Model Stock Market using normal and log-normal distributions

I'm aware that stock prices follow log normal distribution assuming the rate is continuously compounded and the return follows normal distribution, because stock prices cannot be negative.

What I do not understand is that why to model the stock market, it is more reasonable to assert that the rate of change of the stock price has normal distribution (compared to the stock price itself having normal distribution)?

• This question is probably a better fit on quant.stackexchange.com
– Flux
Apr 29 '20 at 5:10
• Stock prices do not follow log normal distribution. They have a reversion to the mean tendency over very long intervals. Log normal distribution does not have that tendency. Apr 29 '20 at 7:29

So, the stock market clearly has neither a normal nor a log-normal distribution. The distribution is a complicated mixture distribution that lacks a "first moment." In other words, it has no mean or variance. If you have enough technical math skills, this video will explain why it does not have that. https://youtu.be/R3fcVUBgIZw

If you do not have the math skills for advanced modeling, I will explain it in simpler terms.

The rate of return is FV/PV-1 for a single period of time. Prices are stochastic. I didn't use the word random because that is not quite correct. Since prices are in the numerator and the denominator, that makes returns a statistic and not data.

It turns out that the ratio of the two distributions has the property that there is no population mean, variance, kurtosis or skew. Furthermore, the distribution, even in log form, lacks a covariance matrix.

Because of this, any form of least squares regression will fail miserably.

The distribution lacks a sufficient statistic. Bayesian regression, if you use the appropriate distributions, will work wonderfully. You can use quantile regression if nothing else is available to you.

Models such as the CAPM or Black-Scholes collapse under this. Do not use them.

Because the distributions involved are outside the exponential family of distributions, there will not exist a point estimator that is a sufficient statistic. The Bayesian likelihood function is always minimally sufficient, so if the application is important you will want to use a Bayesian method.

The reason the normal distribution was chosen has nothing to do with the data. In the 1950s, the Office of Naval Research needed to perform calculations for nuclear weapons testing using Brownian motion and Brownian motion with drift. It was felt that if the stock market were a Brownian motion, then stock data would be cheaper to acquire than data from fission and fusion reactions.

As Wall Street did not then fund economists, it was the only game in town, so to speak. So economists went along and assumed normality. In 1963, the first warning was made by Benoit Mandelbrot that the data had heavy tails and could not have a mean. By 1973, falsification was complete.

The problem is that all econometric tools vanish in that case and no one was up to building new tools, so the assumptions have remained.

For real purposes, never ever use ordinary least squares, sample means or any normality or log-normality based solution.

If you are technically sophisticated, there is a new branch of stochastic calculus that solves all these problems. Go to https://www.datasciencecentral.com/profiles/blogs/a-generalized-stochastic-calculus

To get a visual understanding, consider the two graphs for Carnival Cruise Lines for daily returns (CCL) This first graph does not fit very well compared to the second graph below. The problem, however, isn't the fit. It is that the Cauchy distribution has terrible statistical properties.

Consider this following quote from the National Institute of Standards and Technology:

The Cauchy distribution is important as an example of a pathological case. Cauchy distributions look similar to a normal distribution. However, they have much heavier tails. When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of how sensitive the tests are to heavy-tail departures from normality. Likewise, it is a good check for robust techniques that are designed to work well under a wide variety of distributional assumptions. The mean and standard deviation of the Cauchy distribution are undefined. The practical meaning of this is that collecting 1,000 data points gives no more accurate an estimate of the mean and standard deviation than does a single point.

If you have a good statistical background, but none in Bayesian methods, then you will want to try and mentally wiggle lose from the standard thinking that you were trained in.

In standard Frequentist statistics, the parameters are fixed and the data is considered random due to chance. In Bayesian statistics, the data is considered fixed (you saw it after all, there can be no randomness to an historical event), while the parameters are considered random in the sense of possessing uncertainty. Chance is not part of Bayesian randomness, uncertainty is.

From a Frequentist perspective, a coin is fair or it is not fair. From a Bayesian perspective, the person tossing the coin has to be considered as well, even with a fair coin. A magician, con man or physicist tossing a coin is different from a small child. The magician should have no uncertainty as to what is seen by the audience, even with a perfectly fair coin. There is no randomness, only uncertainty as to how the system is really working.

That can result in radically different parameter estimates and solutions with exactly the same inputs.

A frequentist seeing 7,8,9 as inputs might describe the estimate of the mean as 8. The Bayesian may describe it as 25, even though it is outside the sample, because the Bayesian is allowed to bring in information from prior research. Maybe 7,8,9 are extreme outliers in the normal data collection. If you have seen a hundred other data generation processes, you may know 8 cannot be the answer.

• You are right that stock prices have a long tail and as a consequence that a mean does not really exist and perhaps that a normal standard deviation has less meaning than we think. This would mean that most economic models do not accurately describe reality. But what does this mean for, for example, simulating stock prices using Brownian motion (where we use sigma and mu)? Jun 25 '20 at 14:51

The return on prices are used for modelling purposes as they are constant with the absolute value of the price. This is simple to see as a change of \$1 on \$100 has much more effect on the value than the same change on \$1000.

It has been empirically determined that log returns are close to normally distributed by running tests of normality such as Jaques-Berra, Kolmogorov-Smirnov, or Shapiro-Wilk on historic data. With a little programming and statistics knowledge you can demonstrate this for yourself, indeed there is a DataCamp course in which you get lead through it. I say that they are close to normally distributed since there is a degree of skew to the distribution which means that they are closer to a Student distribution. The Student distribution can be thought of as being normal with skew.

The core reason why the (log) normal distribution of stock price returns is used is that it fits the data well but is not as costly to simulate and parameterize as using a log student distribution.

One partial explanation of why it fits is that it is related to the fact that there are a small number of low probability, high impact negative movements in stock prices giving the distribution fat tails. These are the occasional market crashes. Under non-crash circumstances the distribution is much closer to the normal distribution than it is to the lognormal distribution.

I'll offer a different kind of answer. Others have gone into how the lognormal model breaks down in reality. You already understand why lognormal is used instead of normal (because stock prices cannot be negative).

I think your question is simply why the starting point is to apply a distribution to changes in price (returns) rather than prices themselves. Well, unlike prices, changes can be independent and identically distributed from one period to the next.

Stock prices behave as a martingale because, according to the efficient market hypothesis, the current price determines the expectation of future prices. So stock prices look like some kind of random walk. Prices are (strongly) correlated between times even if returns are uncorrelated.

And if there were a stationary distribution of the price itself, it would mean that prices are mean-reverting (returns are negatively correlated), contrary to the efficient market hypothesis. (A non-tradable index such as the VIX can be mean-reverting, but it is not a price.)