Lognormal Stock Prices
It would be useful at this point to stop for a moment and collect a few important results. We established the process of geometric Brownian motion whereby stock returns follow the process:
We then used Ito's lemma to prove that for , it must be that the logarithm of prices, ln(St), has the following form:
These are both models of stock return dynamics, but they are clearly not the same. The difference is subtle with the log form having a small correction factor. In fact, as the variance of the returns gets arbitrarily small, the two are equivalent. Recall from Chapter 1 that the difference between the arithmetic mean and the geometric mean goes to zero with the variance of the random return. That logic is applicable to explaining the difference here as well and as we shall see, the assumption that the mean of geometric Brownian motion is equal to μ overstates the mean and the correction factor is required. Let's try to sketch out an explanation as to why this is the case. I will do this in two steps.
The first step will derive the mean of a lognormally distributed ...