Post on 22-Jan-2021
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Application of Multifractal Geometry onFinancial Markets
Jan Korbel
Fakulta jaderná a fyzikálne inženýrská CVUT, Praha
1. 9. 2011
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Introduction
random processes - important part of modeling of manyproblems in physics, biology, economy, etc.
Basic model - random walk (Brownian motion)Brownian motion - the simplest process, but cannot beapplied for modeling of complex systemsOur aim is to find generalizations of Brownian motion, thatbetter describe real processesCommon properties of different processes - (multi)fractalgeometry
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Introduction
random processes - important part of modeling of manyproblems in physics, biology, economy, etc.Basic model - random walk (Brownian motion)
Brownian motion - the simplest process, but cannot beapplied for modeling of complex systemsOur aim is to find generalizations of Brownian motion, thatbetter describe real processesCommon properties of different processes - (multi)fractalgeometry
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Introduction
random processes - important part of modeling of manyproblems in physics, biology, economy, etc.Basic model - random walk (Brownian motion)Brownian motion - the simplest process, but cannot beapplied for modeling of complex systems
Our aim is to find generalizations of Brownian motion, thatbetter describe real processesCommon properties of different processes - (multi)fractalgeometry
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Introduction
random processes - important part of modeling of manyproblems in physics, biology, economy, etc.Basic model - random walk (Brownian motion)Brownian motion - the simplest process, but cannot beapplied for modeling of complex systemsOur aim is to find generalizations of Brownian motion, thatbetter describe real processes
Common properties of different processes - (multi)fractalgeometry
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Introduction
random processes - important part of modeling of manyproblems in physics, biology, economy, etc.Basic model - random walk (Brownian motion)Brownian motion - the simplest process, but cannot beapplied for modeling of complex systemsOur aim is to find generalizations of Brownian motion, thatbetter describe real processesCommon properties of different processes - (multi)fractalgeometry
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Concrete example - financial markets
0
10
20
30
40
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60
lem [2008−01−02/2008−12−31]
Last 0.03
Volume (millions):6,557,000
0
100
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Volatility() :0.482
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I 02 2008 III 03 2008 V 01 2008 VII 01 2008 IX 02 2008 XI 03 2008 XII 31 2008
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Concrete example
evolution of Lehman brothers’ shares in 2008 - rapid fallbefore the begin of financial crisisin model with random walk extremely improbablemany times larger standard deviation than normallyneed of other processes - modeling of extreme situations
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Brownian motion - Definition
classical random walk:p - probability of step to the rightq - probability of step to the leftn - number of steps
after n steps is the probability of the walker at position m:p(m,n) = n!
( n+m2 )!( n−m
2 )!p
n+m2 (1− p)
n−m2
limit for n→∞: according to Central limit theorem we getGaussian distribution
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Brownian motion - Definition
classical random walk:p - probability of step to the rightq - probability of step to the leftn - number of steps
after n steps is the probability of the walker at position m:p(m,n) = n!
( n+m2 )!( n−m
2 )!p
n+m2 (1− p)
n−m2
limit for n→∞: according to Central limit theorem we getGaussian distribution
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Brownian motion - Definition
classical random walk:p - probability of step to the rightq - probability of step to the leftn - number of steps
after n steps is the probability of the walker at position m:p(m,n) = n!
( n+m2 )!( n−m
2 )!p
n+m2 (1− p)
n−m2
limit for n→∞: according to Central limit theorem we getGaussian distribution
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
random walk for n=1000
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Wiener process - continuous random walk
Definition: stochastic process ξ(t) is the set of randomvariables parameterized by parameter t (often time)
stochastic process W (t) is called Wiener if:
1 W (0) = 0 almost surely,2 function t 7→W (t) is continuous a.s.,3 for all t , s: W (t)−W (s) ∼ N(0, |t − s|)4 W (t) has independent increments of t
Wiener process is almost nowhere differentiable!
