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## Q. 23A.S-T.2

Black-Scholes OPM: Part II Calculate the Black-Scholes price for a nine-month option given the following:
$S_{0}$=$80 E=$70
σ=.30 (per year)
$R_{f}$=10% (per year)

## Verified Solution

We first calculate $d_{1}$ and $d_{2}$:
$d_{1}=[\ln(S_{0}/E)+(R_{f}+½)×σ^{2}×t]/(σ×\sqrt{t})$
=$[\ln(80/70)+(.10+½×.30^{2})×.75]/(.30×\sqrt{.75})$
=.9325
$d_{2}=d_{1}-σ×\sqrt{t}$
=$.9325-.30×\sqrt{.75}$
=.6727
From Table 23A.1, $N(d_{1})$appears to be roughly .825, and $N(d_{2})$ is about .75.
Plugging these in, we determine that the option’s value is \$17.12. Notice again that we used an annual t value of 9/12=.75 in this case.

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