This is an example of using SmoothKernelDistribution to model stock price returns as discussed in the thread.

To download Mathematica notebook: CLICK HERE

Get some stock data for IBM over the last 10 years







That was just to show we can use the FractionalChange property and it gives us single day returns, including dividends.


Return histogram




Note the high peak and long tails, with much less weight in the midrange flanks than in a normal distribution.


Basic descriptive statistics


Note the high figure for the Kurtosis - a Gaussian distribution would show a 3.00 on that measure.

we can annualize the return using 251 market days in the average year.

For volatility, we annualize using the serial independence assumption, which implies the variation grows as the square root of the time. We have to use log returns to be symmetric in our treatment of proportional rises and falls.


SmoothKernel vs LogNormal



SK sample






Notice the excellent agreement of the smooth kernel distribution with the actual returns on Kurtosis and overall shape of the return histogram. But with only a single sample of the same length taken, the mean can easily "miss" by a significant amount.

It is easy to get excellent agreement on the Mean as well by just using 1000 sample runs of the same length, rather than just one.


LN sample






Notice, the lognormal distribution is able to get the first 3 moments of the distribution approximately correct, but the 4th - Kurtosis - is hopelessly low and the return histogram looks nothing like the real one as a result. It has way too much weight in the midrange flanks and not nearly enough in the small-change peak of the distribution.

@Michael - all true.

I also note that people cared about fitting theoretical distributions in part because they could solve, calculate, and simulate with them. But these days we can just use SmoothKernelDistribution fitted to past empiricals, and we get the tails that are actually there, without needing to fit this or that theoretical distribution that doesn't actually fit - and then sample and simulate and calculate with that fitted empirical.  (In the past people might have just bootstrapped - SmoothKernel gets us the benefits of that while not being tricked by a few outliers etc).

There is still high interest in the modeling, however, to give insight as to how and why we get the shapes we see empirically, where they come from causally, etc.

Sincerely,
Jason

The overall idea is quite promising, but I think 2 idealizations made in the process you outline prevent me from accepting your explanation for the observed tail characteristics.  Also you put in one power law rather than deriving it from the modeled market clearance dynamics, and if you correct the issues I mention below you might instead derive that one too, instead of putting it in.

These are the two idealizations - real orders do not come in a fixed block size.  They come in a distribution of sizes, and all the evidence I've seen (including that from agent based sims of trading systems) indicates that this is a significant contributor to returns kurtosis / derivations from lognormality.  So my first recommendation is to drop the chunking of blocks into 100 shares, that treats large orders as just a large number of 100 share orders, and instead to select order sizes from an empirically fitted order distribution, across all the scales of order sizes actually seen.

The second idealization is that market orders can always be filled without moving the market.  You have it easy due to the above idealization, basically - you always have some bid or ask that will truly fill any market order that hits your imagine system, because your block size is just 100 shares.  What actually happens is you have bids or asks for up to 1000 shares at X, and a market order for 5000 shares arrives, and blows through that bid or ask immediately, being filled only after the market has moved.  Because mechanically the way large market orders are filled is they sweep up all market or limit orders on the other side until they are filled, the price moves to the last required to fill that market order.

In your imagined system, only limit orders result in price movements, but in market reality, large market orders hitting limit orders as the other side / as needed to "fill" that market order, are the main events that move market prices.  And volatility spikes precisely when market orders of larger size than the tight limit bids and asks of the ordinary market makers, hit and exhaust those, and push the price in either direction deeper into the limit order books.

So, why recommendation would be - (1) drop the block size simplifications, (2) get block size and limit order sizes from real order books, (3) simulate with those details empirically right, and see if the rest of your analysis then results in power law tails as an emergent result.

I hope this is helpful.

Sincerely,
Jason Cawley