Power Tail Distributions

When we used a Pareto distribution for the continuous double auction model, we did not develop any rationale or theory about how such distributions might occur in financial markets.  William Reed, a mathematician, at the University of Victoria (see reference page) has developed a distribution which he calls the double Pareto lognormal distribution which may be a better representation of the distributions in order books, but more research is needed on this idea.

The simplest power tail distribution is the Pareto distribution; it is essentially only a tail and a defined starting point, before which the value is zero.  The density function for a Pareto distribution that starts at one and its plot are below.  α is the tail exponent.

f(x)= α (x^(-α-1))

Plot[PDF[ParetoDistribution[1,α]/.α-> 1.5,x],{x,0,5},PlotRange->All,PlotLabel-> "Pareto Distribution\nα = 1.5",Frame->True]

It is easy enough to generate Pareto random variables from a uniform random variable, because the distribution function has a simple formula which can be solved for x and p(x) is then a uniform random variable.

p(x)=(Piecewise[{{1-((((1/x)))^α), x>1}}])

But that doesn't give any insight as to how a Pareto distribution might arise in financial markets.  A Pareto random variable starting at one, can also be generated by taking the exponent (base e) of an Exponential random variable.  The Exponential distribution has a light tail; the mean and variance exist, but that is not true for its exponent when the parameter λ is less than 2 for the variance or 1 for the mean.

PDF[ExponentialDistribution[λ],x]
CDF[ExponentialDistribution[λ],x]
Mean[ExponentialDistribution[λ]]
Variance[ExponentialDistribution[λ]]
CharacteristicFunction[ExponentialDistribution[λ],t]

Here is the proof using characteristic functions.  The characteristic function of the exponent of an exponential random variable is:

cf=ExpectedValue[Exp[I t Exp[x]],ExponentialDistribution[λ],x,Assumptions-> t∈Reals]

This is the characteristic function of the ParetoDistribution[1, λ] as can be seen by doing the inversion of the Fourier Transform to get the density:

InverseFourierTransform[cf,t,x,Assumptions->{ λ>0,x>1},FourierParameters->{1,1}]

Exponential distributions are often used to characterize waiting times.  In the case of markets it might be a waiting time for a transaction, but if the value of the asset is appreciating exponentially while waiting, it is easy to see how prices might have some tendency toward a Pareto distribution.  In the next pages we explore some of the other distributions that Dr. Reed has described.

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