Sums of random variables occur frequently in financial applications. Let's start with an example. Assume that monthly log returns of some asset are independent and normally distributed. The asset's one-year return will be the sum of 12 consecutive monthly returns. Will that one-year return also be normally distributed? The answer is "yes." This is because the 12 monthly returns are independent and normal, so they are joint-normal. A sum of joint-normal random variables must be normal. What if the monthly returns are not normal? Assume that they are independent and identically distributed. Their common distribution is not normal, but its mean and standard deviation exist. Will the annual return also have that same distribution? Now the answer is "no." This follows from the central limit theorem, which tell us that the distribution for the annual return will be approximately normal. This poses a problem for finance. We don't want to have to change our financial models each time we change our unit of time. If we assume that returns have a certain distribution over a day, we would like them to have the same distribution (perhaps with a different mean and standard deviation) over a month. If we assume the returns are normally distributed, there is no problem, but what if we want to assume some other distribution? This is the problem Benoit Mandelbrot contemplated in the early 1960s. It would lead to his groundbreaking (1963) paper suggesting that asset returns be modeled, not with the normal distribution assumed in the then-emerging random walk hypothesis, but with stable Paretian distributions. Mandelbrot was modeling cotton prices. His analysis of historical data indicated that returns had sample distributions that were highly leptokurtic. They had "fat tails" that made extreme market moves more likely than would be predicted by the normal distribution. This phenomena has been observed before, and today we know it is typical of most asset returns—stock, bond, commodity and energy returns routinely exhibit leptokurtosis. This is particularly extreme for energy returns.
Mandelbrot didn't want to have to assume one distribution for daily returns, another for monthly returns, and still another for annual returns. This would reduce any model to mere empirical distribution fitting. He wanted a consistent, flexible model that could be fit to different asset returns irrespective of the unit of time over which returns were calculated. This was possible with a model that assumed normally distributed returns, but normal distributions didn't fit historical data well. The problem was the central limit theorem, which tells us that sums of random variables will converge to a normal random variable. The only way to get around the central limit theorem was to depart from one of its two main assumptions:
Abandoning either would mean departing from the random walk hypothesis. The literature for that hypothesis gave strong empirical support for the first assumption, so Mandelbrot chose to depart from the second and consider distributions whose standard deviations don't exist. A probability distribution with distribution function
also has that same distribution function
As we have already noted, the central limit theorem suggests that the normal distribution is the only stable distribution whose standard deviation is defined. There are others whose standard deviations are not defined (or can be thought of as infinite). Paul Levy (1925, 1937) identified the general class of stable distributions. Just as a normal distribution can be specified with a mean μ and standard deviation σ, stable distributions can be specified with four parameters:
This is one standard parameterization that is commonly used. Other very similar parameterizations are also used, so be careful to check any author's definitions carefully. The distribution's mean exists so long as
There are only three cases in which a closed form expression is known for a stable distribution's probability density function. These are the
For this reason, theoretical work with stable distributions tends to be presented in terms of characteristic functions instead of probability density functions or distribution functions. However, density functions or distribution functions can always be valued using numerical techniques. The general formula for the characteristic function of a stable distribution is
where log denotes a natural logarithm and x/|x| is understood to equal 0 when x = 0. Non-normal stable distributions have "fat tails" that generally satisfy a convergence property defined by Vilfredo Pareto. For this reason, non-normal stable distributions are often called stable Paretian distributions. Despite their merits for modeling asset returns, stable Paretian distributions have remained a fringe topic in finance, occasionally mentioned by researchers but rarely implemented by practitioners. This may be because the mathematics of these distributions is more technical than that of more familiar distributions. Perhaps distributions with undefined standard deviations are simply too counter-intuitive for most people. Modest research is ongoing.
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that the random variables are independent, and
is said to be 
all having that distribution function
determines tail thickness. It satisfies 0 <
2. Generally, as
determines asymmetry. It satisfies –1
is a scale parameter satisfying
0, although the case
is a location parameter and can take on any