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Wiener process - continuous random walk
Definition: stochastic process ξ(t) is the set of randomvariables parameterized by parameter t (often time)stochastic process W (t) is called Wiener if:
1 W (0) = 0 almost surely,2 function t 7→W (t) is continuous a.s.,3 for all t , s: W (t)−W (s) ∼ N(0, |t − s|)4 W (t) has independent increments of t
Wiener process is almost nowhere differentiable!
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Wiener process - continuous random walk
Definition: stochastic process ξ(t) is the set of randomvariables parameterized by parameter t (often time)stochastic process W (t) is called Wiener if:
1 W (0) = 0 almost surely,2 function t 7→W (t) is continuous a.s.,3 for all t , s: W (t)−W (s) ∼ N(0, |t − s|)4 W (t) has independent increments of t
Wiener process is almost nowhere differentiable!
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
representative trajectories of Wiener process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy distribution - definitionstable distribution is such, that :p(a1x + b1) ∗ p(a2x + b2) = p(ax + b)
Examples: Gauss distribution, Cauchy distribution...Theorem: stable distributions are limit distributions ofinfinite sums of random variablesgeneral for of stable distribution in Fourier image (Lévy,Kchintchin):
ln Lαβ(k) = ick − γ|k |α (1 + iβsgn(k)ω(k , α))where: 0 < α ≤ 2, −1 ≤ β ≤ 1, γ ≥ 0, c ∈ R,
ω(k, α) =
{tan(πα/2) if α 6= 12π
ln |k| if α = 1.
for α < 2 is variance infinite - class of α-stable Lévydistribution
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy distribution - definitionstable distribution is such, that :p(a1x + b1) ∗ p(a2x + b2) = p(ax + b)
Examples: Gauss distribution, Cauchy distribution...
Theorem: stable distributions are limit distributions ofinfinite sums of random variablesgeneral for of stable distribution in Fourier image (Lévy,Kchintchin):
ln Lαβ(k) = ick − γ|k |α (1 + iβsgn(k)ω(k , α))where: 0 < α ≤ 2, −1 ≤ β ≤ 1, γ ≥ 0, c ∈ R,
ω(k, α) =
{tan(πα/2) if α 6= 12π
ln |k| if α = 1.
for α < 2 is variance infinite - class of α-stable Lévydistribution
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy distribution - definitionstable distribution is such, that :p(a1x + b1) ∗ p(a2x + b2) = p(ax + b)
Examples: Gauss distribution, Cauchy distribution...Theorem: stable distributions are limit distributions ofinfinite sums of random variables
general for of stable distribution in Fourier image (Lévy,Kchintchin):
ln Lαβ(k) = ick − γ|k |α (1 + iβsgn(k)ω(k , α))where: 0 < α ≤ 2, −1 ≤ β ≤ 1, γ ≥ 0, c ∈ R,
ω(k, α) =
{tan(πα/2) if α 6= 12π
ln |k| if α = 1.
for α < 2 is variance infinite - class of α-stable Lévydistribution
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy distribution - definitionstable distribution is such, that :p(a1x + b1) ∗ p(a2x + b2) = p(ax + b)
Examples: Gauss distribution, Cauchy distribution...Theorem: stable distributions are limit distributions ofinfinite sums of random variablesgeneral for of stable distribution in Fourier image (Lévy,Kchintchin):
ln Lαβ(k) = ick − γ|k |α (1 + iβsgn(k)ω(k , α))where: 0 < α ≤ 2, −1 ≤ β ≤ 1, γ ≥ 0, c ∈ R,
ω(k, α) =
{tan(πα/2) if α 6= 12π
ln |k| if α = 1.
for α < 2 is variance infinite - class of α-stable Lévydistribution
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy distribution - definitionstable distribution is such, that :p(a1x + b1) ∗ p(a2x + b2) = p(ax + b)
Examples: Gauss distribution, Cauchy distribution...Theorem: stable distributions are limit distributions ofinfinite sums of random variablesgeneral for of stable distribution in Fourier image (Lévy,Kchintchin):
ln Lαβ(k) = ick − γ|k |α (1 + iβsgn(k)ω(k , α))where: 0 < α ≤ 2, −1 ≤ β ≤ 1, γ ≥ 0, c ∈ R,
ω(k, α) =
{tan(πα/2) if α 6= 12π
ln |k| if α = 1.
