Hi All,
So today I was goin over Bionic Turtle's material on the Black-Scholes Merton model and I noticed something that seemed inconsistent with other sources and which kind of threw me off. In the video as well as in the textbook David says that the mean of the log of a variable x which is log-normally distributed is log(X) = log(X0)+(u-(σ^2)/2).
This doesn't seem right to me as from previous probability classes I've always been left with the impression that the mean of a lognormally distributed variable would be log(X) = log(X0)+(u+(σ^2)/2). If we go by the second method the d1 and d2 of the bsm model seem to make a lot more sense as well as then d1 would represent the number of standard deviations we expect the spot price to be from the strike price at expiry and d2 would represent d1 minus the volatility we expect to be realized untill expiry. Cross-checking with other sources in the internet, I always encounter the formula for the mean of a log-normal distribution to be u+(σ^2)/2. Where am I wrong?
So today I was goin over Bionic Turtle's material on the Black-Scholes Merton model and I noticed something that seemed inconsistent with other sources and which kind of threw me off. In the video as well as in the textbook David says that the mean of the log of a variable x which is log-normally distributed is log(X) = log(X0)+(u-(σ^2)/2).
This doesn't seem right to me as from previous probability classes I've always been left with the impression that the mean of a lognormally distributed variable would be log(X) = log(X0)+(u+(σ^2)/2). If we go by the second method the d1 and d2 of the bsm model seem to make a lot more sense as well as then d1 would represent the number of standard deviations we expect the spot price to be from the strike price at expiry and d2 would represent d1 minus the volatility we expect to be realized untill expiry. Cross-checking with other sources in the internet, I always encounter the formula for the mean of a log-normal distribution to be u+(σ^2)/2. Where am I wrong?
