In finance, the price of an asset is given by the following formula.
r=returns
Pt = Price at time "t".
P0 = Actual Price.
Then.
$r=\ln (\frac{P_t}{P_0})$
We assume that prices are Log-normally distributed $\sim LogN(0,\sigma^2_p)$.
so, the distribution functions $$ F_r(r) = P(r\le r)=P[(ln(\frac{P_t}{P_0})\le r]=P[P_t\le (P_0e^r)] =F_P(P_0e^y), $$ thus the density can be found as its derivative w.r.t $y$, i.e., $$ f_r(r)=f_P(P_0e^r)*P_0e^r$$ $$=>\frac{P_0e^r}{P_0e^r\sqrt{2\pi \sigma^2}}\exp\{-(\ln P_0e^r)^2/(2\sigma^2)\}$$ $$=>\frac{1}{\sqrt{2\pi \sigma^2}}\exp\{(ln(P_0)-r)^2/(2\sigma^2)\}, $$ hence, $r \sim N(r,\sigma^2)$.
I know that this sequence or derivation is wrong, but I don't know where the error is. I would be very grateful if you could clarify the matter for me. Thank you very much.
When you assume that prices $P_t$ are lognormally distributed you can write this as $$\tag{1} P_t=P_0\,e^{\sigma_P W_t-\frac{\sigma_P^2t}{2}} $$ where $W_t$ is a standard Brownian motion, i.e. $W_t\sim N(0,t)$. This ensures $$ \mathbb E[P_t]=P_0\,,\quad \operatorname{Var}[P_t]=P_0^2\big(e^{2\sigma_p^2 t}-1\big)\,. $$ Then the return $$ \ln(P_t/P_0)=\sigma_P W_t-\frac{\sigma_P^2 t}{2} $$ is clearly normally distributed with mean $-\frac{\sigma_P^2 t}{2}$.
If you want the return to be normally distributed with mean zero you can achieve it in this framework by assuming that the price is $$\tag{2} P_t=P_0\,e^{\sigma_P W_t} $$ which leads to $$ \mathbb E[P_t]=P_0\,e^\frac{\sigma_P^2 t}{2}\,,\quad \operatorname{Var}[P_t]=P_0^2\big(e^{2\sigma_P^2 t}-e^{\sigma_P^2 t}\big). $$