for α < 2 is variance infinite - class of α-stable Lévydistribution
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy distributuion
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractals - introductionfractal geometry deals with objects that have its inner andrepeating structure, that remains apparent when scaling.
informal definition: fractal = self-similar object defined by simplerules
fractal dimension - generalization of topological dimension forother objects than manifolds
possible way of dimension computation: Nδ(F ) - minimal no. ofballs with diameter δ, that cover Ffor manifolds with dimension n: Nδ(F ) ' cδ−n
Box counting dimension:
dimB F = limδ→0
ln Nδ(F )
ln 1δ
formal definition of fractal: fractal dimension is greater thantopological
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractals - introductionfractal geometry deals with objects that have its inner andrepeating structure, that remains apparent when scaling.
informal definition: fractal = self-similar object defined by simplerules
fractal dimension - generalization of topological dimension forother objects than manifolds
possible way of dimension computation: Nδ(F ) - minimal no. ofballs with diameter δ, that cover Ffor manifolds with dimension n: Nδ(F ) ' cδ−n
Box counting dimension:
dimB F = limδ→0
ln Nδ(F )
ln 1δ
formal definition of fractal: fractal dimension is greater thantopological
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractals - introductionfractal geometry deals with objects that have its inner andrepeating structure, that remains apparent when scaling.
informal definition: fractal = self-similar object defined by simplerules
fractal dimension - generalization of topological dimension forother objects than manifolds
possible way of dimension computation: Nδ(F ) - minimal no. ofballs with diameter δ, that cover Ffor manifolds with dimension n: Nδ(F ) ' cδ−n
Box counting dimension:
dimB F = limδ→0
ln Nδ(F )
ln 1δ
formal definition of fractal: fractal dimension is greater thantopological
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractals - introductionfractal geometry deals with objects that have its inner andrepeating structure, that remains apparent when scaling.
informal definition: fractal = self-similar object defined by simplerules
fractal dimension - generalization of topological dimension forother objects than manifolds
possible way of dimension computation: Nδ(F ) - minimal no. ofballs with diameter δ, that cover Ffor manifolds with dimension n: Nδ(F ) ' cδ−n
Box counting dimension:
dimB F = limδ→0
ln Nδ(F )
ln 1δ
formal definition of fractal: fractal dimension is greater thantopological
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractals - introductionfractal geometry deals with objects that have its inner andrepeating structure, that remains apparent when scaling.
informal definition: fractal = self-similar object defined by simplerules
fractal dimension - generalization of topological dimension forother objects than manifolds
possible way of dimension computation: Nδ(F ) - minimal no. ofballs with diameter δ, that cover Ffor manifolds with dimension n: Nδ(F ) ' cδ−n
Box counting dimension:
dimB F = limδ→0
ln Nδ(F )
ln 1δ
formal definition of fractal: fractal dimension is greater thantopological
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractals - introductionfractal geometry deals with objects that have its inner andrepeating structure, that remains apparent when scaling.
informal definition: fractal = self-similar object defined by simplerules
fractal dimension - generalization of topological dimension forother objects than manifolds
possible way of dimension computation: Nδ(F ) - minimal no. ofballs with diameter δ, that cover Ffor manifolds with dimension n: Nδ(F ) ' cδ−n
Box counting dimension:
dimB F = limδ→0
ln Nδ(F )
ln 1δ
formal definition of fractal: fractal dimension is greater thantopological
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
MultifractalsPresence of local fractal dimension - different dimensionsin different part of object
It is constructed on F with some arbitrary measure µ,µ(F ) = 1.Let Ui be minimal cover of F with balls of diameter δ.
