Is return normal or log normal? I read that return is normal and stock price is log normal. But I also read that return is log normal. So I am confused about which it is.
In the 14th Chapter of Options, Futures, and other Derivatives, John C Hull says

... the expected percentage return required by investors from a stock
is independent of stock's price...


A reasonable assumption is that the variability of the return in a short period of time, $\Delta t$, is the same regardless of the stock price.

Based on these two observations, he got the model :
$$\frac{\Delta S}{S}=\mu\Delta t + \sigma\epsilon\sqrt{\Delta t}$$
where $S$ is price, $\epsilon$ has a standard normal distribution. From this model, it seems like return has a normal distribution. This model also says $S$ follows a geometric Brownian motion and so we can apply Ito's lemma on some function of $S, t$. Applying Ito's Lemma to $lnS$ he arrived at the equation
$$lnS_T - lnS_0\sim N((\mu-\frac{\sigma^2}{2})T, \sigma^2 T)$$
His conclusion from this is that price follows a log normal distribution. But it seems that return $r$ is also normal since $lnS_T-lnS_0=ln\frac{S_T}{S_0}=lnr$ is normal.
My question is, assuming the proposed model is right, does return have a normal or log normal distribution? Or is it a matter of time span, for small $\Delta t$ return is normal, and for longer periods $T$ return is log normal?

The actual model here is the one described by the equation with the tilde (~). In that model:

*

*$S_t$ has a lognormal distribution for any $t$

*$S_t/S_0$ has a lognormal distribution for any $t$

*$S_t/S_0$ is approximately normal for any small $t$.

The equation with $\Delta$'s is less a definition of the model, and more a guideline to its numerical implementation.
