application of multifractal geometry on financial...

$: Application of Multifractal Geometry on Financial Marketskmlinux.fjfi.cvut.cz/.../presentations/multifractals.pdf · 2011. 8. 29. · Financial Markets Jan Korbel ... Common properties$
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

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



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



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



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



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



Concrete example - financial markets

0

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lem [2008−01−02/2008−12−31]

Last 0.03

Volume (millions):6,557,000

0

100

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400

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



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



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



random walk for n=1000

Jan Korbel FJFI



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



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



representative trajectories of Wiener process

Jan Korbel FJFI



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




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):


ω(k, α) =


ln |k| if α = 1.


Jan Korbel FJFI




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):


ω(k, α) =


ln |k| if α = 1.


Jan Korbel FJFI




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):


ω(k, α) =


ln |k| if α = 1.


Jan Korbel FJFI



Lévy distributuion

Jan Korbel FJFI



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



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



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}


f (α) = limδ→0

limε→0


− log δ


Jan Korbel FJFI




Nδ,α(F ) = #{Ui |µ(Ui) ≥ δα}


f (α) = limδ→0

limε→0


− 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




Nδ,α(F ) = #{Ui |µ(Ui) ≥ δα}


f (α) = limδ→0

limε→0


− log δ


Jan Korbel FJFI



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



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


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



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


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



Wiener process - example

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-100

Jan Korbel FJFI



Lévy process - example

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Jan Korbel FJFI



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




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


α}

Jan Korbel FJFI




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


α}

Jan Korbel FJFI



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







2 - Wiener process

correlation is not zero:E(WH(t)WH(s)) = 1

2

[|t |2H + |s|2H − |t − s|2H]


Jan Korbel FJFI








2

[|t |2H + |s|2H − |t − s|2H]


Jan Korbel FJFI








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








2

[|t |2H + |s|2H − |t − s|2H]


Jan Korbel FJFI



ukázka fBM

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500 1000 1500 2000

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fBM for H = 0.3; 0.5; 0.6 a 0.7

Jan Korbel FJFI



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



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



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




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"



Jan Korbel FJFI




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"



Jan Korbel FJFI




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"



Jan Korbel FJFI



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




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




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




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



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




0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0


Jan Korbel FJFI



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



Time as multifractal

0.2 0.4 0.6 0.8 1.0

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0.02

0.03

0.04

0.05

0.06

0.07

0.2 0.4 0.6 0.8 1.0

0.2

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0.6

0.8

1.0

times difference dependence of times

Jan Korbel FJFI



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

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-2

2

4

0.2 0.4 0.6 0.8 1.0

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-40

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20

40

We can generate processes from view of trading time andtransform them to clock time

Jan Korbel FJFI



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



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



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



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 multifractal geometry on financial...

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