Consider a 145-day put option at 30 on a stock selling at 27 with an annualized standard deviation of 0.30 when the continuously compounded risk-free rate is 4 percent. The value of the put option is

PT = [Xe-r (T) × (1 - N(d2))] - [ST × (1 - N(d1))]

where:

d1 = [ln(St / X) + [r + σ2/2](T) ] / σ √(T-t)

d2 = d1 - σ √(T)

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-----|-------------------------------------------------------------

0.3 | 0.6406 | 0.6443 | 0.648 | 0.6517

0.4 | 0.6772 | 0.6808 | 0.6844 | 0.6879

0.5 | 0.7123 | 0.7157 | 0.719 | 0.7224

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T=145/365 = 0.39726

d1 = [ln(27/30) + [.04 + .32/2](.39726)] / (.3√.39726)

= (-.10536052 + .0337671) / .18908569

= -.07159342 / .18908569

= -0.37863

d1 = -0.37863 ≈ -0.38 N(d1) = 1 -0.6480 = 0.3520

d2 = -0.37863 - .3√.39726

= -0.37863 - .18908569

= -.56771569

= -.56772

d2 = -0.56772 ≈ -0.57 N(d2) = 1 - 0.7157 = 0.2843

PT = 30e-.04(.39726) (1-.2843) – 27(1-.352)

= (29.527056 × .7157) – 17.496

= 21.1325 – 17.496

p = $3.64

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*closest*to: [round d1 and d2 rather than interpolate for N(.)].PT = [Xe-r (T) × (1 - N(d2))] - [ST × (1 - N(d1))]

where:

d1 = [ln(St / X) + [r + σ2/2](T) ] / σ √(T-t)

d2 = d1 - σ √(T)

**Cumulative Standard Normal Probability:**--------------------------------------------------------------------

0.06 | 0.07 | 0.08 | 0.09

0.3 | 0.6406 | 0.6443 | 0.648 | 0.6517

0.4 | 0.6772 | 0.6808 | 0.6844 | 0.6879

0.5 | 0.7123 | 0.7157 | 0.719 | 0.7224

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**A)**$3.32.**B)**$3.64.**C)**$3.97.**D)**$4.07.**Your answer: B was correct!**T=145/365 = 0.39726

d1 = [ln(27/30) + [.04 + .32/2](.39726)] / (.3√.39726)

= (-.10536052 + .0337671) / .18908569

= -.07159342 / .18908569

= -0.37863

d1 = -0.37863 ≈ -0.38 N(d1) = 1 -0.6480 = 0.3520

d2 = -0.37863 - .3√.39726

= -0.37863 - .18908569

= -.56771569

= -.56772

d2 = -0.56772 ≈ -0.57 N(d2) = 1 - 0.7157 = 0.2843

PT = 30e-.04(.39726) (1-.2843) – 27(1-.352)

= (29.527056 × .7157) – 17.496

= 21.1325 – 17.496

p = $3.64

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Is the above solution correct. If yes I did not understand why N(d1) was calculated as "1-" and again in the Put Price formula it was "1-". Please help - Thanks Nik

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