23black-scholes model v3
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Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Tho
reference number 9 classificatio 40401
Version V3 level basic Instructions
publication date Jul-03 1. The risk source of both the call an
author Hanyang Financial Engineering Lab. 2. Enter the option descriptions
affiliation Hanyang University input option terms and conditi
email address leesb@hanyang.ac.kr output option price and the senlast revised date Jan-04
references Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Lia
Ch. 4 of "The Oxford Guide to Financial Modeling" by Thomas S.Y. Ho an
Descriptions
financial model class equity options
issuer/modeler market participants
model type continuous time model
risk sources stock price
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words the law of one price, no abitrage condition, dynamic hedging, hedge ratio,
Links
data N/A
financial models
Inputs
Stock Price(S) 100
Strike Price(K) 100
Time to expiration(T) 1
Stock volitility() 0.2
Risk-free rate(r) 0.04
Dividend yields N/A
Outputs
Call Put
Price 9.92505 6.00400 C = S* N(d1) - Ke-rT
N(d2)
(Delta) 0.61791 -0.38209 P = -S* N(-d1) + Ke-rT
N(-d2)
(Gamma) 0.01907 0.01907
v (Vega) 38.13878 38.13878
(Theta) -5.88852 -2.04536
(Rho) 51.86609 -44.21286
(Omega) 6.22577 -6.36390
Interim Calcuations
Step 1 Calculate call price and put price.
d1 0.3
d2 0.1
2
1
ln2
SR T
Kd
2 1d d
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Step 2 Calculate option greeks.
Step 3 Show the performance profile of option greeks.
Stock Price 50 51 52 53 54 55
Delta 0.0008 0.0011 0.0015 0.0020 0.0027 0.0036
Gamma 0.0003 0.0004 0.0005 0.0006 0.0008 0.0010
Vega 0.1329 0.1846 0.2523 0.3397 0.4508 0.5901
Theta -0.0148 -0.0206 -0.0282 -0.0380 -0.0506 -0.0664
Rho 0.0367 0.0523 0.0733 0.1013 0.1381 0.1855
v (Vega)
(Omega)
(Theta)
(Rho)
Call
(Delta)
(Gamma)
1( )
CN d
S
P
S
1
2
( )( )
2
rTS N dCr K e N d
t T
P
t
1'( )N d
S S T
S S
1( )S T N d S T
2( )
rTK T e N d K T
1( )
C S SN d
S C C
P
S
0.00.10.20.30.40.5
0.60.70.80.91.0
50
58
66
74
82
90
98
106
114
122
130
138
146
Stock Price
Delta of European Call Option
Delta
0.000
0.005
0.010
0.015
0.020
0.025
50
58
66
74
82
90
98
106
114
122
130
138
Stock Price
Gamma of European Call Option
Gamma
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mas Ho Company, New York, NY. USA www. thomasho.com tom.ho@thomasho.com
put options and the underlying asset is the same.
ns
itivities
ilities, Journal of Political Economy , Vol.81, 637-654.
d Sang Bin Lee, 2004, Oxford University Press
greeks
T
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56 57 58 59 60 61 62 63
0.0047 0.0060 0.0077 0.0097 0.0121 0.0149 0.0183 0.0222
0.0012 0.0015 0.0018 0.0022 0.0026 0.0031 0.0036 0.0042
0.7624 0.9729 1.2269 1.5297 1.8868 2.3031 2.7837 3.3327
-0.0861 -0.1102 -0.1394 -0.1744 -0.2158 -0.2644 -0.3207 -0.3853
0.2462 0.3226 0.4179 0.5353 0.6784 0.8511 1.0574 1.3015
Put
1( ) 1d
1
2
( )( )
2
rTS N d
r K e N d
T
1'( )N d
T
1( )N d
2( )
rTe N d
1( ) 1S
N dP
146
0.00005.0000
10.000015.000020.0000
25.000030.000035.000040.000045.0000
50
58
66
74
82
90
98
106
114
122
130
138
146
Stock Price
Vega of European Call Option
Vega
(7.0)
(6.0)
(5.0)
(4.0)(3.0)
(2.0)
(1.0)
0.0
50
58
66
74
82
90
st
Theta of Euro
Theta
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64 65 66 67 68 69 70 71
0.0267 0.0319 0.0377 0.0443 0.0517 0.0599 0.0690 0.0789
0.0048 0.0055 0.0062 0.0070 0.0078 0.0086 0.0095 0.0104
3.9540 4.6504 5.4240 6.2757 7.2057 8.2125 9.2939 10.4461
-0.4589 -0.5419 -0.6346 -0.7373 -0.8503 -0.9735 -1.1068 -1.2500
1.5879 1.9208 2.3048 2.7440 3.2428 3.8050 4.4342 5.1337
98
106
114
122
130
138
146
ck Price
pean Call Option
0.0
20.0
40.0
60.0
80.0
100.0
50
57
64
71
78
85
92
99
106
113
120
127
134
141
148
Stock Price
Rho of European Call Option
Rho
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72 73 74 75 76 77 78 79
0.0897 0.1014 0.1140 0.1275 0.1418 0.1570 0.1730 0.1898
0.0113 0.0121 0.0130 0.0139 0.0148 0.0156 0.0164 0.0172
11.6645 12.9430 14.2746 15.6514 17.0646 18.5046 19.9614 21.4244
-1.4027 -1.5645 -1.7347 -1.9125 -2.0971 -2.2876 -2.4829 -2.6819
5.9063 6.7544 7.6798 8.6837 9.7668 10.9290 12.1697 13.4876
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80 81 82 83 84 85 86 87
0.2073 0.2255 0.2444 0.2638 0.2837 0.3041 0.3249 0.3459
0.0179 0.0185 0.0191 0.0197 0.0202 0.0206 0.0209 0.0212
22.8829 24.3261 25.7432 27.1238 28.4577 29.7353 30.9477 32.0866
-2.8835 -3.0865 -3.2896 -3.4918 -3.6918 -3.8884 -4.0806 -4.2674
14.8809 16.3470 17.8830 19.4852 21.1495 22.8714 24.6461 26.4682
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88 89 90 91 92 93 94 95
0.3672 0.3887 0.4103 0.4319 0.4535 0.4749 0.4963 0.5174
0.0214 0.0215 0.0216 0.0216 0.0215 0.0214 0.0212 0.0210
33.1446 34.1153 34.9931 35.7734 36.4527 37.0284 37.4989 37.8636
-4.4478 -4.6208 -4.7859 -4.9421 -5.0890 -5.2260 -5.3528 -5.4690
28.3323 30.2327 32.1635 34.1189 36.0927 38.0791 40.0723 42.0665
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96 97 98 99 100 101 102 103
0.5382 0.5587 0.5789 0.5986 0.6179 0.6367 0.6551 0.6728
0.0207 0.0203 0.0200 0.0195 0.0191 0.0186 0.0181 0.0175
38.1228 38.2776 38.3299 38.2826 38.1388 37.9026 37.5784 37.1711
-5.5745 -5.6692 -5.7530 -5.8261 -5.8885 -5.9406 -5.9825 -6.0148
44.0561 46.0359 48.0006 49.9455 51.8661 53.7581 55.6176 57.4411
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104 105 106 107 108 109 110 111
0.6901 0.7068 0.7229 0.7384 0.7533 0.7676 0.7813 0.7944
0.0170 0.0164 0.0158 0.0152 0.0146 0.0140 0.0134 0.0128
36.6859 36.1284 35.5044 34.8196 34.0801 33.2918 32.4606 31.5923
-6.0376 -6.0516 -6.0571 -6.0546 -6.0448 -6.0280 -6.0050 -5.9761
59.2254 60.9677 62.6654 64.3165 65.9191 67.4716 68.9730 70.4223
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112 113 114 115 116 117 118 119
0.8069 0.8189 0.8302 0.8411 0.8513 0.8610 0.8702 0.8790
0.0122 0.0117 0.0111 0.0105 0.0100 0.0095 0.0090 0.0085
30.6927 29.7672 28.8212 27.8598 26.8876 25.9093 24.9291 23.9510
-5.9420 -5.9032 -5.8603 -5.8136 -5.7638 -5.7113 -5.6565 -5.6000
71.8190 73.1627 74.4533 75.6910 76.8760 78.0091 79.0908 80.1220
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120 121 122 123 124 125 126 127
0.8872 0.8949 0.9022 0.9091 0.9155 0.9216 0.9272 0.9326
0.0080 0.0075 0.0071 0.0067 0.0062 0.0059 0.0055 0.0051
22.9784 22.0149 21.0634 20.1265 19.2068 18.3063 17.4270 16.5703
-5.5420 -5.4830 -5.4233 -5.3632 -5.3031 -5.2432 -5.1837 -5.1249
81.1038 82.0373 82.9236 83.7642 84.5604 85.3137 86.0254 86.6973
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128 129 130 131 132 133 134 135
0.9375 0.9422 0.9465 0.9505 0.9543 0.9578 0.9611 0.9641
0.0048 0.0045 0.0042 0.0039 0.0036 0.0034 0.0031 0.0029
15.7376 14.9301 14.1485 13.3937 12.6660 11.9658 11.2932 10.6482
-5.0670 -5.0101 -4.9544 -4.9000 -4.8471 -4.7956 -4.7458 -4.6976
