INTRODUCTION TO

STABLE DISTRIBUTIONS

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Riccardo Donati fondatore Redexe R.M. & Finance

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"Insanity: doing the same thing over and over again and expecting different results",

A. Einstein

Vilfredo Federico Damaso Pareto (15 July 1848 – 19 August 1923), was an Italian engineer, sociologist, economist, and philosopher. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices.

He introduced the concept of Pareto efficiency and helped develop the field of microeconomics. He also was the first to discover that income follows a Pareto distribution, which is a power law probability distribution.

Paul Pierre Lévy (15 September 1886 – 15 December 1971) was a French mathematician who was active especially in probability theory. Lévy attended the École Polytechnique and published his first paper in 1905 at the age of 19, while still an undergraduate.

In 1920 he was appointed Professor of Analysis at the École Polytechnique, where his students included Benoît Mandelbrot. He remained at the École Polytechnique until his retirement in 1959.

DEFINITION

Let 𝑋1…𝑋𝑁 be independent copies of the random variable X;

let X distribution being non-degenerate;

X is said to be stable if there exist constants 𝑎 > 0 and 𝑏 such that

𝑋1 +⋯+ 𝑋𝑁 = a X + b (equality of distributions) for every 𝑋1…𝑋𝑁 (Feller 1971)

«A distribution is stable if its form is invariant under addition»

The stable distribution class is also sometimes referred to as the Lévy alpha-stable distribution or Pareto-Levy stable distribution.

PARAMETERIZATION

The Probability Density Function PDF of a stable distribution is not analytically expressible.

A parameterization of the Characteristic Function CF is (Nolan 2009, Voit 2003)

φ 𝑡, μ, 𝑐, α, β = exp [ 𝑖 𝑡μ − 𝑐 𝑡 α(1−i β sgn(t) Φ) +

Φ = tan (π

α

2) α ≠ 1

−2

πlog 𝑡 α = 1

α 𝜖 ]0,2] is the stability parameter

β𝜖 [−1,1] is the skewness parameter

𝑐 𝜖 ]0,∞[ is a scale parameter

μ 𝜖 ] − ∞,∞[ is the location parameter

Probability Density Function is the inverse Fourier transform of the CF

𝑓 𝑥 =1

2π φ 𝑡, μ, 𝑐, α, β 𝑒−𝑖 𝑥 𝑡∞

−∞

STABILITY PARAMETER

α measures “concentration”, α ∈]0,2]

PDF. Log Plot. β, μ = 0; c = 1 ; α = 2 α = 1.7 α = 1.5

PDF. β, μ = 0; c = 1 ; α = 2 α = 1.7 α = 1.5

For α = 2 the distribution reduces to a Normal distribution with variance σ2 = 2𝑐2, and β has no effect

𝑓 𝑥 =1

2 𝑐 π 𝑒−(𝑥−μ)2

4 𝑐2

For α =1 and β = 0 the distribution reduces to a Cauchy distribution

𝑓 𝑥 =𝑐

π

1

𝑐2+(𝑥−μ)2

SKEWNESS, SCALE AND LOCATION

β measures “asymmetry”, β ∈ [−1,1]

PDF. Log Plot. α = 1.5; μ = 0; c = 1 ; β = 0 β = −1 β = 1

For β = 0 the distribution is symmetric.

In such a case, φ 𝑡, μ, 𝑐, α, β = exp [ 𝑖 𝑡μ − 𝑐 𝑡 α+ and 𝑓 𝑥 =1

π 𝑒−(𝑐 𝑡)

αcos 𝑥 − μ 𝑡 𝑑𝑡

∞

0

c is just a scale parameter (dilatation)

μ is a just location parameter (translation)

PDF. α = 1.5; μ = 0; c = 1 ; β = 0 β = −1 β = 1

PROPERTIES

Descriptive Statistics

Mean: μ if α > 1, otherwise it is undefined.

Median, mode: μ if β = 0, otherwise they are not analytically expressible.

