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Summary of Variation of Certain Speculative Prices by Benoit Mandelbrot. Journal of Business (1963) pp. 394-419.

Mandelbrot cites the work of Louis Bachelier which formulated the random walk models for stock prices around 1900. Bachelier's work assumed a normal (Gaussian) distribution of relative price changes. Mandelbrot praises Bachelier's work but asserts that

the empirical distributions of price changes are usually too 'peaked' to be relative to samples from Gaussian populations.

Mandelbrot credits Wesley Claire Mitchell as the first (in 1915) to note this.

Mandelbrot formulates a theory based upon more general family of probability distributions discovered by the French mathematician Paul Levy.

Stable Distributions

Levy's distributions are stable in the sense that if random variables x and y come from the same type of Levy distribution then the random variable which is their sum, i.e. x+y, has a Levy distribution of the same type. The normal (Gaussian) distribution is a special case of a Levy distribution. It has the special distinction of being the only one with a finite variance.

Unfortunately the general form of the Levy distributions is not available. What is available is their characteristic function, which is also known as a moment generating function. For a distribution whose probability density function is f(x) the characteristic function is

Mf(z) = ∫−∞exp(izx)f(x)dx,

where i = [-1]1/2.

For the normal distribution with mean μ and variance σ2,

ln(M(z)) = iμz-½σ2.

If x and y are random variables whose characteristic functions are Mx(z) and My(z), respectively, then the characteristic function for x+y is Mx(z)My(y).

For the Levy distributions:

ln(M(z)) = iμz - va|z|a[1 -ib(sgn(z))tan(πa)/2)]

for a not equal to 1.

ln(M(z)) = iμz - v|z|[1+ib(sgn(z))(2/π)ln(|z|)]

for a equal to 1.

The parameter a is an index of the peakedness of a distribution and must be greater than 0 and less than or equal to 2. The Gaussian or normal distribution corresponds to a=2. The parameter b is an index of the skewness of a distribution and can range from -1 to +1.

If a=1 then b must be 0. The parameter v is a scale factor which must be nonnegative. In the case of the Gaussian or normal distribution v is one half of the variance. The remaining parameter is μ which corresponds to the mean of the distribution if a is greater than 1 and less than 2.

For the normal distribution the parameters are a=2, b=0, v= σ2/2, and μ = mean.

One case of a non-Gaussian Levy distribution is the Cauchy distribution, for which a=1, b=0.

f(x) = 1/[π(1+x2)].

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