07          0.487137
 0.05          -19.3041
 0.03          -46.1886
 0.01          -75.4486
-0.01          -105.822
-0.03          -137.098

AMERICAN SWAPTION IN INTEREST RATE MODEL

exercise  times:
[0] = 0.125
[1] = 0.25
[2] = 0.375
[3] = 0.5
[4] = 0.625
[5] = 0.75
[6] = 0.875
[7] = 1
[8] = 1.125
[9] = 1.25
[10] = 1.375
[11] = 1.5

swap parameters:
notional = 1000
period between payments = 0.25
number of payments = 6
rate = 0.07
we pay float and receive fixed

RISK REPORT: 

price = 11.8387
delta = 631.992
one percent gamma = 228.659

OPTION VALUES VERSUS SHORT RATE:

 rate            option
 0.17       0.000156823
 0.15        0.00461072
 0.13         0.0719297
 0.11          0.635945
 0.09           3.40136
 0.07           11.8387
 0.05           29.2269
 0.03           55.5509
 0.01           85.1441
-0.01           115.762
-0.03           147.406

AMERICAN SWAPTION IN INTEREST RATE MODEL

exercise  times:
[0] = 0.125
[1] = 0.25
[2] = 0.375
[3] = 0.5
[4] = 0.625
[5] = 0.75
[6] = 0.875
[7] = 1
[8] = 1.125
[9] = 1.25
[10] = 1.375
[11] = 1.5

swap parameters:
notional = 1000
period between payments = 0.25
number of payments = 6
rate = 0.07
we pay fixed and receive float

RISK REPORT: 

price = 12.8513
delta = -627.299
one percent gamma = 200.437

OPTION VALUES VERSUS SHORT RATE:

 rate            option
 0.17           126.436
 0.15           102.972
 0.13           78.7089
 0.11           53.6295
 0.09           29.5541
 0.07           12.8513
 0.05           4.07575
 0.03          0.852369
 0.01          0.108563
-0.01        0.00786831
-0.03       0.000303541

AMERICAN PUT ON FUTURES PRICE OF ZERO-COUPON BOND IN INTEREST RATE MODEL

maturity of futures = 0.25
number of futures times = 10
maturity of bond = 0.35
notional of bond = 1000
strike = 993.049

RISK REPORT: 

price = 0.404202
delta = -49.141
one percent gamma = 35.0809

OPTION VALUES VERSUS SHORT RATE:

 rate            option
 0.17           9.71832
 0.15           7.79122
 0.13           5.85838
 0.11           3.91979
 0.09           1.97672
 0.07          0.404202
 0.05        0.00922398
 0.03       4.04426e-06
 0.01      -2.31244e-13
-0.01      -1.19201e-15
-0.03      -6.14456e-18

PUTABLE AND CALLABLE BOND IN INTEREST RATE MODEL

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

redemption price for put = 0.95
repurchase price for call = 1.05

RISK REPORT: 

price = 999.13
delta = 1383.82
one percent gamma = 19.5621

OPTION VALUES VERSUS SHORT RATE:

 rate            option
 0.17           927.387
 0.15           932.007
 0.13           936.903
 0.11           948.069
 0.09           971.883
 0.07            999.13
 0.05           1027.15
 0.03           1052.59
 0.01           1064.51
-0.01           1070.04
-0.03           1075.37

