From Quantum Randomness to Fat-Tails
and High-Peaked Distributions in
Financial Markets
Espen Gaarder Haug
www.espenhaug.com
NASDAQ MARKET FORUM
MILANO 21 Mai 2015
1
Property of probability
distribution
• Fat-Tail/High-peaked distribution
• Heavy Tailed distribution
• Leptokurtic distribution
• Non-Gasussian distribution
2
Relay too much on what we
think we know!
•  The Turkey Problem/ Juleribbe
problemet
•  Focus (also) on what you do not know!
3
The Gauss Curve/Math
•  1733 Abraham de Moivre
•  1783 Laplace extended on Moivre
•  1794 Carl Friedrich Gauss ( published
1809 )
4
Esprit Jouffret coined it the Bell curve in 1872
5
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.”
6
The Distribution of Galaxies
1990: finds high peak (and likely
fat-tails). Claims data must be
wrong!
2000: Finds out model is wrong
7
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
8
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.”
9
10
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
Number of Observations
85
Real returns
Normal
65
45
25
5.
0
-4 0%
1.
0
-3 0%
7.
0
-3 0%
3.
0
-2 0%
9.
0
-2 0%
5.
0
-2 0%
1.
0
-1 0%
7.
0
-1 0%
3.
00
-9 %
.0
0
-5 %
.0
0
-1 %
.0
0
3. %
00
7. %
00
11 %
.0
15 0%
.0
19 0%
.0
23 0%
.0
27 0%
.0
31 0%
.0
35 0%
.0
39 0%
.0
43 0%
.0
0%
5
-4
-15
Daily Returns
11
Figure 2: Year 2004 Forward/Swap Daily Returns
100
90
80
70
60
50
40
30
20
10
0
Daily returns
12
5.50%
5.00%
4.50%
4.00%
3.50%
3.00%
2.50%
2.00%
1.50%
1.00%
0.50%
0.00%
-0.50%
-1.00%
-1.50%
-2.00%
-2.50%
-3.00%
-3.50%
-4.00%
-4.50%
-5.00%
Real returns
Normal
-5.50%
Number of observations
Daily data from Jan 3 -2002 to Nov 3 -2004,
Volatility 15.4%, Skewness 0.15, Pearson kurtosis 4.9
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
Fischer Kurtosis = Pearson - 3 (Normal =0)
Excess Kurtosis
Kurtosis in practice very unstable: Mandelbrot
13
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 )
14
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…
15
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"
16
17
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.
18
Professor Bronzin 1908, Option Pricing:
19
Professor Bronzin 1908, Option Pricing:
20
Osborne (Physicists)
1958
Brownian Motion in
Stock Market
21
Osborne (1959) detects fat-tails in price
data, but basically ignores them and seems
to be a strong believer in normal distributed
returns."
22
Mandelbrot 1962
The Variation of Certain
Speculative Prices
23
If Gaussian
We can measure all risk by variance/
standard deviation σ.
Easy to make models.
Consistent models
Cost: we are loosing out on important
information if non-Gaussian
24
Some of the BIG Ideas in
Finance
CAPM: Based on Gaussian!
Sharpe Ratio: Based on Gaussian!
Black-Scholes-Merton: Based on Gaussian!
THEORY OF EVERYTHING (In finance)
But based on fantasy assumptions! 25
What Sharp Ratio Do You
Want Sir?
•  The problem with carry trades:
•  Sharpe ratio not well suited for options
in particular.
26
Trading Implications:
Trading:
Take advantage of extreme events.
Kurtosis trading typically involves
instruments with strongly convex payoff,
options.
Hedging Risk/Management: Avoid blow-ups
due to fat-tails.
27
Cause of Fat-Tails
•  Stochastic volatility
•  Jumps in assed prices (most important)
(systematic jumps versus nonsystematic jumps) (Merton 1976, Bates)
28
Derivatives Valuation
Merton (1976) jump diffusion, one of the first
to take into account fat-tails.
His model unfortunately not very practical!
But Yes it gives some great insight!
