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YouTube T4-10: Lognormal property of stock prices assumed by Black-Scholes

Nicole Seaman

Chief Admin Officer
Staff member
Thread starter #1
Although the Black-Scholes option pricing model makes several assumptions, the most important is the first assumption that stock prices follow a lognormal distribution (and that volatility is constant). Specifically, the model assumes that log RETURNS (aka, continuously compounded returns) are normally distributed, such that asset PRICES are lognormally distributed.

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Since ST follows log normal and LN(ST) follows normal distribution, so while calculating lower and upper limit for LN(ST), a Z score of 1.64 should be used as 1.64 is Z - score for 95 pc confidence level of a normal distribution. In the video ,You have used Z-score of 1.96 , which is applicable to log normal distribution i.e of ST. Here whole calculation is done by converting ST to LN(ST) so Z-scores of normal distribution should apply.

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @anind1 If we are crafting a two-sided confidence interval, the normal deviate is 1.96 because 2.5% of the area lies in each (up and down) rejection region, per NORM.S.INV(0.975) = 1.96.

If we want a one-sided CI, then we can use 1.645. Here is an example of a one-sided confidence interval, in this context: There is a 95.0% probability the future stock price, S(t), will be greater than $34.00 because $34.00 = exp(3.759 - 1.645*sqrt(0.020).

But the video is showing a (slightly more typical) two-sided CI such that, we would say, there is a 95.0% probability the future stock price, S(t), will lie in between two values: $32.51 < S(t) < $56.60.

Of course I agree that "Since ST follows log normal and LN(ST) follows normal distribution," as we are applying the normal Z deviate to the LN(X). Thank you!