87.3307 87.9273 88.4886 89.0161 89.5115 89.9762 90.4117 90.8195
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136 137 138 139 140 141 142 143
0.9669 0.9695 0.9720 0.9742 0.9763 0.9782 0.9800 0.9816
0.0027 0.0025 0.0023 0.0022 0.0020 0.0018 0.0017 0.0016
10.0307 9.4405 8.8770 8.3400 7.8289 7.3431 6.8819 6.4447
-4.6511 -4.6063 -4.5633 -4.5221 -4.4825 -4.4447 -4.4087 -4.3743
91.2009 91.5575 91.8904 92.2012 92.4909 92.7608 93.0120 93.2458
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144 145 146 147 148 149 150
0.9831 0.9845 0.9858 0.9870 0.9881 0.9891 0.9900
0.0015 0.0013 0.0012 0.0011 0.0010 0.0010 0.0009
6.0306 5.6390 5.2690 4.9198 4.5905 4.2805 3.9887
-4.3416 -4.3105 -4.2810 -4.2530 -4.2265 -4.2015 -4.1779
93.4631 93.6649 93.8523 94.0261 94.1872 94.3365 94.4748
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Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Tho
reference number 9 classificatio 40401
Version 1.0 level basic Instructions
publication date Jul-03 1. The risk source of both the call an
author Hanyang Financial Engineering Lab. 2. Enter the option descriptions
affiliation Hanyang University input option terms and conditi
email address leesb@hanyang.ac.kr output option price and the senlast revised date Jan-04
references Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Lia
Ch. 4 of "The Oxford Guide to Financial Modeling" by Thomas S.Y. Ho an
Descriptions
financial model class equity options: the Black-Scholes model
issuer/modeler market participants
model type continuous time model
risk sources stock price
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words the law of one price, no abitrage condition, dynamic hedging, hedge ratio,
Links
data N/A
financial models
Inputs
Stock Price (S) 100
Mean () 0.15
Volitility () 0.10
time increment (t) 0.004
Outputs
average of stock return -0.000315
0.00058
variance of stock return 3.758E-05
annualized var 0.009395
annualized std. 0.0969278
max return 0.0176123
min return -0.016722
max price 106.27154
min price 91.101883
Interim Calcuations
time Z stock price stock return exp(return)
0 100
1 -0.441778 99.77884 -0.002214 0.9977884
2 1.0650047 100.51147 0.0073157 1.0073425
3 1.3170848 101.41102 0.00891 1.0089498
90
92
94
96
98
100
102
104
106
108
0 50 100 150 200
Stock
Price
Day
Simulated Stock Price Path
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4 0.3237642 101.67785 0.0026277 1.0026311
5 -1.502613 100.77458 -0.008923 0.9911163
6 0.8786537 101.39494 0.0061371 1.006156
7 -0.564692 101.09208 -0.002991 0.997013
8 0.1378658 101.23896 0.0014519 1.001453
9 0.5404524 101.64454 0.0039981 1.0040061
10 0.6559503 102.12631 0.0047286 1.0047398
11 0.1863918 102.3061 0.0017588 1.0017604
12 -0.011207 102.3582 0.0005091 1.0005092
13 -0.660891 101.99039 -0.0036 0.9964066
14 0.3455284 102.27281 0.0027653 1.0027691
15 -0.420621 102.06028 -0.00208 0.9979219
16 0.6767721 102.55753 0.0048603 1.0048721
17 -0.539302 102.26762 -0.002831 0.9971732
18 0.7624112 102.82155 0.0054019 1.0054165
19 -1.433418 101.95273 -0.008486 0.9915502
20 -0.934497 101.41074 -0.00533 0.9946839
21 0.5106779 101.79783 0.0038098 1.0038171
22 0.7357755 102.33198 0.0052335 1.0052472
23 1.1852762 103.16179 0.0080763 1.008109
24 -0.450482 102.92798 -0.002269 0.9977335
25 0.7881374 103.50233 0.0055646 1.0055801
26 -1.195062 102.78258 -0.006978 0.9930461
27 -0.635221 102.42987 -0.003437 0.9965684
28 -1.190233 101.72068 -0.006948 0.9930764
29 -2.002525 100.49878 -0.012085 0.9879876
30 -0.394685 100.30639 -0.001916 0.9980856
31 -0.222639 100.22336 -0.000828 0.9991723
32 -1.148258 99.555878 -0.006682 0.9933401
33 0.5046741 99.932095 0.0037718 1.003779
34 -1.604585 98.980476 -0.009568 0.9904773
35 0.5570814 99.387457 0.0041033 1.0041117
36 -0.77423 98.959359 -0.004317 0.9956926
37 0.401144 99.268303 0.0031171 1.003121938 0.738587 99.790955 0.0052512 1.005265
39 0.6410346 100.25449 0.0046343 1.004645
40 0.3907189 100.56084 0.0030511 1.0030558
41 0.4826381 100.92679 0.0036325 1.0036391
42 -1.483954 100.042 -0.008805 0.9912333
43 0.1826561 100.21574 0.0017352 1.0017367
44 -0.571704 99.91197 -0.003036 0.9969688
45 -0.179391 99.856577 -0.000555 0.9994456
46 0.9245991 100.50049 0.0064277 1.0064484
47 0.4090327 100.81927 0.003167 1.003172
48 -1.351741 100.01902 -0.007969 0.9920625
49 0.0376849 100.10091 0.0008183 1.0008187
50 2.6930502 101.87953 0.0176123 1.017768451 -1.709577 100.84239 -0.010232 0.9898199
52 0.2454397 101.05764 0.0021323 1.0021346
53 1.0116287 101.7653 0.0069781 1.0070025
54 1.3201861 102.67809 0.0089296 1.0089696
55 -0.73137 102.26354 -0.004046 0.9959626
56 -0.151239 102.22504 -0.000377 0.9996236
57 0.229702 102.43305 0.0020328 1.0020348
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58 -0.429496 102.21445 -0.002136 0.9978659
59 -0.08444 102.21914 4.596E-05 1.000046
60 0.3540674 102.50774 0.0028193 1.0028233
61 1.6620652 103.65107 0.0110918 1.0111536
62 -0.233709 103.55802 -0.000898 0.9991023
63 0.5219533 103.96072 0.0038811 1.0038887
64 0.8616735 104.58947 0.0060297 1.0060479
65 -0.963419 104.01443 -0.005513 0.994502
66 0.4227945 104.35344 0.003254 1.0032593
67 -1.656913 103.32552 -0.009899 0.9901496
68 1.353078 104.274 0.0091376 1.0091795
69 1.0498843 105.02958 0.0072201 1.0072462
70 -0.910497 104.4871 -0.005178 0.9948349
71 -2.286633 103.04663 -0.013882 0.986214
72 0.7432163 103.59221 0.0052805 1.0052945
73 -0.981586 103.01082 -0.005628 0.9943877
74 -0.819737 102.5376 -0.004604 0.9954061
75 -0.793261 102.08365 -0.004437 0.9955728
76 1.3008662 102.98671 0.0088074 1.0088463
77 0.9538508 103.66999 0.0066127 1.0066346
78 -0.963298 103.10008 -0.005512 0.9945027
79 2.4409383 104.76482 0.0160178 1.0161468
80 1.4768602 105.80931 0.0099205 1.0099699
81 0.2508841 106.03882 0.0021667 1.0021691
82 -0.016562 106.08923 0.0004753 1.0004754
83 0.1797798 106.27154 0.001717 1.0017185
84 -0.988171 105.67072 -0.00567 0.9943463
85 -1.292533 104.87122 -0.007595 0.9924341
86 -1.129963 104.18484 -0.006567 0.993455
87 -1.145304 103.49291 -0.006664 0.9933586
88 0.0665648 103.59655 0.001001 1.0010015
89 0.4335266 103.94126 0.0033219 1.0033274
90 -0.949519 103.37888 -0.005425 0.9945894
91 0.0403634 103.46526 0.0008353 1.000835692 0.0159503 103.53573 0.0006809 1.0006811
93 -0.890191 103.01419 -0.00505 0.9949627
94 0.0351588 103.09688 0.0008024 1.0008027
95 1.7700222 104.31798 0.0117746 1.0118442
96 -0.752018 103.88323 -0.004176 0.9958325
97 -0.193636 103.81629 -0.000645 0.9993555
98 1.4359875 104.82422 0.009662 1.0097088
99 -0.125133 104.80206 -0.000211 0.9997886
100 -0.357979 104.62572 -0.001684 0.9983174
101 1.5695514 105.73078 0.0105067 1.0105621
102 -0.1408 105.69796 -0.00031 0.9996896
103 -1.024473 105.07625 -0.005899 0.994118
104 -2.339912 103.59275 -0.014219 0.9858817105 -1.291298 102.80978 -0.007587 0.9924418
106 0.0707814 102.91549 0.0010277 1.0010282
107 0.3345053 103.19328 0.0026956 1.0026992
