Black-Scholes OPM: Part I Calculate the Black-Scholes price for a sixmonth option given the following:
S_{0}=$60
E=$50
R_{f}=6%(per year)
d_{1}=.82
d_{2}=.74
Chapter 23
Q. 23A.S-T.1
Step-by-Step
Verified Solution
We need to evaluate the following:
C_{0}=\$60×N(.82)-50/1.06^{.5}×N(.74)
From Table 23A.1, the values for N(.82) and N(.74) are .7939 and .7704, respectively.
The value of the option is about $10.22. Notice that because the interest rate is quoted on an annual basis, we used a t value of .50, representing a half year, in calculating the present value of the exercise price.
Table 23A.1 Cumulative normal distribution
| N(d) | d | N(d) | d | N(d) | d | N(d) | d | N(d) | d | N(d) | d |
| .9515 | 1.66 | .8051 | 0.86 | .5239 | 0.06 | .2236 | -0.76 | .0571 | -1.58 | .0013 | -3.00 |
| .9535 | 1.68 | .8106 | 0.88 | .5319 | 0.08 | .2297 | -0.74 | .0594 | -1.56 | .0016 | -2.95 |
| .9554 | 1.70 | .8159 | 0.90 | .5398 | 0.10 | .2358 | -0.72 | .0618 | -1.54 | .0019 | -2.90 |
| .9573 | 1.72 | .8212 | 0.92 | .5478 | 0.12 | .2420 | -0.70 | .0643 | -1.52 | .0022 | -2.85 |
| .9591 | 1.74 | .8264 | 0.94 | .5557 | 0.14 | .2483 | -0.68 | .0668 | -1.50 | .0026 | -2.80 |
| .9608 | 1.76 | .8315 | 0.96 | .5636 | 0.16 | .2546 | -0.66 | .0694 | -1.48 | .003 | -2.75 |
| .9625 | 1.78 | .8365 | 0.98 | .5714 | 0.18 | .2611 | -0.64 | .0721 | -1.46 | .0035 | -2.70 |
| .9641 | 1.80 | .8414 | 1.00 | .5793 | 0.20 | .2667 | -0.62 | .0749 | -1.44 | .004 | -2.65 |
| .9656 | 1.82 | .8461 | 1.02 | .5871 | 0.22 | .2773 | -0.60 | .0778 | -1.42 | .0047 | -2.60 |
| .9671 | 1.84 | .8508 | 1.04 | .5948 | 0.24 | .2810 | -0.58 | .0808 | -1.40 | .0054 | -2.55 |
| .9686 | 1.86 | .8554 | 1.06 | .6026 | 0.26 | .2877 | –0.56 | .0838 | -1.38 | .0062 | -2.50 |
| .9699 | 1.88 | .8599 | 1.08 | .6103 | 0.28 | .2946 | -0.54 | .0869 | -1.36 | .0071 | -2.45 |
| .9713 | 1.90 | .8643 | 1.10 | .6179 | 0.30 | .3015 | -0.52 | .0901 | -1.34 | .0082 | -2.40 |
| .9726 | 1.92 | .8686 | 1.12 | .6255 | 0.32 | .3085 | -0.50 | .0934 | -1.32 | .0094 | -2.35 |
| .9738 | 1.94 | .8729 | 1.14 | .6331 | 0.34 | .3156 | -0.48 | .0968 | -1.30 | .0107 | -2.30 |
| .9750 | 1.96 | .8770 | 1.16 | .6406 | 0.36 | .3228 | -0.46 | .1003 | -1.28 | .0122 | -2.25 |
| .9761 | 1.98 | .8810 | 1.18 | .6480 | 0.38 | .3300 | -0.44 | .1038 | -1.26 | .0139 | -2.20 |
| .9772 | 2.00 | .8894 | 1.20 | .6554 | 0.40 | .3373 | -0.42 | .1075 | -1.24 | .0158 | -2.15 |
| .9798 | 2.05 | .8888 | 1.22 | .6628 | 0.42 | .3446 | -0.40 | .1112 | -1.22 | .0179 | -2.10 |
| .9821 | 2.10 | .8925 | 1.24 | .6700 | 0.44 | .3520 | -0.38 | .1151 | -1.20 | .0202 | -2.5 |
| .9842 | 2.15 | .8962 | 1.26 | .6773 | 0.46 | .3594 | -0.36 | .119 | -1.18 | .0228 | -2.00 |
| .9861 | 2.20 | .8997 | 1.28 | .6844 | 0.48 | .3669 | -0.34 | .123 | -1.16 | .0239 | -1.98 |
| .9878 | 2.25 | .9032 | 1.30 | .6915 | 0.50 | .3745 | -0.32 | .1271 | -1.14 | .025 | -1.96 |
| .9893 | 2.30 | .9066 | 1.32 | .6985 | 0.52 | .3821 | -0.30 | .1314 | -1.12 | .0262 | -1.94 |
| .9906 | 2.35 | .9099 | 1.34 | .7054 | 0.54 | .3897 | -0.28 | .1357 | -1.10 | .0274 | -1.92 |
| .9918 | 2.40 | .9131 | 1.36 | .7123 | 0.56 | .3994 | -0.26 | .1401 | -1.80 | .0287 | -1.90 |
| .9929 | 2.45 | .9162 | 1.38 | .7191 | 0.58 | .4052 | -0.24 | .1446 | -1.60 | .0301 | -1.88 |
| .9938 | 2.50 | .9192 | 1.40 | .7258 | 0.60 | .4129 | -0.22 | .1492 | -1.40 | .0314 | -1.86 |
| .9946 | 2.55 | .9222 | 1.42 | .7324 | 0.62 | .4207 | -0.20 | .1539 | -1.20 | .0329 | -1.84 |
| .9953 | 2.60 | .9251 | 1.44 | .7389 | 0.64 | .4286 | -0.18 | .1587 | -1.00 | .0344 | -1.82 |
| .9960 | 2.65 | .9279 | 1.46 | .7454 | 0.66 | .4365 | -0.16 | .1635 | -0.98 | .0359 | -1.80 |
| .9965 | 2.70 | .9306 | 1.48 | .7518 | 0.68 | .4443 | -0.14 | .1685 | -0.96 | .0375 | -1.78 |
| .9970 | 2.75 | .9332 | 1.50 | .7580 | 0.70 | .4523 | -0.12 | .1736 | -0.94 | .0392 | -1.76 |
| .9974 | 2.80 | .9357 | 1.52 | .7642 | 0.72 | .4602 | -0.10 | .1788 | -0.92 | .0409 | -1.74 |
| .9978 | 2.85 | .9382 | 1.54 | .7704 | 0.74 | .4681 | -0.80 | .1841 | -0.90 | .0427 | -1.72 |
| .9981 | 2.90 | .9406 | 1.56 | .7764 | 0.76 | .4761 | -0.60 | .1894 | -0.88 | .0446 | -1.70 |
| .9984 | 2.95 | .9429 | 1.58 | .7823 | 0.78 | .4841 | -0.40 | .1949 | -0.86 | .0465 | -1.68 |
| .9986 | 3.00 | .9452 | 1.60 | .7882 | 0.80 | .4920 | -0.20 | .2005 | -0.84 | .0485 | -1.66 |
| .9989 | 3.05 | .9474 | 1.62 | .7939 | 0.82 | .5000 | -0.00 | .2061 | -0.82 | .0505 | -1.64 |
| .9495 | 1.64 | .7996 | 0.84 | .5080 | -0.20 | .2119 | -0.80 | .0526 | -1.62 | ||
| .5160 | -0.04 | .2177 | -0.78 | .0548 | -1.60 |