Nδ,α(F ) = #{Ui |µ(Ui) ≥ δα}
We define f (α) - multifractal spectrum:
f (α) = limδ→0
limε→0
log(Nδ,α+ε(F )− Nδ,α−ε(F ))
− log δ
multifractal spectrum reflects particular weight of fractaldimensionsFor such α0 for that f ′(α0) = 0 holds, that f (α0) = dimB(F )
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
MultifractalsPresence of local fractal dimension - different dimensionsin different part of objectIt is constructed on F with some arbitrary measure µ,µ(F ) = 1.Let Ui be minimal cover of F with balls of diameter δ.
Nδ,α(F ) = #{Ui |µ(Ui) ≥ δα}
We define f (α) - multifractal spectrum:
f (α) = limδ→0
limε→0
log(Nδ,α+ε(F )− Nδ,α−ε(F ))
− log δ
multifractal spectrum reflects particular weight of fractaldimensionsFor such α0 for that f ′(α0) = 0 holds, that f (α0) = dimB(F )
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
MultifractalsPresence of local fractal dimension - different dimensionsin different part of objectIt is constructed on F with some arbitrary measure µ,µ(F ) = 1.Let Ui be minimal cover of F with balls of diameter δ.
Nδ,α(F ) = #{Ui |µ(Ui) ≥ δα}
We define f (α) - multifractal spectrum:
f (α) = limδ→0
limε→0
log(Nδ,α+ε(F )− Nδ,α−ε(F ))
− log δ
multifractal spectrum reflects particular weight of fractaldimensionsFor such α0 for that f ′(α0) = 0 holds, that f (α0) = dimB(F )
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
MultifractalsPresence of local fractal dimension - different dimensionsin different part of objectIt is constructed on F with some arbitrary measure µ,µ(F ) = 1.Let Ui be minimal cover of F with balls of diameter δ.
Nδ,α(F ) = #{Ui |µ(Ui) ≥ δα}
We define f (α) - multifractal spectrum:
f (α) = limδ→0
limε→0
log(Nδ,α+ε(F )− Nδ,α−ε(F ))
− log δ
multifractal spectrum reflects particular weight of fractaldimensions
For such α0 for that f ′(α0) = 0 holds, that f (α0) = dimB(F )
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
MultifractalsPresence of local fractal dimension - different dimensionsin different part of objectIt is constructed on F with some arbitrary measure µ,µ(F ) = 1.Let Ui be minimal cover of F with balls of diameter δ.
Nδ,α(F ) = #{Ui |µ(Ui) ≥ δα}
We define f (α) - multifractal spectrum:
f (α) = limδ→0
limε→0
log(Nδ,α+ε(F )− Nδ,α−ε(F ))
− log δ
multifractal spectrum reflects particular weight of fractaldimensionsFor such α0 for that f ′(α0) = 0 holds, that f (α0) = dimB(F )
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy processLévy analog Wiener process
Wiener process can be written as t1/2W , whereW ∼ N(0,1). Then
t1/21 W + t1/2
2 W d= (t1 + t2)1/2W
Strict α-stability criterion:
t1/α1 Y + t1/α
2 Y d= (t1 + t2)1/αY
Lévy α-stable process:
1 Lα(0) = 0 a.s.2 Lα(t) has independent increments of t3 Lα(t) is strictly α-stable process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy processLévy analog Wiener processWiener process can be written as t1/2W , whereW ∼ N(0,1). Then
t1/21 W + t1/2
2 W d= (t1 + t2)1/2W
Strict α-stability criterion:
t1/α1 Y + t1/α
2 Y d= (t1 + t2)1/αY
Lévy α-stable process:
1 Lα(0) = 0 a.s.2 Lα(t) has independent increments of t3 Lα(t) is strictly α-stable process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy processLévy analog Wiener processWiener process can be written as t1/2W , whereW ∼ N(0,1). Then
t1/21 W + t1/2
2 W d= (t1 + t2)1/2W
Strict α-stability criterion:
t1/α1 Y + t1/α
2 Y d= (t1 + t2)1/αY
Lévy α-stable process:
