“probability” distribution

26 Pages·2015·7.3 MB·English

From Quantum Randomness to Fat-Tails
and High-Peaked Distributions in
Financial Markets

Espen Ga arder Haug
www.espenhaug.com

NASDAQ MARKET FORUM
MILANO 21 Mai 2015

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Property of (cid:2)probability(cid:3)
distribution
• Fat-Tail/High-peaked distribution
• Heavy Tailed distribution
• Leptokurtic distribution
• Non-Gasussian distribution

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Relay too much on what we
think we know!
•  The Turkey Problem/ Juleribbe
problemet
•  Focus (also) on what you do not know!
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The Gauss Curve/Math
•  1733 Abraham de Moivre
•  1783 Laplace extended on Moivre
•  1794 Carl Friedrich Gauss ( published
1809 )
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Esprit Jouffret coined it the Bell curve in 1872
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Swedish astronomer and mathematician
Carl Vilhelm Ludwig Charlier 1920:

“The responsibility for stagnation in the development of
mathematical statistics until recent years rests
principally upon Gauss. The great mathematician
believed it possible to demonstrate that the fluctuations
of the items of a statistical series – he was concerned
chiefly with astronomical and geodesic observations –
followed the simple law which was called after him, the
Gaussian law of error. He believed the deviations from
the law were accidental and would disappear if the
observations increased.”
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The Distribution of Galaxies
1990: finds high peak (and likely
fat-tails). Claims data must be
wrong!

2000: Finds out model is wrong
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Arne Fisher 1922
“The great German mathematician -- or
rather the dogmatic faith in his authority
as a mathematician -- proved thus for a
number of years a veritable stumbling
block to a fruitful development of
mathematical statistics. ``

Fisher Kurtosis
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Arne Fisher 1922
“The Gaussian dogma held sway despite
the fact that the Danish actuary,
Opperman, and the French
mathematicians, Binemaye and Cournot,
have pointed out that several statistical
series, despite all efforts to the contrary,
offered a persistent defiance to the
Gaussian law.”
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Figure 1: Electricity Spot Daily Returns
Daily data from Nov 5- 2001 to Nov 3- 2004
Volatility 114.99%, Skewness 0.96, Pearson kurtosis 11.12
105
85
s
n
o
ati 65 Real returns
v
er Normal
s
b
O 45
of
r
be 25
m
u
N
5
% % % % % % % % % % % % % % % % % % % % % % %
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-015 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5. 1. 7. 3. 9. 5. 1. 7. 3. 9. 5. 1. 3. 7. 1. 5. 9. 3. 7. 1. 5. 9. 3.
4 4 3 3 2 2 2 1 1 - - - 1 1 1 2 2 3 3 3 4
- - - - - - - - -
Daily Returns
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Figure 2: Year 2004 Forward/Swap Daily Returns
Daily data from Jan 3 -2002 to Nov 3 -2004,
Volatility 15.4%, Skewness 0.15, Pearson kurtosis 4.9
100
90
s
n
o 80
ati 70 Real returns
v
r Normal
e 60
s
b 50
o
of 40
er 30
b
m 20
u
N 10
0
% % % % % % % % % % % % % % % % % % % % % % %
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5
5. 5. 4. 4. 3. 3. 2. 2. 1. 1. 0. 0. 0. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5.
- - - - - - - - - - -
Daily returns
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KURTOSIS was used
by Karl Pearson in 1905

Pearson Kurtosis (Normal =3)
If more flat-topped I term them platykurtic, if less flat-topped leptokurtic, and if

equally flat-topped mesokurtic(cid:2)
Fischer Kurtosis = Pearson - 3 (Normal =0)
Excess Kurtosis

Kurtosis in practice very unstable: Mandelbrot

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NOT ALL TAIL EVENTS ARE
EQUAL
Probability : low, but much higher
then most people expect.

Impact:
Typical very low for most real life
phenomena.
Finance: tail events typically MASSIVE
IMPACT ( Black Swan )
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FAT-TAILS BIG IMPACT EVENTS:

Tulip Mania - Crash 1637
South Sea Bubble - Crash 1720
Panic of 1819
Panic of 1873 Long Depression
Florida Real Estate Crash 1925
Crash of 1929 Great Depression
1973 Oil Crisis a quadrupling of oil prices
Crash of 1987
Asian financial crisis 1997, Thai currency
collapse
Argentina Peso Crash 2001
Financial crisis 2008 – 2011
Ruble 2014, Oil 2014…
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Fat-Tails in Price Data

Wesley Clair Mitchell
1874-1948
National Bureau of Economics Research

“The Making and Using of Index
Numbers” published in 1915"
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Mills rejects the Gaussian hypothesis."
"
“A distribution may depart widely from the
Gaussian type because the influence of
one or two extreme price changes.” "
"
Mills, F. C. (1927): The Behaviour of Prices. New York: National" Bureau of Economic Research, Albany: The
Messenger Press.
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Professor Bronzin 1908, Option Pricing:
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Professor Bronzin 1908, Option Pricing:
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