In 1980s 1990s and up to today massive
research into fixing our Gaussian models!29
Theoretical models
•  Stochastic volatility models and jumpdiffusion models are not robust, but give
us some insight.
•  More important than modeling is how to
build portfolio and how to hedge.
30
• 
Implied Volatility and logchanges JPYUSD 1W
USDJPY 1W ATM 1997- 2006
Log-Changes of Implied Volatilities
160
140
120
Real data
Gaussian
100
80
60
40
20
0
Daily log returns
31
Derivatives Valuation
Not Enough to Have Correct fat-tail
distribution.
Stable Paretian, Levy…
Also need a way to remove risk.
Dynamic replication fails completely for jumps.
32
Merton 1998
A broader, and still open, research issue is
the robustness of the pricing formula in the
absence of a dynamic portfolio strategy that
exactly replicates the payoffs to the option
security. "
33
What about jumps
Jumps at random time during option lifetime
100 000 simulations per run
34
Before Black-Scholes-Merton
Option valuation by discounting expected value
Bachelier/Sprenkle/Boness/Thorp (1964/1969)
formula: Exact Boness formula
35
Before Black-Scholes-Merton
Option valuation by discounting expected value
Bachelier/Sprenkle/Boness/Thorp (1964/1969)
formula: Exact Boness formula
36
Five attempts 1857, two in 1858,
1865, 1866 lasting connection
37
Arnold Bernhard & Co., 1970"
38
Options Arbitrage Between
London and New York
(Nelson 1904)
Up to 500 messages per hour and "
typically 2,000 to 3,000 messages per "
day where sent between the London and "
the New York market through the cable "
companies. Each message flashed over "
the wire system in less than a minute."
39
The Bachelier-Thorp formula
•  Haug and Taleb (2008): “Option Traders
Use (very) Sophisticated Heuristics,
Never the Black–Scholes–Merton
Formula”
•  Bachelier, Nelson, Higgins, Bronzin,
Sprenkel, Thorp,
40
We Do Not Know The Tail
Probability
The further out in tail, the less we know!
Higher uncertainty in probability.
Bigger Impact!
The more important the less we know!
41
Implications Risk Management
1. Diversify (systematic jumps happens less
often than non-systematic). Systematic jumps
somewhat smaller.
2. Reduce leverage to acceptable level (have
excess reserves)
3. Buy tail-insurance if necessary and possible.
4. Antifragile strategy. Taleb
42
Increases
collateral value
Can give bigger
mortgage
43
SOROS REFLEXIVITY
Cause of Fat-tails
Stochastic volatility, jump models are mainly curve fitting, do
not give us deep insight. But practical useful for modeling and
stress testing.
Jumps: partly due to government collect lots of information
before releasing information. (Haug: Taming the tails)..
Ultimately: all human behavior is related to physics.
44
Deeper cause of fat-tails in finance:
must be related to quantum
randomness, but how?
Atomism can explain everything
about: light, Doppler shift, energy,
matter, causality (and likely gravity).
What about fat-tails in financial
markets?
Low latency trading!
45
46
•  Single photon experiment: Perfect
binomial distributed. Perfect unbiased
coin-flipping machine.
•  All macroscopic phenomena we observe
must be aggregates of large numbers of
particles.
•  A little similar to stock index, 50 stocks up,
50 stocks down, index unchanged.
47
48
Causes of Fat-tails
1 Stochastic Time intervals
•  Not observing with uniform time interval.
•  Professor Clark 1970: Stochastic time
interval leads to fat-tails and high peak
relative to Gaussian. (see also Haug
2004).
2: Stochastic space window, stochastic
number of particles? Haug 2014
49
In Finance: Main driver for fat-tails
is stochastic space (number of
particles).
•  Obama speak we pay attention to a
“small” number of particles
•  Unemployment number consist of much
larger number of particles.
•  We are mixing “Gaussian” distributions.
•  Stochastic number of particles
•  Stochastic space size, stochastic
50
number of particles.
51
Fat-tails in art
52