1 Lα(0) = 0 a.s.2 Lα(t) has independent increments of t3 Lα(t) is strictly α-stable process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy processLévy analog Wiener processWiener process can be written as t1/2W , whereW ∼ N(0,1). Then
t1/21 W + t1/2
2 W d= (t1 + t2)1/2W
Strict α-stability criterion:
t1/α1 Y + t1/α
2 Y d= (t1 + t2)1/αY
Lévy α-stable process:
1 Lα(0) = 0 a.s.2 Lα(t) has independent increments of t3 Lα(t) is strictly α-stable process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy processLévy analog Wiener processWiener process can be written as t1/2W , whereW ∼ N(0,1). Then
t1/21 W + t1/2
2 W d= (t1 + t2)1/2W
Strict α-stability criterion:
t1/α1 Y + t1/α
2 Y d= (t1 + t2)1/αY
Lévy α-stable process:1 Lα(0) = 0 a.s.2 Lα(t) has independent increments of t3 Lα(t) is strictly α-stable process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Wiener process - example
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-200
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Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Lévy process - example
-100 100 200 300
-300
-200
-100
100
200
300
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractal dimension of Lévy processes
representative trajectory of Wiener process in Rn (n ≥ 2)has fractal dimension 2
representative trajectory of Lévy α process in Rn (n ≥ 2)has dimension max{1, α}graph of random function t 7→W (t) has dimension 3
2
graph of random function t 7→ Lα(t) has fractal dimensionmax{1,2− 1
α}
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractal dimension of Lévy processes
representative trajectory of Wiener process in Rn (n ≥ 2)has fractal dimension 2representative trajectory of Lévy α process in Rn (n ≥ 2)has dimension max{1, α}
graph of random function t 7→W (t) has dimension 32
graph of random function t 7→ Lα(t) has fractal dimensionmax{1,2− 1
α}
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractal dimension of Lévy processes
representative trajectory of Wiener process in Rn (n ≥ 2)has fractal dimension 2representative trajectory of Lévy α process in Rn (n ≥ 2)has dimension max{1, α}graph of random function t 7→W (t) has dimension 3
2
graph of random function t 7→ Lα(t) has fractal dimensionmax{1,2− 1
α}
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractal dimension of Lévy processes
representative trajectory of Wiener process in Rn (n ≥ 2)has fractal dimension 2representative trajectory of Lévy α process in Rn (n ≥ 2)has dimension max{1, α}graph of random function t 7→W (t) has dimension 3
2
graph of random function t 7→ Lα(t) has fractal dimensionmax{1,2− 1
α}
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractional Brownian motion
process WH(t) se is called fractional Brownian motion(fBM), if:
1 WH(0) = 0 a.s.,2 WH(t) has independent increments of t3 WH(t)−WH(s) ' N(0, |t − s|2H)
H is called Hurst exponentFor H = 1
2 - Wiener processcorrelation is not zero:E(WH(t)WH(s)) = 1
2
[|t |2H + |s|2H − |t − s|2H]
fBM brings memory to random walkgraph of random function t 7→WH(t) has dimension 2− H-analogue to Lévy process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractional Brownian motion
process WH(t) se is called fractional Brownian motion(fBM), if:
1 WH(0) = 0 a.s.,2 WH(t) has independent increments of t3 WH(t)−WH(s) ' N(0, |t − s|2H)
H is called Hurst exponentFor H = 1
2 - Wiener processcorrelation is not zero:E(WH(t)WH(s)) = 1
2