108 -0.823488 102.71678 -0.004628 0.9953825
109 -1.014696 102.11892 -0.005838 0.9941795
110 0.9484393 102.79292 0.0065785 1.0066001
111 -1.193346 102.07921 -0.006967 0.9930568
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112 -0.864673 101.5814 -0.004889 0.9951233
113 -1.692993 100.55783 -0.010127 0.9899237
114 -1.994631 99.354857 -0.012035 0.988037
115 0.3677837 99.644009 0.0029061 1.0029103
116 -0.974183 99.089418 -0.005581 0.9944343
117 0.9107293 99.719636 0.00634 1.0063601
118 0.2593401 99.941281 0.0022202 1.0022227
119 -0.474661 99.699514 -0.002422 0.9975809
120 0.1856915 99.874582 0.0017544 1.001756
121 -0.517225 99.606159 -0.002691 0.9973124
122 -2.139313 98.324555 -0.01295 0.9871333
123 -0.004858 98.378578 0.0005493 1.0005494
124 0.5604807 98.785207 0.0041248 1.0041333
125 -2.735651 97.147077 -0.016722 0.9834173
126 -1.347557 96.378523 -0.007943 0.9920888
127 -0.91514 95.877902 -0.005208 0.9948057
128 0.5091101 96.242921 0.0037999 1.0038071
129 0.7931567 96.783042 0.0055964 1.0056121
130 -1.241826 96.081596 -0.007274 0.9927524
131 -0.376474 95.908706 -0.001801 0.9982006
132 0.0018736 95.965486 0.0005918 1.000592
133 -0.354801 95.805937 -0.001664 0.9983374
134 -0.02758 95.8448 0.0004056 1.0004056
135 0.3097693 96.088474 0.0025392 1.0025424
136 -0.689524 95.725856 -0.003781 0.9962262
137 -1.24697 95.028981 -0.007307 0.9927201
138 -0.154644 94.991162 -0.000398 0.999602
139 1.0955796 95.707141 0.0075091 1.0075373
140 -0.464496 95.481755 -0.002358 0.997645
141 -0.648756 95.145956 -0.003523 0.9964831
142 -0.016104 95.191461 0.0004782 1.0004783
143 -0.793575 94.769841 -0.004439 0.9955708
144 0.1947274 94.941678 0.0018116 1.0018132
145 -2.404959 93.562765 -0.01463 0.9854762146 -0.327152 93.423546 -0.001489 0.998512
147 -1.03239 92.869381 -0.005949 0.9940683
148 -0.897605 92.397234 -0.005097 0.994916
149 0.0600494 92.485958 0.0009598 1.0009602
150 -2.009965 91.370685 -0.012132 0.9879412
151 1.8393803 92.493462 0.0122133 1.0122881
152 -0.385193 92.321937 -0.001856 0.9981455
153 -0.153638 92.285783 -0.000392 0.9996084
154 -0.559138 92.013361 -0.002956 0.9970481
155 -0.241184 91.926414 -0.000945 0.9990551
156 0.0734667 92.022495 0.0010446 1.0010452
157 0.9658561 92.64006 0.0066886 1.006711
158 -1.362114 91.898702 -0.008035 0.9919974159 0.6476239 92.32942 0.0046759 1.0046869
160 -0.522655 92.078113 -0.002726 0.9972782
161 -0.709063 91.719295 -0.003905 0.9961031
162 0.0346995 91.79265 0.0007995 1.0007998
163 -1.270621 91.110779 -0.007456 0.9925716
164 0.1299918 91.238619 0.0014021 1.0014031
165 0.547725 91.608345 0.0040441 1.0040523
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166 0.7667154 92.107052 0.0054291 1.0054439
167 -0.579739 91.823193 -0.003087 0.9969182
168 -1.091685 91.244297 -0.006324 0.9936955
169 -0.103803 91.237317 -7.65E-05 0.9999235
170 -0.326587 91.101883 -0.001486 0.9985156
171 1.0645999 91.770564 0.0073131 1.0073399
172 0.4256412 92.071328 0.003272 1.0032774
173 -1.039587 91.521019 -0.005995 0.994023
174 -0.298152 91.4016 -0.001306 0.9986952
175 1.1495899 92.121986 0.0078506 1.0078815
176 0.1524768 92.264364 0.0015443 1.0015455
177 -0.528892 92.009605 -0.002765 0.9972388
178 -0.365902 91.850183 -0.001734 0.9982673
179 1.2313103 92.621963 0.0083675 1.0084026
180 -1.41009 91.852874 -0.008338 0.9916965
181 1.2141845 92.614645 0.0082592 1.0082934
182 0.2973525 92.842815 0.0024606 1.0024637
183 0.3619789 93.109597 0.0028694 1.0028735
184 -0.238429 93.023235 -0.000928 0.9990725
185 0.458964 93.347776 0.0034827 1.0034888
186 -0.158284 93.308478 -0.000421 0.999579
187 -1.104545 92.712677 -0.006406 0.9936147
188 0.0271984 92.782425 0.000752 1.0007523
189 -0.630454 92.466822 -0.003407 0.9965985
190 2.4424662 93.960775 0.0160275 1.0161566
191 -0.153948 93.923794 -0.000394 0.9996064
192 1.3673571 94.794527 0.0092279 1.0092706
193 -0.199095 94.730166 -0.000679 0.999321
194 -0.055758 94.751706 0.0002274 1.0002274
195 1.6121157 95.778263 0.0107759 1.0108342
196 -2.245743 94.482293 -0.013623 0.9864691
197 0.6246796 94.911347 0.0045308 1.0045411
198 -0.508501 94.661486 -0.002636 0.9973674
199 -0.251121 94.566094 -0.001008 0.9989923200 -1.140602 93.940837 -0.006634 0.9933882
201 -1.381518 93.177634 -0.008157 0.9918757
202 0.8225231 93.717956 0.0057821 1.0057988
203 -0.09527 93.715844 -2.25E-05 0.9999775
204 1.4807392 94.652499 0.009945 1.0099946
205 2.5392728 96.240672 0.0166398 1.016779
206 -0.682555 95.881705 -0.003737 0.9962701
207 -1.304797 95.14889 -0.007672 0.9923571
208 -0.240871 95.059169 -0.000943 0.999057
209 -0.296793 94.935949 -0.001297 0.9987038
210 0.947241 95.561816 0.0065709 1.0065925
211 0.1678973 95.718845 0.0016419 1.0016432
212 2.2773202 97.163802 0.014983 1.0150958213 0.6441383 97.617046 0.0046539 1.0046647
214 -1.056947 97.022936 -0.006105 0.9939139
215 -0.486854 96.780766 -0.002499 0.997504
216 -1.459994 95.946857 -0.008654 0.9913835
217 -0.605729 95.635443 -0.003251 0.9967543
218 1.5993419 96.663766 0.0106951 1.0107525
219 -0.01771 96.709014 0.000468 1.0004681
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220 -0.019088 96.753441 0.0004593 1.0004594
221 -1.734857 95.753166 -0.010392 0.9896616
222 0.9308337 96.374419 0.0064671 1.0064881
223 -0.070629 96.387267 0.0001333 1.0001333
224 -0.888599 95.902698 -0.00504 0.9949727
225 0.3244099 96.155422 0.0026317 1.0026352
226 -0.349347 95.998868 -0.001629 0.9983719
227 -1.712551 95.019798 -0.010251 0.9898012
228 0.4212063 95.328537 0.0032439 1.0032492
229 -0.392371 95.147436 -0.001902 0.9981002
230 0.2409479 95.347826 0.0021039 1.0021061
231 0.4046285 95.647602 0.0031391 1.003144
232 -0.161538 95.605368 -0.000442 0.9995584
233 0.164174 95.760215 0.0016183 1.0016196
234 -0.828438 95.315057 -0.00466 0.9953513
235 -0.155418 95.276658 -0.000403 0.9995971
236 -1.313811 94.543078 -0.007729 0.9923005
237 -1.193814 93.886372 -0.00697 0.9930539
238 -0.331836 93.743893 -0.001519 0.9984824
239 -0.583364 93.452848 -0.00311 0.9968953
240 -1.027579 92.901336 -0.005919 0.9940985
241 -0.289845 92.784991 -0.001253 0.9987476
242 0.4260596 93.089326 0.0032746 1.00328
243 0.7991434 93.615293 0.0056342 1.0056501
244 -0.429245 93.415657 -0.002135 0.9978675
245 -1.48857 92.594008 -0.008835 0.9912044
246 1.2685655 93.394041 0.0086031 1.0086402
247 -0.353498 93.239535 -0.001656 0.9983457
248 -0.096192 93.23689 -2.84E-05 0.9999716
249 -1.090575 92.649731 -0.006317 0.9937025
250 -0.453786 92.437807 -0.00229 0.9977126
92.437807
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mas Ho Company, New York, NY. USA www. thomasho.com tom.ho@thomasho.com
put options and the underlying asset is the same.
ns
itivities
ilities, Journal of Political Economy , Vol.81, 637-654.
d Sang Bin Lee, 2004, Oxford University Press
greeks
bins frequency bins frequency
-0.04 0 73 0
-0.038 0 74 0
-0.036 0 75 0
stock return stock price
250 300
0
5
10
15
2025
30
35
40
-0.