Variance: 2 𝑐2 when α = 2, otherwise it is infinite

Skewness, kurtosis: 0 if α = 2, otherwise they are undefined

If α < 2 (distribution is not normal), then distributions is leptokurtic and heavy-tailed.

Convolution of stable distributions (the probability distribution of the sum of two independent random stable variables ) having a fixed α is a stable itself with the some α.

Generalized central limit theorem. The sum of random variables with power tail distribution, decreasing as 1/ 𝑥 α +1 where 0 < α < 2 (so having infinite variance), will tend to a stable distribution with stability parameter α as the number of variables grows.

STABLE PROCESS - TIME SCALING

Let 𝑃𝑡 be a price time series; let log-returns of period 1, 𝑟𝑡 = log 𝑃𝑡 − log 𝑃𝑡−1 , be i.i.d. according to a symmetric stable distribution with zero mean, scale parameter c and stability parameter α.

How are log-returns 𝑅𝑡 of period T distributed?

𝑅𝑡 = log 𝑃𝑡 − log 𝑃𝑡−𝑇 =log 𝑃𝑡 − log 𝑃𝑡−1 + log 𝑃𝑡−1 − log 𝑃𝑡−2 +⋯+ log 𝑃𝑡−𝑇−1 − log 𝑃𝑡−𝑇

𝑅𝑡 = 𝑟𝑡 + 𝑟𝑡−1 +…+ 𝑟𝑡−𝑇

So 𝑅𝑡 is distributed according to a stable distribution, as a convolution of identical stable distributions.

It is easy to show that PDF of R (call it 𝑓𝑇 ) is 𝑓𝑇 (𝑥) = 𝑇−1

α𝑓(𝑇−1

α 𝑥), and the scale

parameter 𝑐𝑇 = 𝑇1

α 𝑐

It is a generalization of the Wiener scaling law 𝑓𝑇 (𝑥) =1

𝑇𝑓(

1

𝑇𝑥)

A RISK MANAGEMENT EXAMPLE

𝐴𝑛 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑜𝑓 𝑠𝑡𝑎𝑏𝑙𝑒 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 𝑖𝑛 𝐹𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙 𝑅𝑖𝑠𝑘 𝑀𝑎𝑛𝑎𝑔𝑒𝑚𝑒𝑛𝑡.

NORMAL FITTING

Let’s consider S&P 500 Index, daily close price log-returns from 3/1/1928 to 19/10/2010.

Mean is 1.922 10−4, standard deviation is 1.166 10−2.

Normal Distribution does not fit well empirical log-returns, standard error is 1.1 10−1

Empirical distribution normal best fit.

HUNTING FOR THE DISTRIBUTION

We don’t use a “try and see” approach, because it would be fragile.

Instead, we are seeking for a simple hypothesis and deduce consequences.

So let’s assume financial time series are invariant under time scaling (that is you can’t say if you have daily or weekly log-returns).

Given such assumption, the log-return distribution has to be stable.

B. Mandelbrot in 1963 has been the first to use stable distributions and statistics to study price time series (he focused on cotton price and deduced price is fractal and returns are stably distributed).

STABLE FITTING OVERVIEW

Let’s consider S&P 500 Index, daily close price log-returns from 3/1/1928 to 19/10/2010.

Calculate empirical distribution ED;

Fast Fourier Transform (ED);

Maximum Likelihood estimator;

Find out stable parameters.

STABLE FITTING

Maximum Likelihood method

α = 1.51, β = −8.35 10−2, c = 5.27 10−3, μ = 1.27 10−4

Standard error is 3.5 10−2 .

Empirical distribution normal best fit stable best fit.

CONVERGENCE OF PARAMETERS

Let’s consider the window *lastDate, lastDate - sessionsBack] in S&P 500 time series;

for each possible sessionsBack calculate the stable fitting and parameters;

Plot the chart (sessionsBack , parameter) to see how time series length influences the fitting.

α β

c μ

INFINITE VARIANCE

Let’s consider the previously defined window and calculate the Standard Deviation for

S&P 500 log-returns;

100’000 simulated log-returns (stable distributed i.i.d. with the same S&P’s stable parameters).

