- #1

Master1022

- 611

- 117

- TL;DR Summary
- The stock price follows a log-normal distribution; the linear fractional change of the stock price follows a normal distribution. What about the volatility of the stock?

Hi,

I am not sure whether this is the right forum to post this. Please let me know if I should move and will do so.

[tex] m_1 = \frac{c_{t} - c_{t - 1}}{c_{t-1}} [/tex]

where ##c_t## is the closing price of the stock on day ##t##. The distribution came out looking something like: (

This looks somewhat fairly normal, at least from a visual standpoint. Then I thought:

After some searching around, it seems like yes, I do expect this value to be normally distributed. One explanation was provided on stack exchange (

So now I am calculating the volatility using the standard deviation of the log(returns) (i.e. ## log \left( \frac{c_t}{c_{t-1}} \right) ##). Then I used a rolling window of about 10 days for the standard deviation (and also annualized it, etc. - this part doesn't matter as much in terms of the shape of the distribution as its just a scaling factor). When I plotted this out in Python, I was getting a distribution looking like this:

This seems lognormal to me. However,

Any help would be greatly appreciated.

I am not sure whether this is the right forum to post this. Please let me know if I should move and will do so.

**Overall question:**Does the 'volatility' (i.e. standard deviation of the log returns) follow any sort of statistical distribution - maybe normal or log-normal?**Background/ Context:**I was looking at some stock data from yahoo finance and was plotting out some metrics for the data. For example, I looked at a group of technology companies, calculated a metric (let us call it ##m_1##) which was the linear change in closing price from the previous day:[tex] m_1 = \frac{c_{t} - c_{t - 1}}{c_{t-1}} [/tex]

where ##c_t## is the closing price of the stock on day ##t##. The distribution came out looking something like: (

*Note: it will take a very long time to make/read this post if I have to post all the code and the cleaning/processing steps)*This looks somewhat fairly normal, at least from a visual standpoint. Then I thought:

*does that make sense that I see a normal-looking distribution?*After some searching around, it seems like yes, I do expect this value to be normally distributed. One explanation was provided on stack exchange (

__here__) and it ends up (after some brief mathematics) by saying that stock returns are normally distributed.*From the post*So now I am calculating the volatility using the standard deviation of the log(returns) (i.e. ## log \left( \frac{c_t}{c_{t-1}} \right) ##). Then I used a rolling window of about 10 days for the standard deviation (and also annualized it, etc. - this part doesn't matter as much in terms of the shape of the distribution as its just a scaling factor). When I plotted this out in Python, I was getting a distribution looking like this:

This seems lognormal to me. However,

**is this what we expect?**I couldn't find any answers online (at least that were comprehensible by someone like me who has no stochastic/financial mathematics background). I didn't really have time to learn stochastic calculus just for this, but am just trying to understand if this aligns with expectation or not.Any help would be greatly appreciated.