040
-0.
036
-0.
032
-0.
028
-0.
024
-0.
020
-0.
016
-0.
012
-0.
008
-0.
004
0.
000
0.
004
0.
008
0.
012
0.
016
0.
020
0.
024
0.
028
0.
032
0.
036
frequen
cy
stock return
distribution of stock return
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-0.034 0 76 0
-0.032 0 77 0
-0.03 0 78 0
-0.028 0 79 0
-0.026 0 80 0
-0.024 0 81 0
-0.022 0 82 0
-0.02 0 83 0
-0.018 1 84 0
-0.016 2 85 0
-0.014 6 86 0
-0.012 4 87 0
-0.01 10 88 0
-0.008 21 89 0
-0.006 26 90 0
-0.004 27 91 0
-0.002 36 92 17
2.082E-17 31 93 27
0.002 30 94 21
0.004 18 95 13
0.006 15 96 31
0.008 12 97 13
0.01 5 98 4
0.012 1 99 5
0.014 1 100 17
0.016 4 101 16
0.018 0 102 15
0.02 0 103 28
0.022 0 104 23
0.024 0 105 12
0.026 0 106 6
0.028 0 107 3
0.03 0 108 0
0.032 00.034 0
0.036 0
0.038 0
0.04 0
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0.
040
0
5
10
15
2025
30
35
73 75 77 79 81 83 85 87 89 91 93 95 97 99 101103105107
stock price
distribution of stock price
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Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Tho
reference number 9 classificatio 40401
Version 1.0 level basic Instructions
publication date Jul-03 1. The risk source of both the call an
author Hanyang Financial Engineering Lab. 2. Enter the option descriptions
affiliation Hanyang University 3. Use "data-table" to generate the p
email address leesb@hanyang.ac.kr input option terms and conditilast revised date Jan-04 output option price and the sen
references Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Lia
Ch. 4 of "The Oxford Guide to Financial Modeling" by Thomas S.Y. Ho an
Descriptions
financial model class equity options: the Black-Scholes model
issuer/modeler market participants
model type continuous time model
risk sources stock price
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words the law of one price, no abitrage condition, dynamic hedging, hedge ratio,
Links
data N/A
financial models
Inputs
Stock Price (S) 100
Strike Price (K) 100
Time to expiration (T) 1
Stock volitility () 0.2
Risk-free rate (r) 0.04
Dividend yields () 0.01
Outputs
Call Put
Price 9.3197 6.3937 C = S*N(d1) - Ke
-rTN(d2)
(Delta) 0.5927 -0.3973 P = -S*N(-d1) + Ke
-rTN(-d2)
(Gamma) 0.0191 0.0191 S*= S e
-T
v (Vega) 38.2821 38.2821
(Theta) -5.2337 -2.3806
(Rho) 49.9552 -46.1238
(Omega) 6.3601 -6.2139
ensitivity to dividend yield (f/ -59.27491 39.730074
Interim Calcuations
Step 1 Calculate call price and put price.
d1 0.2500
d2 0.0500
21
ln 2S K r T d
T
2 1d d
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Step 2 Calculate option greeks.
Step 3 Show the performance profile of option greeks.
Stock Price 50 51 52 53 54
Delta 0.5927 0.0006 0.0009 0.0013 0.0017 0.0023
Gamma 0.0191 0.0002 0.0003 0.0004 0.0005 0.0007
Vega 38.2821 0.1122 0.1566 0.2151 0.2909 0.3879
Theta -5.2337 -0.0121 -0.0169 -0.0233 -0.0316 -0.0422
Rho 49.9552 0.0306 0.0438 0.0617 0.0856 0.1171
(Omega) 6.3601 19.5637 19.1222 18.6913 18.2706 17.8597
ensitivity to dividend yield (f/ -59.2749 -0.0322 -0.0462 -0.0651 -0.0905 -0.1241
Call
(Delta)
(Gamma)
v (Vega)
sensitivity to dividend yield C/ = -TSe-T
N(d1) P/ =
(Omega)
(Theta)
(Rho)
1( )TC
e N dS
Pe
S
1
1 2
( )( ) ( )
2
T
T rTSe N d C
Se N d rKe N d t T
P
t
1'( )T N d
eS S T
e
S
1( )
TCSe T N d
P
2( )rTC
K Te N d r
P
1( )
C S STN d
S C Ce
P S
S P
0.10.20.30.40.50.60.70.80.91.0
Delta of European Call Option
Delta 0.005
0.010
0.015
0.020
0.025
Gamma of European Call Option
Gamma
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.
50
58
66
74
82
90
98
106
114
122
130
138
146
Stock Price
.
50
58
66
74
82
90
98
106
114
122
130
138
Stock Price
0
5
10
15
20
25
30
35
40
45
50
58
66
74
82
90
98
106
114
122
130
138
146
Stock Price
Vega of European Call Option
Vega
(6.0)
(5.0)
(4.0)
(3.0)
(2.0)
(1.0)
0.0
50
58
66
74
82
90
98
106
114
122
130
138
stock Price
Theta of European Call Option
Theta
0.010.020.030.040.050.060.070.080.090.0
100.0
5
0
5
8
6
6
7
4
8
2
9
0
9
8
10
6
11
4
12
2
13
0
13
8
14
6
Stock Price
Rho of European Call Option
Rho
0
5
10
15
20
25
50 70 90 110 130Stock Price
(Omega)
(Omega)
-160
-140
-120
-100
-80
-60
-40
-20
050 100 150
Stock Price
sensitivity to dividend yield (f/)
sensitivity to
dividend yield
(f/)
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mas Ho Company, New York, NY. USA www. thomasho.com tom.ho@thomasho.com
put options and the underlying asset is the same.
rformance profiles of the option greeks
nsitivities
ilities, Journal of Political Economy , Vol.81, 637-654.