[|t |2H + |s|2H − |t − s|2H]
fBM brings memory to random walkgraph of random function t 7→WH(t) has dimension 2− H-analogue to Lévy process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractional Brownian motion
process WH(t) se is called fractional Brownian motion(fBM), if:
1 WH(0) = 0 a.s.,2 WH(t) has independent increments of t3 WH(t)−WH(s) ' N(0, |t − s|2H)
H is called Hurst exponentFor H = 1
2 - Wiener process
correlation is not zero:E(WH(t)WH(s)) = 1
2
[|t |2H + |s|2H − |t − s|2H]
fBM brings memory to random walkgraph of random function t 7→WH(t) has dimension 2− H-analogue to Lévy process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractional Brownian motion
process WH(t) se is called fractional Brownian motion(fBM), if:
1 WH(0) = 0 a.s.,2 WH(t) has independent increments of t3 WH(t)−WH(s) ' N(0, |t − s|2H)
H is called Hurst exponentFor H = 1
2 - Wiener processcorrelation is not zero:E(WH(t)WH(s)) = 1
2
[|t |2H + |s|2H − |t − s|2H]
fBM brings memory to random walkgraph of random function t 7→WH(t) has dimension 2− H-analogue to Lévy process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractional Brownian motion
process WH(t) se is called fractional Brownian motion(fBM), if:
1 WH(0) = 0 a.s.,2 WH(t) has independent increments of t3 WH(t)−WH(s) ' N(0, |t − s|2H)
H is called Hurst exponentFor H = 1
2 - Wiener processcorrelation is not zero:E(WH(t)WH(s)) = 1
2
[|t |2H + |s|2H − |t − s|2H]
fBM brings memory to random walk
graph of random function t 7→WH(t) has dimension 2− H-analogue to Lévy process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Fractional Brownian motion
process WH(t) se is called fractional Brownian motion(fBM), if:
1 WH(0) = 0 a.s.,2 WH(t) has independent increments of t3 WH(t)−WH(s) ' N(0, |t − s|2H)
H is called Hurst exponentFor H = 1
2 - Wiener processcorrelation is not zero:E(WH(t)WH(s)) = 1
2
[|t |2H + |s|2H − |t − s|2H]
fBM brings memory to random walkgraph of random function t 7→WH(t) has dimension 2− H-analogue to Lévy process
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
ukázka fBM
500 1000 1500 2000
-10
10
20
30
40
50
60
500 1000 1500 2000
-10
10
20
30
40
50
60
500 1000 1500 2000
-10
10
20
30
40
50
60
500 1000 1500 2000
-10
10
20
30
40
50
60
fBM for H = 0.3; 0.5; 0.6 a 0.7
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Real behavior of financial marketsObserved properties of financial markets :
Large fluctuations
Memory
Different behavior for different seasons (prosperity, crisis,...)
Estimation of α and H by index S&P 500 in years 1985-2010:
1985 1990 1995 2000 2005 2010
1.5
1.6
1.7
1.8
1.9
2.0
1990 1995 2000 2005 2010
0.4
0.5
0.6
0.7
0.8
α parameter Hurst exponent
Necessity of processes with time-dependent parameters
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Real behavior of financial marketsObserved properties of financial markets :
Large fluctuations
Memory
Different behavior for different seasons (prosperity, crisis,...)
Estimation of α and H by index S&P 500 in years 1985-2010:
1985 1990 1995 2000 2005 2010
1.5
1.6
1.7
1.8
1.9
2.0
1990 1995 2000 2005 2010
0.4
0.5
0.6
0.7
0.8
α parameter Hurst exponent
Necessity of processes with time-dependent parameters
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Real behavior of financial marketsObserved properties of financial markets :
Large fluctuations
Memory
Different behavior for different seasons (prosperity, crisis,...)
