# Price is Log-normal distributed, yet the return is non-normal

I have a price series. The natural logarithm of the price shows good normality. As shown in the standardized normal probability plot below:

However, by viewing the standardized normal probability plot, the returns (or say, change of the price), do not show good normality.

My questions are why the price is log-normal, yet the return can be non-normal? And given the situation, what can be the feasible model to describe the price process?

Thank you in advance.

If I'm understanding you correctly, the log returns are normal, but the simple returns are not. While I'm surprised your plots are that different, simple returns will not be normal even if the log returns are normal; they will instead be (shifted) log-normal. If $S_t =S_0 e^{y\sqrt{t}}$ where $y$ is Gaussian, then the simple return is $\frac{S_{t+1}-S_{t}}{S_{t}}=\frac{S_{t+1}}{S_t}-1=e^{y}-1$, which is also log-normal (though shifted by 1).
It will provide you a methodology to calculate the distribution that should be present in section two of the paper. It does depend upon whether you calculate returns as $$r=\frac{p_{t+1}}{p_t},$$ or $$r=\log(p_{t+1})-\log(p_t)$$ or $$p_{t+1}=rp_t+\epsilon_{t+1}.$$ Each one generates a different answer.