d Sang Bin Lee, 2004, Oxford University Press
greeks
T
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55 56 57 58 59 60 61 62
0.0030 0.0040 0.0052 0.0066 0.0084 0.0105 0.0130 0.0160
0.0008 0.0011 0.0013 0.0016 0.0019 0.0023 0.0027 0.0032
0.5101 0.6620 0.8486 1.0747 1.3457 1.6668 2.0431 2.4794
-0.0557 -0.0724 -0.0930 -0.1181 -0.1482 -0.1840 -0.2261 -0.2751
0.1581 0.2106 0.2772 0.3605 0.4635 0.5897 0.7426 0.9259
17.4584 17.0665 16.6835 16.3093 15.9436 15.5861 15.2367 14.8952
-0.1677 -0.2237 -0.2948 -0.3840 -0.4945 -0.6301 -0.7947 -0.9926
Put
TSe-T
N(-d1)
1( ) 1T N d
1
1 2
( )( ) ( )
2
T
T rTe N d
Se N d rKe N d
T
1'( )T N d
S T
1( )
Te T N d
2( )
rTK Te N d
1( ) 1T S
e N dP
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146
146
150
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63 64 65 66 67 68 69 70
0.0195 0.0235 0.0282 0.0335 0.0395 0.0462 0.0537 0.0620
0.0038 0.0043 0.0050 0.0056 0.0063 0.0071 0.0079 0.0087
2.9803 3.5498 4.1913 4.9072 5.6992 6.5680 7.5131 8.5330
-0.3315 -0.3959 -0.4688 -0.5505 -0.6413 -0.7413 -0.8508 -0.9695
1.1438 1.4004 1.6999 2.0466 2.4448 2.8986 3.4120 3.9887
14.5613 14.2349 13.9158 13.6038 13.2987 13.0005 12.7089 12.4238
-1.2282 -1.5062 -1.8315 -2.2090 -2.6436 -3.1402 -3.7034 -4.3378
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71 72 73 74 75 76 77 78
0.0711 0.0811 0.0919 0.1036 0.1162 0.1296 0.1439 0.1589
0.0095 0.0104 0.0113 0.0121 0.0130 0.0138 0.0147 0.0155
9.6250 10.7852 12.0086 13.2892 14.6200 15.9929 17.3993 18.8297
-1.0973 -1.2340 -1.3790 -1.5319 -1.6920 -1.8584 -2.0304 -2.2070
4.6321 5.3452 6.1308 6.9909 7.9271 8.9405 10.0315 11.2000
12.1451 11.8726 11.6062 11.3458 11.0914 10.8426 10.5996 10.3621
-5.0477 -5.8369 -6.7088 -7.6666 -8.7126 -9.8488 -11.0765 -12.3963
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79 80 81 82 83 84 85 86
0.1748 0.1914 0.2087 0.2267 0.2453 0.2644 0.2840 0.3040
0.0162 0.0170 0.0177 0.0183 0.0189 0.0194 0.0198 0.0202
20.2742 21.7226 23.1645 24.5893 25.9866 27.3463 28.6587 29.9145
-2.3871 -2.5698 -2.7538 -2.9380 -3.1214 -3.3027 -3.4809 -3.6550
12.4452 13.7658 15.1598 16.6246 18.1573 19.7543 21.4114 23.1244
10.1300 9.9033 9.6819 9.4656 9.2544 9.0482 8.8469 8.6504
-13.8083 -15.3119 -16.9059 -18.5884 -20.3571 -22.2088 -24.1401 -26.1470
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87 88 89 90 91 92 93 94
0.3244 0.3451 0.3660 0.3871 0.4082 0.4294 0.4505 0.4716
0.0205 0.0208 0.0210 0.0211 0.0212 0.0212 0.0211 0.0210
31.1051 32.2228 33.2603 34.2115 35.0711 35.8349 36.4993 37.0620
-3.8238 -3.9865 -4.1422 -4.2901 -4.4295 -4.5598 -4.6805 -4.7912
24.8882 26.6979 28.5481 30.4331 32.3474 34.2852 36.2406 38.2079
8.4586 8.2715 8.0889 7.9108 7.7372 7.5679 7.4028 7.2419
-28.2251 -30.3695 -32.5752 -34.8368 -37.1488 -39.5053 -41.9007 -44.3291
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95 96 97 98 99 100 101 102
0.4925 0.5131 0.5336 0.5537 0.5734 0.5927 0.6117 0.6301
0.0208 0.0205 0.0203 0.0199 0.0196 0.0191 0.0187 0.0182
37.5216 37.8775 38.1299 38.2801 38.3300 38.2821 38.1397 37.9067
-4.8916 -4.9814 -5.0605 -5.1288 -5.1865 -5.2337 -5.2705 -5.2971
40.1815 42.1557 44.1253 46.0850 48.0298 49.9552 51.8566 53.7301
7.0851 6.9324 6.7836 6.6386 6.4975 6.3601 6.2264 6.0963
-46.7847 -49.2618 -51.7547 -54.2580 -56.7665 -59.2749 -61.7786 -64.2730
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103 104 105 106 107 108 109 110
0.6481 0.6655 0.6825 0.6988 0.7146 0.7299 0.7445 0.7586
0.0177 0.0172 0.0166 0.0161 0.0155 0.0149 0.0144 0.0138
37.5875 37.1866 36.7092 36.1607 35.5466 34.8727 34.1446 33.3682
-5.3141 -5.3216 -5.3202 -5.3102 -5.2923 -5.2668 -5.2343 -5.1954
55.5718 57.3782 59.1464 60.8735 62.5570 64.1950 65.7855 67.3271
5.9697 5.8466 5.7268 5.6103 5.4971 5.3871 5.2801 5.1762
-66.7538 -69.2172 -71.6594 -74.0772 -76.4675 -78.8278 -81.1555 -83.4488
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111 112 113 114 115 116 117 118
0.7721 0.7850 0.7974 0.8092 0.8204 0.8311 0.8412 0.8508
0.0132 0.0126 0.0121 0.0115 0.0109 0.0104 0.0099 0.0094
32.5492 31.6934 30.8061 29.8929 28.9588 28.0088 27.0477 26.0799
-5.1506 -5.1005 -5.0455 -4.9862 -4.9232 -4.8570 -4.7880 -4.7166
68.8186 70.2592 71.6482 72.9853 74.2704 75.5035 76.6851 77.8155
5.0752 4.9771 4.8819 4.7894 4.6996 4.6124 4.5278 4.4457
-85.7057 -87.9249 -90.1052 -92.2457 -94.3456 -96.4045 -98.4222 -100.3987
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119 120 121 122 123 124 125 126
0.8600 0.8686 0.8767 0.8844 0.8917 0.8985 0.9049 0.9109
0.0089 0.0084 0.0079 0.0075 0.0070 0.0066 0.0062 0.0058
25.1095 24.1404 23.1762 22.2201 21.2752 20.3441 19.4293 18.5329
-4.6434 -4.5688 -4.4931 -4.4167 -4.3400 -4.2632 -4.1866 -4.1106
78.8955 79.9258 80.9075 81.8416 82.7292 83.5717 84.3704 85.1265
4.3660 4.2887 4.2137 4.1410 4.0704 4.0019 3.9355 3.8711
-102.3341 -104.2288 -106.0832 -107.8978 -109.6736 -111.4111 -113.1115 -114.7756
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127 128 129 130 131 132 133 134
0.9166 0.9219 0.9268 0.9315 0.9358 0.9399 0.9436 0.9472
0.0055 0.0051 0.0048 0.0045 0.0042 0.0039 0.0036 0.0034
17.6567 16.8023 15.9712 15.1644 14.3829 13.6274 12.8983 12.1962
-4.0353 -3.9609 -3.8877 -3.8158 -3.7453 -3.6763 -3.6090 -3.5434
85.8417 86.5173 87.1549 87.7559 88.3219 88.8543 89.3548 89.8246
3.8087 3.7481 3.6894 3.6325 3.5773 3.5238 3.4718 3.4215
-116.4045 -117.9994 -119.5614 -121.0916 -122.5912 -124.0615 -125.5037 -126.9190
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135 136 137 138 139 140 141 142
0.9504 0.9535 0.9563 0.9589 0.9614 0.9637 0.9658 0.9677
0.0032 0.0029 0.0027 0.0025 0.0024 0.0022 0.0020 0.0019
11.5210 10.8729 10.2518 9.6575 9.0897 8.5480 8.0318 7.5407
-3.4796 -3.4177 -3.3576 -3.2995 -3.2433 -3.1889 -3.1365 -3.0860
90.2654 90.6785 91.0653 91.4273 91.7656 92.0816 92.3766 92.6516
3.3727 3.3254 3.2795 3.2350 3.1918 3.1499 3.1093 3.0699
-128.3086 -129.6737 -131.0154 -132.3350 -133.6335 -134.9121 -136.1717 -137.4135
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143 144 145 146 147 148 149 150
0.9695 0.9712 0.9727 0.9741 0.9754 0.9766 0.9778 0.9788
0.0017 0.0016 0.0015 0.0014 0.0013 0.0012 0.0011 0.0010
7.0740 6.6311 6.2112 5.8136 5.4375 5.0823 4.7470 4.4308
-3.0373 -2.9905 -2.9455 -2.9022 -2.8606 -2.8206 -2.7823 -2.7455
92.9080 93.1467 93.3688 93.5754 93.7673 93.9456 94.1110 94.2644
3.0316 2.9945 2.9585 2.9236 2.8897 2.8567 2.8248 2.7937
-138.6385 -139.8475 -141.0416 -142.2216 -143.3883 -144.5426 -145.6853 -146.8170
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Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Tho
reference number 9 classificatio 40401
Version 1.0 level basic Instructions
publication date Jul-03 1. The risk source of both the call an
author Hanyang Financial Engineering Lab. 2. Enter the option descriptions
affiliation Hanyang University input option terms and conditi
email address leesb@hanyang.ac.kr output option price and the senlast revised date Jan-04
references Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Lia
Ch. 4 of "The Oxford Guide to Financial Modeling" by Thomas S.Y. Ho an
Descriptions
financial model class equity options: the Black-Scholes model
issuer/modeler market participants
model type continuous time model
risk sources stock price
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words the law of one price, no abitrage condition, dynamic hedging, hedge ratio,
Links
data N/A
financial models
Inputs
Stock Price (S) 27.43
Strike Price (K) 27.5
Time to expiration (T) 1
Stock volitility () 0.2
bid price: PV(100) 97.40625
Risk-free rate (r) 0.0263
Dividend schedule (T) quarterly 0.25 0.5 0.75 1Dividend amount 0.16 0.16 0.16 0.16 0.16
PV(Div.) 0.6295909 0.1589523 0.1579114 0.1568773 0.15585
Div. Adjusted Stock Price (S*) 26.800409
Outputs
2.14111 2.12742
0 2.49611 1.85283
0.16 2.14111 2.12742
Dividend Call Put
Call Put C = S*N(d1) - Ke
-rTN(d2)
European Option Price 2.14111 2.12742 P = -S*N(-d1) + Ke
-rTN(-d2)
Interim Calcuations
Step 1 Calculate call price and put price.
