Another old question on this site (How to simulate stock prices with a Geometric Brownian Motion?) inspired me to ask the following question: if we assume that regular returns could be normally distributed, doesn't that entirely invalidate the idea behind the GBM model?
And vice versa, if we like the GBM model and we assume that stock-prices are log-normally distributed, doesn't that imply that regular returns cannot be normally distributed?
Specifically:
Let's denote $R_i$ as regular returns and let's assume that these are normally distributed:
$$R_i=\frac{S_{i+1}-S_i}{S_i}=\mu \Delta t + \sigma W(t)$$.
Let's denote $r_i$ as log-returns, defined as $r_i = ln \left( \frac{S_{i+1}}{S_i} \right)$. Then:
$$ R_i = e^{r_i} - 1 $$
$$ r_i=ln(R_i+1) $$
If we assume that $R_i$ are normally distributed, then $ln(R_i+1)$ is undefined, because Normal distribution produces negative values and $ln(negative)$ is undefined.
(Edit: as per the comments below, I now realize this is a "stupid" thought since regular returns are trivially bounded below by -1, so the log can never be negative: I initially just focused on the hypothetical idea of regular returns being normally distributed, i.e. unbounded.
However the following point is still valid: if $R_i$ is assumed approximately "normally" distributed but bounded by -1 from below, then $ln(R_1 +1)$ still won't be log-normally distributed, so the claim that "assuming $R_i$ to be normally distributed invalidates the assumptions of the GBM model" still holds).
So by this reasoning, believers in the GBM model would argue: regular returns cannot be normally distributed, because we like the idea of stock prices being log-normal (i.e. we like that the future stock-price distribution conditioned on today's value is log-normal: cannot be negative & doesn't have an upper bound, which reflects the real-world behavior we'd expect from stocks). Therefore, based on the GBM model, regular returns have to be log-normally distributed (shifted by "-1").
Reasoning the other way, I am pretty sure that I have seen some papers (apologies, don't have a link and can't remember the name of the authors) that argue that empirical evidence suggests that regular returns are normally distributed. In fact, just a quick philosophical thought: why shouldn't they be? Human beings use regular returns to look at investments, NOT log returns. It would seem sensible at first thought that these regular returns can be negative as well as positive, with a large probability mass centered on zero (or inflation, if $\mu$= inflation): i.e. a "normal" distribution. So if we entertain the idea of regular returns to be normally distributed, that would seem to invalidate the idea of the GBM model.