You'll find the workshop material, including some videos, here: http://www.redexe.net/riskmanagement/workshopStableStatistics/ Riccardo Donati, founding member of Redexe Risk Management & Finance, held this workshop on November the 1st in Parma. He talked about Pareto-Lévy stable distributions, giving some examples of usage in Financial Risk Management: VAR, Shortfall Probability, and Probability Field. Normal statistics, even if still the core of most financial and risk management models, should be abandoned, as it dramatically underestimates ruin probability and rare events. When rare events occur, log-normal fans, asset managers and risk managers, talk about black swans and unpredictable conditions, but they are simply relying on a statistics far from empirical evidence. Audio track is in Italian Language.

1. INTRODUCTION TO
STABLE DISTRIBUTIONS
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1 dec 2010, Parma
Riccardo Donati fondatore Redexe R.M. & Finance
it.linkedin.com/in/riccardodonati
"Insanity: doing the same thing over and over again and expecting different results",
A. Einstein

2. 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…𝑋 𝑁 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 + ⋯ + 𝑋 𝑁 = a X + b (equality of distributions)
for every 𝑋1…𝑋 𝑁 (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.

3. 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
π
log 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
∞
−∞

4. 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

5. 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)α
cos 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

6. 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.

7. 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
α 𝑓(𝑇−
1
α 𝑥), and the scale
parameter 𝑐𝑇 = 𝑇
1
α 𝑐
 It is a generalization of the Wiener scaling law 𝑓𝑇 (𝑥) =
1
𝑇
𝑓(
1
𝑇
𝑥)

8. A RISK MANAGEMENT EXAMPLE
𝐴𝑛 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑜𝑓 𝑠𝑡𝑎𝑏𝑙𝑒 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠 𝑖𝑛 𝐹𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙 𝑅𝑖𝑠𝑘 𝑀𝑎𝑛𝑎𝑔𝑒𝑚𝑒𝑛𝑡.

9. 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.

10. 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).

11. 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.

12. 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.

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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.

19. 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.

20. 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.

21. 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).
 info@redexe.net