d1 0.1025548
d2 -0.097445
* 21
ln 2S K r T d
T
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2 1d d T
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mas Ho Company, New York, NY. USA www. thomasho.com tom.ho@thomasho.com
put options and the underlying asset is the same.
ns
itivities
ilities, Journal of Political Economy , Vol.81, 637-654.
d Sang Bin Lee, 2004, Oxford University Press
greeks
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Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Tho
reference number 9 classificatio 40401
Version 1.0 level basic Instructions
publication date Jul-03 1. The risk source of both the call an
author Hanyang Financial Engineering Lab. 2. Enter the option descriptions
affiliation Hanyang University input option terms and conditi
email address leesb@hanyang.ac.kr output option price and the senlast revised date Jan-04
references Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Lia
Ch. 4 of "The Oxford Guide to Financial Modeling" by Thomas S.Y. Ho an
Descriptions
financial model class equity options: the Black-Scholes model
issuer/modeler market participants
model type continuous time model
risk sources stock price
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words the law of one price, no abitrage condition, dynamic hedging, hedge ratio,
Links
data N/A
financial models
Inputs
Stock Price (S) 27.29
Strike Price (K) 30
Time to expiration (T) 0.1360
Stock volitility () 0.2231 Changing cell for solver
Risk-free rate (r) 0.0085
Dividend schedule (T) quarterly 0.25 0.5 0.75
Dividend amount 0 0 0 0PV(Div.) 0 0 0 0
Div. Adjusted Stock Price (S*) 27.29
Option premium (market value) 0.15
Outputs
Call Put C = S*N(d1) - Ke
-rTN(d2)
European Option Price 0.15 2.83 P = -S*N(-d1) + Ke
-rTN(-d2)
(market value - BSOPM)2 4.572E-13 Target cell for solver
Interim Calcuations
Step 1 Calculate call price and put price.
d1 -1.095529
d2 -1.177806
* 21
2 1
ln 2S K r T d
T
d d T
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Historical volatility for the last 34 days
daily average -0.000617
daily standard deviation 0.0101464
annual average -0.154241
annual standard deviation 0.1604282
Date Open High Low Close Volume Adj. Close*
3-Feb-04 27.4 27.55 27.18 27.29 47993800 27.29
2-Feb-04 27.61 27.8 27.24 27.4 62891800 27.4
30-Jan-04 27.84 27.9 27.55 27.65 40528700 27.65
29-Jan-04 27.81 27.95 27.57 27.91 63748400 27.91
28-Jan-04 28.3 28.44 27.47 27.71 71336000 27.71
27-Jan-04 28.64 28.72 28.22 28.25 63196200 28.25
26-Jan-04 28.49 28.83 28.32 28.8 58299600 28.8
23-Jan-04 28.28 28.76 28.22 28.48 127259104 28.48
22-Jan-04 28.36 28.44 27.94 28.01 78425200 28.01
21-Jan-04 28.13 28.3 27.85 28.3 53570600 28.3
20-Jan-04 27.98 28.2 27.93 28.1 63068500 28.1
16-Jan-04 27.71 27.88 27.53 27.81 63983400 27.81
15-Jan-04 27.55 27.72 27.42 27.54 58504100 27.54
14-Jan-04 27.52 27.73 27.47 27.7 43907000 27.7
13-Jan-04 27.55 27.64 27.26 27.43 51555900 27.43
12-Jan-04 27.67 27.73 27.35 27.57 55845200 27.57
9-Jan-04 28.03 28.06 27.59 27.66 67079900 27.66
8-Jan-04 28.39 28.48 28 28.16 58810800 28.16
7-Jan-04 28.17 28.31 28.01 28.21 54298200 28.21
6-Jan-04 28.19 28.28 28.07 28.24 46950800 28.24
5-Jan-04 27.73 28.18 27.72 28.14 67333696 28.14
2-Jan-04 27.58 27.77 27.33 27.45 44487700 27.45
31-Dec-03 27.42 27.55 27.23 27.37 42198900 27.37
30-Dec-03 27.41 27.55 27.4 27.52 34406600 27.5229-Dec-03 27.21 27.53 27.16 27.46 40426700 27.46
26-Dec-03 27.05 27.25 27 27.21 12076900 27.21
24-Dec-03 27.14 27.16 27 27.04 19924200 27.04
23-Dec-03 27.17 27.34 27.01 27.15 39609000 27.15
22-Dec-03 27.16 27.43 26.97 27.18 43894800 27.18
19-Dec-03 27.49 27.55 27.19 27.36 76856400 27.36
18-Dec-03 27.1 27.45 27.05 27.4 61177800 27.4
17-Dec-03 27.04 27.08 26.85 27.04 53712300 27.04
16-Dec-03 26.83 27.16 26.77 27.06 73978000 27.06
15-Dec-03 27.05 27.1 26.68 26.74 88860600 26.74
2
( ) ( ) exp
~ (0,1)
ln ( ) ( )
ln ( ) ( )
S t t S t t e t
e N
E S t t S t t
Var S t t S t t
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put options and the underlying asset is the same.
ns
itivities
ilities, Journal of Political Economy , Vol.81, 637-654.
d Sang Bin Lee, 2004, Oxford University Press
greeks
1
00
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0.0040227
0.0090827
0.0093593
-0.007192
0.0193001
0.0192819
-0.011173
-0.01664
0.0103002
-0.007092
-0.010374
-0.009756
0.0057929
-0.009795
0.0050909
0.0032591
0.0179152
0.001774
0.0010629
-0.003547
-0.024826
-0.002919
0.0054655-0.002183
-0.009146
-0.006267
0.0040598
0.0011044
0.0066007
0.0014609
-0.013226
0.0007394
-0.011896
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Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Tho
reference number 9 classificatio 40401
Version 1.0 level basic Instructions
publication date Jul-03 1. The risk source of both the call an
author Hanyang Financial Engineering Lab. 2. Enter the option descriptions
affiliation Hanyang University 3. Use "data-table" to generate the p
email address leesb@hanyang.ac.kr input option terms and conditilast revised date Jan-04 output option price and the sen
references Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Lia
Ch. 4 of "The Oxford Guide to Financial Modeling" by Thomas S.Y. Ho an
Descriptions
financial model class equity options: the Black-Scholes model
issuer/modeler market participants
model type continuous time model
risk sources stock price
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words the law of one price, no abitrage condition, dynamic hedging, hedge ratio,
Links
data N/A
financial models
Inputs
Stock Price (S) 100
Strike Price (K) 100
Time to expiration (T) 0.25
Stock volitility () 0.2
Risk-free rate (r) 0.1
Dividend yields () 0.03
Outputs
Call Put
Price 4.8444 3.1226
(Delta) 0.5846 -0.4079
(Gamma) 0.0386 0.0386
v (Vega) 19.3032 19.3032
(Theta) -11.3291 -4.5536
(Rho) 13.4041 -10.9786
(Omega) 12.0677 -13.0634
ensitivity to dividend yield (f/ -14.61523 10.197969
Interim Calcuations
d1 0.2250
d2 0.1250
Option Value
stock price 50 51 52 53 54
long call 4.8444 6.982E-12 2.787E-11 1.044E-10 3.683E-10 1.227E-09
P S
21
ln 2S K r T d
T
2 1
d d T
1 2( ) ( )
T rTC S e N d K e N d
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short call -4.8444 -6.98E-12 -2.79E-11 -1.04E-10 -3.68E-10 -1.23E-09
covered call 95.1556 50 51 52 53 54
long put 3.1226 47.904588 46.91206 45.919532 44.927004 43.934476
short put -3.1226 -47.90459 -46.91206 -45.91953 -44.927 -43.93448
protective put 103.1226 97.904588 97.91206 97.919532 97.927004 97.934476
straddle 7.9670 47.904588 46.91206 45.919532 44.927004 43.934476
3.1226 50 51 52 53 54
put (K=90) 90.0000 38.151489 37.158961 36.166433 35.173905 34.181378
strangle 38.151489 37.158961 36.166433 35.173905 34.181378
Intrinsic Value
long call 0 0 0 0 0
short call 0 0 0 0 0
covered call 50 51 52 53 54
long put 50 49 48 47 46
short put -50 -49 -48 -47 -46
protective put 100 100 100 100 100
straddle 50 49 48 47 46
strangle 40 39 38 37 36
Profit Diagram
stock price 50 51 52 53 54
long call -4.844417 -4.844417 -4.844417 -4.844417 -4.844417
short call 4.844417 4.844417 4.844417 4.844417 4.844417
covered call -45.15558 -44.15558 -43.15558 -42.15558 -41.15558
long put 46.877397 45.877397 44.877397 43.877397 42.877397
short put -46.8774 -45.8774 -44.8774 -43.8774 -42.8774
protective put -3.122603 -3.122603 -3.122603 -3.122603 -3.122603
straddle 42.03298 41.03298 40.03298 39.03298 38.03298
strangle 34.662336 33.662336 32.662336 31.662336 30.662336
-60
-40
-20
0
20
40
60
0 50 100 150 200Profit/Loss
Stock Price
Profit Diagram
long call
short call
covered call
long put
short put
protective put
straddle
strangle
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rate of return
stock price 100 101 102 103 104 105
long stock 0 0.01 0.02 0.03 0.04 0.0
long call 0 0.1246156 0.2569039 0.3965157 0.5430543 0.6960
covered call 0 0.0041649 0.0079391 0.0113405 0.0143893 0.0171
0
10
20
30
40
50
60
50 70 90 110 130 150
stock price
long call
long call
-60
-50
-40
-30
-20
-10
050 70 90 110 130
stock price
short call
short call
0
20
40
60
80
100
120
50 100 150
stock price
covered call
covered call
0
10
20
30
40
50
60
50 100stock price
long put
-60
-50
-40
-30
-20
-10
050 100 150
stock price
short put
short
0
50
100
150
200
50 100
stock price
protective put
protect
0
10
20
30
40
50
60
50 100 150stock price
straddle
strad
0
10
20
30
40
50
60
50 100stock price
strangle
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0
0.02
0.04
0.06
0.08
0.1
0.12
95 100 105 110 115
Rate of Return
long stock
covered call
protective put
0
0.2
0.4
0.6
0.8
1
1.2
1.41.6
1.8
95 100 105
Rate of Ret
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mas Ho Company, New York, NY. USA www. thomasho.com tom.ho@thomasho.com
put options and the underlying asset is the same.
