Homework 3 by kramkov

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

EUROPEAN STRADDLE OPTION IN ASSET MODEL

strike = 100
maturity = 0.5

RISK REPORT: 

price = 11.0622
delta = 18.9767
one percent gamma = 5.39883

OPTION VALUES VERSUS SPOT:

    spot            option
 90.4837           11.7418
 92.3116           11.2193
 94.1765           10.8778
 96.0789           10.7313
 98.0199           10.7904
     100           11.0622
  102.02           11.5498
 104.081           12.2521
 106.184           13.1645
 108.329            14.279
 110.517            15.585

AMERICAN CALL ON FORWARD IN ASSET MODEL

forward price = 100
time to maturity = 0.5

exercise 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 = 7.92769
delta = 62.3099
one percent gamma = 2.95554

OPTION VALUES VERSUS SPOT:

    spot            option
 90.4837           3.16264
 92.3116           3.88749
 94.1765           4.72263
 96.0789           5.67275
 98.0199           6.74076
     100           7.92769
  102.02           9.23273
 104.081           10.6534
 106.184           12.1857
 108.329           13.8245
 110.517           15.5638

DOWN-AND-REBATE OPTION IN ASSET MODEL

lower barrier = 90
notional = 100

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 = 27.9394
delta = -356.367
one percent gamma = 32.4473

OPTION VALUES VERSUS SPOT:

    spot            option
 90.4837           77.1707
 92.3116           66.3357
 94.1765           55.2199
 96.0789           44.8544
 98.0199           35.7249
     100           27.9394
  102.02           21.4493
 104.081           16.1544
 106.184           11.9306
 108.329           8.63728
 110.517           6.12799

UP-AND-IN AMERICAN PUT OPTION IN ASSET MODEL

strike = 100
upper barrier = 110

exercise 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

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 = 0.238012
delta = 3.86599
one percent gamma = 0.542512

OPTION VALUES VERSUS SPOT:

    spot            option
 90.4837         0.0389199
 92.3116         0.0577431
 94.1765         0.0842673
 96.0789          0.120978
 98.0199          0.170944
     100          0.238012
  102.02          0.326728
 104.081          0.440146
 106.184          0.573026
 108.329           0.70323
 110.517          0.793241