Lare jumps lead to sudden changes in the variance and it does not converge as the sample size grows. Althrough the variance does not converge, the distribution has a defined shape and you can calculate probabilities.

S&P 500 100′000 Steps Simulation

A BIT OF RISK MANAGEMENT

Consider the previous S&P 500 fitting.

The probability for -0.229 log-return is

2.45 10−84 % with normal distribution;

7.06 10−2 % with stable distribution.

Such event happened in 1987 (19/10/1987).

Var 1D 1%

normal distribution: −2.69 10−2

stable distribution: −4.13 10−2

Expected Shortfall 1D 1%

normal distribution: −3.09 10−2

stable distribution: −9.35 10−2

S&P 500, NASDAQ 100

S&P 500 Daily Close 3/1/1928 8/11/2010

𝜶 𝜷 𝒄 𝝁

1.51 −8.35 10−2 5.27 10−3 1.27 10−4

q = 0.05 q = 0.01 q = 0.001

V.A.R. −1.60 10−2 −4.13 10−2 −1.82 10−1

E.S. −3.73 10−2 −9.35 10−2 −3.15 10−1

Nasdaq 100 Daily Close 1/10/1985 8/11/2010

𝜶 𝜷 𝒄 𝝁

1.56 −1.78 10−1 9.11 10−3 4.07 10−6

q = 0.05 q = 0.01 q = 0.001

V.A.R. −2.74 10−2 −6.81 10−2 −2.86 10−1

E.S. −6.18 10−2 −1.52 10−1 −5.12 10−1

EURO BUND, COTTON

Bund Daily Close 4/1/1994 8/11/2010

𝜶 𝜷 𝒄 𝝁

1.79 −2.96 10−1 2.23 10−3 1.96 10−5

q = 0.05 q = 0.01 q = 0.001

V.A.R. −5.72 10−3 −1.05 10−2 −3.28 10−2

E.S. −9.81 10−3 −2.01 10−2 −6.21 10−2

Cotton Daily Close 2/1/1980 5/11/2010

𝜶 𝜷 𝒄 𝝁

1.70 3.74 10−2 9.40 10−3 2.26 10−4

q = 0.05 q = 0.01 q = 0.001

V.A.R. −2.45 10−2 −4.76 10−2 −1.65 10−1

E.S. −4.45 10−2 −9.65 10−2 −3.09 10−1

PROBABILITY CONE - STABLE

Let’s consider S&P 500, calculate stable parameters;

Consider the interval from 25/10/2006 to 19/10/2010 and plot:

log-time series,

CDF scaled according to Stable Scaling Law, colors are symmetric with respect to quantile 0.5.

PROBABILITY CONE - WIENER

Let’s consider S&P 500, calculate mean and standard deviation of log returns;

Consider the interval from 25/10/2006 to 19/10/2010 and plot:

log-time series,

CDF scaled according to Wiener Scaling Law, colors are symmetric with respect to quantile 0.5.

REMARKS

You can use stable distributions as a risk management framework for descriptive statistics of log returns.

Replace classic volatility definition (st. dev. of log returns) with the scale parameter.

You usually haven't got much data to calculate stable parameters, so stronger hypotheses are required

β = 0,

α comes from clustering and similarity.

Dynamics is not well described by an i.i.d. stable process, use instead

Energy driven ISING SPIN Models (Econophysics),

Mandelbrot Cartoons…

in order to take into account of other time series features such as heteroscedasticity, infrared dominance, jumps and so on.

Stable statistics is a good approximation, but it fails under certain conditions and time frames, when time series are no more invariant under time scaling, or higher precision is required.

Risk manager expertise is the most important thing.

WORKSHOP MATERIAL

www.redexe.net/riskmanagement/workshopStableStatistics

Download the Workshop Slides (EN) (4.5M).

Download the Introduction To Stable Distributions Workbook (EN) (0.04M).

Download the Example of Stable Distributions on Risk Management Workbook (EN) (0.3M).

Download the Invitation (IT) (0.07M).

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