rformance profiles of the profit diagrams.
nsitivities
ilities, Journal of Political Economy , Vol.81, 637-654.
d Sang Bin Lee, 2004, Oxford University Press
greeks
55 56 57 58 59 60 61 62
3.869E-09 1.158E-08 3.301E-08 8.975E-08 2.333E-07 5.809E-07 1.388E-06 3.191E-06
1 2( ) ( )
T rTe N d K e N d
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-3.87E-09 -1.16E-08 -3.3E-08 -8.97E-08 -2.33E-07 -5.81E-07 -1.39E-06 -3.19E-06
55 56 57 58 59 59.999999 60.999999 61.999997
42.941948 41.94942 40.956892 39.964364 38.971836 37.979308 36.986781 35.994255
-42.94195 -41.94942 -40.95689 -39.96436 -38.97184 -37.97931 -36.98678 -35.99425
97.941948 97.94942 97.956892 97.964364 97.971836 97.979308 97.986781 97.994255
42.941948 41.94942 40.956892 39.964364 38.971836 37.979309 36.986783 35.994258
55 56 57 58 59 60 61 62
33.18885 32.196324 31.203801 30.211284 29.218778 28.226296 27.233859 26.241502
33.18885 32.196324 31.203801 30.211284 29.218778 28.226297 27.23386 26.241505
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
55 56 57 58 59 60 61 62
45 44 43 42 41 40 39 38
-45 -44 -43 -42 -41 -40 -39 -38
100 100 100 100 100 100 100 100
45 44 43 42 41 40 39 38
35 34 33 32 31 30 29 28
55 56 57 58 59 60 61 62
-4.844417 -4.844417 -4.844417 -4.844417 -4.844417 -4.844417 -4.844417 -4.844417
4.844417 4.844417 4.844417 4.844417 4.844417 4.844417 4.844417 4.844417
-40.15558 -39.15558 -38.15558 -37.15558 -36.15558 -35.15558 -34.15558 -33.15558
41.877397 40.877397 39.877397 38.877397 37.877397 36.877397 35.877397 34.877397
-41.8774 -40.8774 -39.8774 -38.8774 -37.8774 -36.8774 -35.8774 -34.8774
-3.122603 -3.122603 -3.122603 -3.122603 -3.122603 -3.122603 -3.122603 -3.122603
37.03298 36.03298 35.03298 34.03298 33.03298 32.03298 31.03298 30.03298
29.662336 28.662336 27.662336 26.662336 25.662336 24.662336 23.662336 22.662336
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106 107 108 109 110
0.06 0.07 0.08 0.09 0.1
0.8551562 1.0197897 1.1895113 1.3638489 1.5423425
0.0195182 0.0216457 0.0235142 0.0251477 0.0265696
0.0406076 0.0484141 0.0564596 0.064722 0.0731796
-50
-40
-30
-20-10
0
10
20
30
40
50
0 50 100 150 200profit/loss
stock price
ro agram
long call
short call
-60
-40
-20
0
20
40
60
0 50 100
-60
-40
-200
20
40
60
0 50 100 150 200Profit/Loss
Stock Price
Profit Diagram
covered call
protective put
-20
-10
0
10
20
30
40
50
0 50 100
Profit Diagram
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110 115
urn
long
call
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63 64 65 66 67 68 69 70
7.066E-06 1.51E-05 3.117E-05 6.228E-05 0.0001206 0.0002267 0.0004141 0.0007358
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-7.07E-06 -1.51E-05 -3.12E-05 -6.23E-05 -0.000121 -0.000227 -0.000414 -0.000736
62.999993 63.999985 64.999969 65.999938 66.999879 67.999773 68.999586 69.999264
35.001731 34.009211 33.016699 32.024202 31.031732 30.03931 29.046969 28.054763
-35.00173 -34.00921 -33.0167 -32.0242 -31.03173 -30.03931 -29.04697 -28.05476
98.001731 98.009211 98.016699 98.024202 98.031732 98.03931 98.046969 98.054763
35.001738 34.009226 33.01673 32.024264 31.031853 30.039537 29.047384 28.055499
63 64 65 66 67 68 69 70
25.249287 24.257313 23.265734 22.274784 21.284803 20.296276 19.309865 18.326458
25.249294 24.257328 23.265765 22.274846 21.284924 20.296502 19.310279 18.327194
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
63 64 65 66 67 68 69 70
37 36 35 34 33 32 31 30
-37 -36 -35 -34 -33 -32 -31 -30
100 100 100 100 100 100 100 100
37 36 35 34 33 32 31 30
27 26 25 24 23 22 21 20
63 64 65 66 67 68 69 70
-4.844417 -4.844417 -4.844417 -4.844417 -4.844417 -4.844417 -4.844417 -4.844417
4.844417 4.844417 4.844417 4.844417 4.844417 4.844417 4.844417 4.844417
-32.15558 -31.15558 -30.15558 -29.15558 -28.15558 -27.15558 -26.15558 -25.15558
33.877397 32.877397 31.877397 30.877397 29.877397 28.877397 27.877397 26.877397
-33.8774 -32.8774 -31.8774 -30.8774 -29.8774 -28.8774 -27.8774 -26.8774
-3.122603 -3.122603 -3.122603 -3.122603 -3.122603 -3.122603 -3.122603 -3.122603
29.03298 28.03298 27.03298 26.03298 25.03298 24.03298 23.03298 22.03298
21.662336 20.662336 19.662336 18.662336 17.662336 16.662336 15.662336 14.662336
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150 200
long put
short put
150 200
straddle
strangle
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71 72 73 74 75 76 77 78
0.0012734 0.002149 0.0035401 0.005698 0.0089697 0.0138224 0.0208702 0.0309009
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-0.001273 -0.002149 -0.00354 -0.005698 -0.00897 -0.013822 -0.02087 -0.030901
70.998727 71.997851 72.99646 73.994302 74.99103 75.986178 76.97913 77.969099
27.062773 26.07112 25.079983 24.089613 23.100357 22.112681 21.127201 20.144704
-27.06277 -26.07112 -25.07998 -24.08961 -23.10036 -22.11268 -21.1272 -20.1447
98.062773 98.07112 98.079983 98.089613 98.100357 98.112681 98.127201 98.144704
27.064046 26.073269 25.083523 24.095311 23.109326 22.126504 21.148071 20.175605
71 72 73 74 75 76 77 78
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79 80 81 82 83 84 85 86
0.0449024 0.0640857 0.0899027 0.1240561 0.1684998 0.2254284 0.2972537 0.386569
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79 80 81 82 83 84 85 86
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87 88 89 90 91 92 93 94
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95 96 97 98 99 100 101 102
2.4203981 2.8243152 3.268755 3.7538507 4.2793058 4.844417 5.4481069 6.0889666
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95 96 97 98 99 100 101 102
1.2648047 1.0595986 0.8826342 0.731078 0.602165 0.4932466 0.4018279 0.3255936
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103 104 105 106 107 108 109 110
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103 104 105 106 107 108 109 110
0.2624255 0.21041 0.1678395 0.1332072 0.1051976 0.0826739 0.064663 0.0503394
7.0277299 7.6856087 8.3843925 9.1203573 9.8899012 10.689579 11.516133 12.366506
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100 100 100 100 100 100 100 100
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111 112 113 114 115 116 117 118
13.198857 14.097513 15.010234 15.935265 16.870996 17.815971 18.76888 19.728557
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111 112 113 114 115 116 117 118
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111 112 113 114 115 116 117 118
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119 120 121 122 123 124 125 126
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119 120 121 122 123 124 125 126
0.004348 0.0032465 0.0024151 0.0017902 0.0013223 0.0009734 0.0007141 0.0005222
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119 120 121 122 123 124 125 126
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127 128 129 130 131 132 133 134
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28.538559 29.527033 30.516359 31.506368 32.496921 33.487906 34.479233 35.470828
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27 28 29 30 31 32 33 34
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135 136 137 138 139 140 141 142
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35 36 37 38 39 40 41 42
