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

enter image description here

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

enter image description here

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.

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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).

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  • $\begingroup$ Thanks for the answer. The price is log-normally distributed. The returns (both in % changes and in natural logarithm) are not normally distributed. The returns show a high kurtosis with fat tails. $\endgroup$ – Alfred Dec 6 '17 at 20:57
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There is an extensive paper on this. It is at

Harris, D.E.(2017) The Distribution of Returns. Journal of Mathematical Finance , 7, 769-804.

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.

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