Estimation of α and H by index S&P 500 in years 1985-2010:
1985 1990 1995 2000 2005 2010
1.5
1.6
1.7
1.8
1.9
2.0
1990 1995 2000 2005 2010
0.4
0.5
0.6
0.7
0.8
α parameter Hurst exponent
Necessity of processes with time-dependent parametersJan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Generating of processes with time-dependent Hurstexponent
volatility as stochastic process (double stochastic equation)
volatility as random variable with given distribution(superstatistics)Time as multifractal (stochastic) process"Trader’s time"and "Clock time"
Many trades are made just just the stock is open or beforethe stock is closedSudden losses cause sales (black days on stocks..)Volume differs over time
generation of multifractal stochastic time using brownianpatterns
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Generating of processes with time-dependent Hurstexponent
volatility as stochastic process (double stochastic equation)volatility as random variable with given distribution(superstatistics)
Time as multifractal (stochastic) process"Trader’s time"and "Clock time"
Many trades are made just just the stock is open or beforethe stock is closedSudden losses cause sales (black days on stocks..)Volume differs over time
generation of multifractal stochastic time using brownianpatterns
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Generating of processes with time-dependent Hurstexponent
volatility as stochastic process (double stochastic equation)volatility as random variable with given distribution(superstatistics)Time as multifractal (stochastic) process
"Trader’s time"and "Clock time"
Many trades are made just just the stock is open or beforethe stock is closedSudden losses cause sales (black days on stocks..)Volume differs over time
generation of multifractal stochastic time using brownianpatterns
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Generating of processes with time-dependent Hurstexponent
volatility as stochastic process (double stochastic equation)volatility as random variable with given distribution(superstatistics)Time as multifractal (stochastic) process"Trader’s time"and "Clock time"
Many trades are made just just the stock is open or beforethe stock is closedSudden losses cause sales (black days on stocks..)Volume differs over time
generation of multifractal stochastic time using brownianpatterns
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Generating of processes with time-dependent Hurstexponent
volatility as stochastic process (double stochastic equation)volatility as random variable with given distribution(superstatistics)Time as multifractal (stochastic) process"Trader’s time"and "Clock time"
Many trades are made just just the stock is open or beforethe stock is closedSudden losses cause sales (black days on stocks..)Volume differs over time
generation of multifractal stochastic time using brownianpatterns
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Wiener fractal pattern
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
1.Initiator
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Wiener fractal pattern
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
2.Generator
∆t = ∆x2
∆x = {23 ,
13 ,
23},∆t = {4
9 ,19 ,
49}
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Wiener fractal pattern
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
3.Recursive iteration
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Wiener fractal pattern
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
4.Fractal structure
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Multifractal pattern
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
other generators, random choice between generators in everyiteration
∆t = ∆xH(t)
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Multifractal pattern
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Multifractal pattern
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Time as multifractal
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Wiener pattern generates trading timeMultifractal pattern generates clock timethe shift of appropriate points in time generatesdependence of both times
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Time as multifractal
0.2 0.4 0.6 0.8 1.0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
times difference dependence of times
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Processes generated by multifractal patterns
Wiener process Multifractal process
0.2 0.4 0.6 0.8 1.0
0.995
1.000
1.005
0.2 0.4 0.6 0.8 1.0
1.000
1.005
1.010
1.015
1.020
1.025
0.2 0.4 0.6 0.8 1.0
-4
-2
2
4
0.2 0.4 0.6 0.8 1.0
-60
-40
-20
20
40
We can generate processes from view of trading time andtransform them to clock time
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Processes generated by fractal patterns
0.2 0.4 0.6 0.8 1.0
0.5
0.6
0.7
Hurst exponent
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Processes generated by fractal patterns
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.700.0
0.2
0.4
0.6
0.8
1.0
Multifractal spectrum
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Conclusion
Brownian motion an illustrative process, but it is not thebest for modeling of complex processesbetter description - Lévy process, fractional Brownianmotion...common properties of different processes - fractalgeometryMultifractal processes - easy modeling of difficultprocesses
Jan Korbel FJFI
Application of Multifractals on Financial Markets
Introduction Brownian motion Lévy distribution Fractal geometry Generalized random walks Multifractal processes
Kenneth Falconer.Fractal Geometry: Mathematical Foundations and Applications.Wiley, Inc., 1990.
Benoit B. Mandelbrot.Self-affine fractals and fractal dimension.Physica Scripta, 32:257–260, 1985.
Benoit B. Mandelbrot.Fractal financial fluctuations; do the threaten sustainability?In Science for Survival and Sustainable Development. Pontificia AcademiaScientiarum, 1999.
Rosario N. Mantegna and H. Eugene Stanley.An Introduction to Econophysics.CUP, Cambridge, 2000.
Wolfgang Paul and Jörg Baschangel.Stochastic Processes: From Physics to Finance.Springer, Berlin, 1999.
Jan Korbel FJFI
Application of Multifractals on Financial Markets