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143 144 145 146 147 148 149 150
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98.599234 98.606768 98.614287 98.621794 98.629292 98.636783 98.64427 98.651753
0.0002454 0.0001833 0.0001366 0.0001015 7.53E-05 5.573E-05 4.116E-05 3.033E-05
-0.0002454 -0.0001833 -0.0001366 -0.0001015 -7.53E-05 -5.573E-05 -4.116E-05 -3.033E-05
143.00025 144.00018 145.00014 146.0001 147.00008 148.00006 149.00004 150.00003
44.401011 45.393415 46.38585 47.378308 48.370783 49.363272 50.355771 51.348278
143 144 145 146 147 148 149 150
1.663E-06 1.16E-06 8.081E-07 5.617E-07 3.897E-07 2.699E-07 1.865E-07 1.287E-07
44.400768 45.393233 46.385714 47.378207 48.370709 49.363217 50.35573 51.348247
43 44 45 46 47 48 49 50
-43 -44 -45 -46 -47 -48 -49 -50
100 100 100 100 100 100 100 100
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
143 144 145 146 147 148 149 150
43 44 45 46 47 48 49 50
43 44 45 46 47 48 49 50
143 144 145 146 147 148 149 150
38.155583 39.155583 40.155583 41.155583 42.155583 43.155583 44.155583 45.155583
-38.155583 -39.155583 -40.155583 -41.155583 -42.155583 -43.155583 -44.155583 -45.155583
4.844417 4.844417 4.844417 4.844417 4.844417 4.844417 4.844417 4.844417
-3.1226027 -3.1226027 -3.1226027 -3.1226027 -3.1226027 -3.1226027 -3.1226027 -3.1226027
3.1226027 3.1226027 3.1226027 3.1226027 3.1226027 3.1226027 3.1226027 3.1226027
39.877397 40.877397 41.877397 42.877397 43.877397 44.877397 45.877397 46.877397
35.03298 36.03298 37.03298 38.03298 39.03298 40.03298 41.03298 42.03298
37.662336 38.662336 39.662336 40.662336 41.662336 42.662336 43.662336 44.662336
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Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Thom
reference number 9 classification 40401
Version 1.0 level basic Instructions
publication date Jul-03 1. The risk source of both the call and
author Hanyang Financial Engineering Lab. 2. Enter the option descriptions
affiliation Hanyang University input option terms and conditio
email address leesb@hanyang.ac.kr output option price and the sensilast revised date Jan-04
references Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Liabiliti
Ch. 4 of "The Oxford Guide to Financial Modeling" by Thomas S.Y. Ho and S
Descriptions
financial model class equity options: the Black-Scholes model
issuer/modeler market participants
model type continuous time model
risk sources stock price
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words the law of one price, no abitrage condition, dynamic hedging, hedge ratio, gre
Links
data N/A
financial models
Inputs
Stock Price (S) 100
Strike Price (K) 100
Time to expiration (T) 0.25
Stock volitility () 0.2
Risk-free rate (r) 0.1
Dividend yields () 0.03
Outputs
Dividend yields () 0 0.02
Call value 4.8444 5.2954 4.9920
Interim Calcuations
d1 0.2250
d2 0.1250
Call PutPrice 4.8444 3.1226
1( )
C S S
N dS C C
P
21
ln 2S K r T d
T
2 1d d T
1 2( ) ( )T rT
C S e N d K e N d
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as Ho Company, New York, NY. USA www. thomasho.com tom.ho@thomasho.com
put options and the underlying asset is the same.
s
tivities
ies, Journal of Political Economy , Vol.81, 637-654.
ang Bin Lee, 2004, Oxford University Press
eks
1( )
S S
N dC C
1 2( ) ( )T rT
P S e N d K e N d
1 2( ) ( )T rT
e N d K e N d
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Black-Scholes Option Pricing Model copyright THC 2003. All rights reserve
reference number 9 classification 40401
Version 1.0 level basic Instructions
publication date Jul-03 1. The risk source of both t
author Hanyang Financial Engineering Lab. 2. Enter the option descripti
affiliation Hanyang University input option terms
email address leesb@hanyang.ac.kr output option price
last revised date Jan-04
references Black, F. and M. Scholes, 1973, The Pricing of Options and Corpo
Ch. 4 of "The Oxford Guide to Financial Modeling" by Thomas S.Y
Descriptions
financial model class equity options: the Black-Scholes model
issuer/modeler market participants
model type continuous time model
risk sources stock price
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words the law of one price, no abitrage condition, dynamic hedging, hedg
Links
data N/A
financial models
Inputs
Stock Price (S) 100
Strike Price (K) 100
Time to expiration (T) 1
Stock volitility () 0.2
Risk-free rate (r) 0.04
Dividend yields () 0
Outputs
stock price 100 100.1
call value 9.9250537 9.9250537 9.9869401
S +( S2)/2 0.0618865
C 0.0618864
Interim Calcuations
d1 0.3000
d2 0.1000
Call
21
ln 2S K r T d
T
2 1d d T
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Price 9.9251
(Delta) 0.6179
(Gamma) 0.0191
v (Vega) 38.1388
(Theta) -5.8885
(Rho) 51.8661
(Omega) 6.2258sensitivity to dividend yield (f/) -61.7911
1 2( ) ( )
T rTC S e N d K e N d
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d by Thomas Ho Company, New York, NY. USA www. thomasho.com tom.ho@thomasho.com
e call and put options and the underlying asset is the same.
ons
and conditions
nd the sensitivities
ate Liabilities, Journal of Political Economy , Vol.81, 637-654.
. Ho and Sang Bin Lee, 2004, Oxford University Press
ratio, greeks
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1 2( ) ( )
T rTP S e N d K e N d
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Black-Scholes Option Pricing Model copyright THC 2003. All rights reserved by Tho
reference number 9 classificatio 40401
Version 1.0 level basic Instructions
publication date Jul-03 1. The risk source of both the call an
author Hanyang Financial Engineering Lab. 2. Enter the option descriptions
affiliation Hanyang University input option terms and conditi
email address leesb@hanyang.ac.kr output option price and the sen
last revised date Jan-04
references Black, F. and M. Scholes, 1973, The Pricing of Options and Corporate Lia
Ch. 4 of "The Oxford Guide to Financial Modeling" by Thomas S.Y. Ho an
Descriptions
financial model class equity options: the Black-Scholes model
issuer/modeler market participants
model type continuous time model
risk sources stock price
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words the law of one price, no abitrage condition, dynamic hedging, hedge ratio,
Links
data N/A
financial models
Inputs
Stock Price (S) 100
Strike Price (K) 100
Time to expiration (T) 0.25
Stock volitility () 0.2
Risk-free rate (r) 0.1
Dividend yields () 0.03
Outputs
4.8444 0.25 10
0.04 4.0782 21.2461
0.05 4.2005 24.2314
0.0300 0.1405
Interim Calcuations
d1 0.2250
d2 0.1250
Call Put
Price 4.8444 3.1226P
21
ln 2S K r T d
T
2 1d d T
1 2( ) ( )
T rTC S e N d K e N d
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as Ho Company, New York, NY. USA www. thomasho.com tom.ho@thomasho.com
put options and the underlying asset is the same.
ns
itivities
ilities, Journal of Political Economy , Vol.81, 637-654.
Sang Bin Lee, 2004, Oxford University Press
reeks
1 2( ) ( )
T rTP S e N d K e N d
1 2( ) ( )
T rTe N d K e N d
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7/29/2019 23Black-Scholes Model V3
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