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I calculated the distribution of the stock price changes (diffs). The diffs are multiplicative, $d_t=p_{t} / p_{t-1}$.

As far as I know the distribution should look like Power law distribution (Pareto distribution). With CDF being a line on log-log plot.

But the actual CDF is not looking like a line on log-log plot. Why?

I wonder could it be caused that price diffs distribution has two tails instead of one? It has two types of rare events, rare huge daily price drops with $d < 0.7$ and rare huge daily price rises with $d > 1.4$.

As far as I know the linear test for Power Law is used for one-tailed distributions. Like wealth distribution. Could it be also used for two-tailed distribution?

Example

The daily prices for 4 stocks for couple of years, normalised to be equal to 1 for the first day.

enter image description here

The CDF of daily diffs. The x axis is log scale, so the changes would look symmetrical around x = 1.

enter image description here

Let's plot it on log-log scale, and it's not looking like a line at all, nether one tail nor another.

enter image description here

On the previous log-log chart, one tail got crushed. So what I did instead I calculated two different CDFs, one for $d < 1$ and another for $d > 1$ and plotted it on log-log scale, so both tails could be seen. And there's same problem it's not looking like a line. Why?

enter image description here

P.S.

If it's not Pareto, what kind of Distribution could it be?

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For ~2 years of daily data for 3 stocks (SPY, AMD, BYND) a histogram of the ratio ($r_t=\frac{P_t}{P_{t-1}}$), simple returns ($r_t=\frac{P_t-P_{t-1}}{P_{t-1}}$), and log returns ($r_t=\log(P_t)-\log(P_{t-1})$) gives the following below. Fitting the distributions results in a variety of distributions, and 2 were Cauchy.

enter image description here enter image description here

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