# An alternative to the Gaussian distribution to describe/fit market stock returns

After the financial crisis in 2008, many people (including me) don't really believe that stock returns can be described in terms of the normal distribution (Gaussian distribution).

But besides the Gaussian distribution, is there any other distribution that was found to be a better way of describing how the stock market behaves?

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You could try a Levy skew alpha-stable distribution with α = 1.8 and β = 0.931. Snark aside, it depends on your use-case , asset, etc. – Bob Jansen Mar 5 '14 at 6:19
@Gravitron Cont 2000 – user25064 Mar 6 '14 at 15:24
another nice paper :) – Probilitator Mar 26 '14 at 15:30

I think one has to distinguish between pricing and fitting/reproducing empirical stock returns. A model might fit the empirical stock returns extremly well but fail to reproduce derivative prices. In my answer I will assume that you are interested in reproducing the empirical stock returns.

Mandelbrot and the Stable Paretian Hypothesis

The most salient difference between the Normal distribution and the real world data are the heavy tails. This has been first noted (or at least publicly asserted) by Bernoit Mandelbrot and resulted in his Stable Paretian Hypothesis. Mandelbrot argued that contrary to the Gaussian-Haypothesis the variance of stock returns can be infinite resulting in heavy tails. He proposes the use of stable distributions with infinite variance - that also known as stable paretian distributions. (for more on the stable paretian distributions see once again the paper on elaborating on the Stable Paretian Hypothesis)

Thus most contemporary approaches focuse on designing distributions (or processes with transition densities) that feature heavy tails.

Concrete suggestions

First of all I don't think "the one" distribution for stock returns actually exists. Heavy-Tail ones will in most cases outperform the Normal distribution. The concrete choice may depend on the data one is working with. Still, there is a couple of examples worth mentioning.

The Heston-Model does a decent job in capturing the high kurtosis and the fat tails of stock returns. The paper Goodness-of-fit of the Heston model provides a detailes analysis. As it turns out the Heston-Model is well suited for reproducing daily and up to monthly returns but doesn't perform as well with annual stock returns. The original paper on the transition density of the HM can be found here. The authors also have their own paper discussing how well the Heston-Model performs with respect to empirical stock returns (Comparison between the probability distribution of returns in the Heston model and empirical data for stock indexes)

I can also wholeheartedly recommend the paper on the Goodness-of-fit of the Heston, Variance-Gamma and Normal-Inverse Gaussian Models It is very extensive, comprehensive and simultaneously provides a good introduction to the respective models

Due to it's fat tails the student-t distribution also performes reasonably well at reproducing market stock returns. As Aksakal has already mentioned in the comments below Student t is not a stable distribution. Still, there are some references using it to fit stock returns and even to construct a random walk. For a reference on the prior confer the paper by Eckhard Platen on Empirical Evidence on Student-tLog-Returns of Diversied World Stock Indices. A construction of student t random walk can be found in A Benchmark Approach to Quantitative Finance by Platen and Heath.

How to go from transition density to the distribution of returns

My explanation above did not calrify how knowing the transition density of a process also gives information on the distribution/density of the returns.

Assume a process $\ln(X_t)$ has the transition density $p(s,x,t,y)$. $p(s,x,t,y)$ can be interpreted as the probability of the process to move from $X_s=x$ at time $s$ to $X_t=y$ at time $t$. Thus if we set $s=0, x=0$ and $t=\Delta t$ (e.g. one day, one weel etc) we arrive at a density $p(0,0,\Delta t,y)$. This function describes the probability of the process to increass by $y$ within a time step of $\Delta t$. Thus depending on choice of $\Delta t$ wone is able to retrieve the theoretical. distribution of daily, montly, etc. returns.

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Nice references! Why are you not even mentioning Student t? Are there any major drawbacks? Do you know of comparisons with NIG and VG? – Quartz Mar 5 '14 at 14:49
@Quarts thanks for bringig up student t. I decided to stick to densitites/distributions that result from relevant processes that can be used to model stock returns. Still, there are some cases when the student-t happens to be the stationary density of such a process - I will edit the post accordingly :) – Probilitator Mar 5 '14 at 17:08
Student t isn't stable. – Aksakal Mar 6 '14 at 0:51
@Aksakal and why would that be a disadvantage? On the countrary, empirical distributions are well known to be far from stable, that's why one has to resort at least to hacks like tempered stable distributions. – Quartz Mar 6 '14 at 8:50
@Quartz, one issue is that random walk is defined as $x_t-x_{t-\Delta t}=\varepsilon_t$, so if you change the interval $\Delta t$ non-stable distributions change too. e.g. for some reason you decided to use Student t for monthly returns, now annual returns or weekly or daily are not Student t anymore. what is the theory here? why suddenly a monthly is Student t and all others are not? how dailies combine into monthly this way? when you think of this it becomes clear that you picked Student t simply because it's fat tailed – Aksakal Mar 6 '14 at 14:41

My take on the whole issue is as follows: We cannot be sure to find the one and only true model, the only thing we can do is to identify the most prevalent so called stylized facts and try to model them parsimoniously. The following paper was already mentioned in the comments:

Empirical properties of asset returns: stylized facts and statistical issues by Rama Cont (2000)

One route to go is to use more complicated yet better fitting distributions, the other route (the one I prefer and actively do research on) is to use mixtures of simpler distributions (e.g. normal distributions again). Most of the stylized facts mentioned in the abovementioned paper (p. 224) can be modeled by a simple mixture of normals. As a starting point I would recommend the following excellent paper:

Regime Changes and Financial Markets by Andrew Ang and Allan G. Timmermann (2011)

From the abstract:

[...] In empirical estimates, the regime switching means, volatilities, autocorrelations, and cross-covariances of asset returns often differ across regimes, which allow regime switching models to capture the stylized behavior of many financial series including fat tails, heteroskedasticity, skewness, and time-varying correlations. [...]

Have e.g. a look at figure 1 (p. 27): There you see how a mixture of just two normals can very well and parsimoniously model fat tails.

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I forgot about gaussian mixture models :) they are quite en vouge nowadays - +1 – Probilitator Mar 9 '14 at 15:51