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Financial Modelling using the
Fractal Market HypothesisJ M Blackledge
Stokes ProfessorDublin Institute of Technologyhttp://eleceng.dit.ie/blackledge
Distinguished Professor
Warsaw University of Technology
Lectures co-financed by the European Union in scope of the European Social Fund
Friday 5th March, 2010: 11:00-13:00
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What is the Problem?
Financial indices are digital signalscomposed of tick datacorresponding todifferent measures of an economy over arange of scales
Is it possible to analyse these signals insuch a way that a prediction can be made
on their likely future behaviour or trend?- Time Series Forecasting
- Systemic Risk Assessment
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Financial Risk Management ?
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What is Capitalism?
FearGreed
Balance
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The Price of Greed
Greed Fear
Balance
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FMH Approach to a Solution
The FMH operator:
q(t)=1.1q(t)=1.8
q(t)=1.1
q(t)Hypothesis: A change in q(t) precedes a
change in a financial signal
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Principal Papers
Application of the Fractal Market Hypothesis for ModellingMacroeconomic Time Series
Blackledge J M, ISAST Transactions on Electronics andSignal Processing, No.1, Vol. 2, 89-110, 2008
ISSN:1797-2329 http://eleceng.dit.ie/papers/106.pdf
Systemic Risk Assessment using a Non-stationaryFractional Dynamic Stochastic Model for the Analysis of
Economic Times Series, Blackledge J M, ISASTTransactions on Electronics and Signal Processing,
2010 (Submitted) http://eleceng.dit.ie/papers/148.pdf
http://eleceng.dit.ie/papers/106.pdfhttp://eleceng.dit.ie/papers/148.pdfhttp://eleceng.dit.ie/papers/148.pdfhttp://eleceng.dit.ie/papers/106.pdf8/6/2019 Fractal Markets
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Contents of Presentation I
Part I:
What are Fractals?
Random Walks and Hurst Processes
Random Walks and the Fractional Diffusion Equation
Fractional Calculus
Basic Model for a Stationary Process
Levy statistics and the Fractional Diffusion Equation
Questions
Interval (10 Minutes)
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Contents of Presentation II
Part II:
The Efficient Market Hypothesis
Properties of Financial Signals
The Fractal Market Hypothesis
Numerical Algorithms
Example Results
Case Study: ABX index analysis for the Bank of England
Summary and Research Project Proposals
Questions
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Part 1: What are Fractals ?
The term fractalis derived from the Latin adjectivefractus. The corresponding Latin verb frangeremeans to break, to create irregular fragments.In addition to fragmaneted fractus should also
mean irregular, both meanings being preservedin fragment.
B Mandelbrot
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Euclidean & Fractal Dimensions
co ri ht
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Fundamental Definitionof the Fractal Dimension
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Fractal Types
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Underlying Philosophy
co ri ht
In every way one can see the shape of the sea
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Self-AffineStructures
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Fractals and Texture
co ri ht
Much of Fractal Geometry can be consideredto be an intrinsic study of textureB Mandelbrot
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Fractal Clouds: D=2.1
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Fractal Clouds: D=2.2
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Fractal Clouds: D=2.4
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Fractal Clouds: D=2.5
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Fractal Clouds: D=2.6
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Fractal Clouds: D=2.8
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Fractal Clouds: D=2.9
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Random Walks & Hurst Processes
Brownian
Motion
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Random Fractal Walks with aVariable Hurst Exponent
H=0.6 H=0.7
H=0.8 H=0.9
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Random Walks and theFractional Diffusion Equation
q=1: Diffusion Equation
q=2: Wave Equation
1
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Fractional Calculus
Lhospital to Leibnitz (1695):
Given that dnf/dtn exists for all integer n,
what if n be 1/2'.
Leibnitz to Lhospital:
It will lead to a paradox ... From thisparadox, one day useful consequenceswill be drawn'.
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Fractional Integration
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Basic Model for a
Stationary Process
Let n(t) be a white noisesource and
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Transformation Equation
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Greens Function Solution
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Power Spectrum Characteristics
log(Power Spectral Density Function)
= log(constant) + qlog(frequency)
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Fundamental Solution(ignoring scaling constants)
Solution isstatisticallyself-affine
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Characteristic Noise
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Levy Statistics and theFractional Diffusion Equation
Levys question:
Under what circumstances does the
distribution associated with a random
walk of a few steps look the same as
the distribution after many steps
(except for scaling)?
Under what circumstances do we
obtain a random walk that is
statistically self-affine?
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Evolution Equation for aBrownian Process
Describes the concentration of particles thatmove over a distance xwith probability p(x).
In Fourier space this equation is
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Fractional Diffusion Equation
Consider the characteristic function
We can then write
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Relationship between the LevyIndex and the Fourier Dimension
Greens function is given by
Implies the following relationship:
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Summary
Random walks with a directional bias areHurst process fractional Brownian motion
Hurst processes are a generalisation ofBrownian motion and classical diffusion
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Summary (continued)
We have considered a model for a financial signalcompounded in the fractional PDE
The solution to this equation is, for
which is a random fractal signal with property
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In the Following Lecture
We shall investigate the properties offinancial signals
Consider a non-stationary model based on
the operator
Develop a moving window based algorithmto compute q(t) for a financial signal
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Questions
+Interval (10 Minutes)
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Part II: Contents
The Efficient Market Hypothesis
Properties of Financial Signals
The Fractal Market Hypothesis
Numerical Algorithms
Example Results
Case Study: ABX indices
Questions
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The Efficient Market Hypothesis
The Theory ofSpeculation
Louis Bachelier, PhD
Thesis, 1900
Used Brownian motion to
evaluate stock options
Basis for the EMF
Efficient Market Hypothesis
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Example of an EMH Model:The Black-Scholes Equation
Assumes market is astationary Gaussianprocess!
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EMH .v. FMH:Principal Differences
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Is an Economy Based on
Stationary Gaussian Processes?
FTSE: 1984-2007
Log price difference
White Gaussian noise
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Does an Economy have Memory?
Absolute Log price difference
Autocorrelation function
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Memory and FractionalDifferentiation
Differentiation of a function is a localised operation;it measures the gradient of a function at a point
Fractional differentiation depends on the historyofthe function its memory
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Does an Economy HaveRepeating Patterns? (Elliot Waves)
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 00 . 4 5
0 . 5
0 . 5 5
0 . 6
0 . 6 5
0 . 7
0 . 7 5
0 . 8
0 . 8 5
0 . 9
0 . 9 5
2004-08 (April)
1984-88 (April)
1994-98 (April)
Normalised FTSE (COD) Macrotrends
I E C ti l
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Is an Economy ContinuouslyStable at all Scales?
0 200 400 600 800 1000 12001700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
HilbertTransform
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Does the FMH work?
FMH Simulation
Dow Jones Reality
EMH
Pr[q(t)]=Gauss(x)
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Non-stationary Signal Processing
Requires application of moving windows to
compute q(t):Time-frequency methods in DSP
Hypothesis:A change in q(t) precedes a change in amacroeconomic index (e.g. FTSE)
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Computing q(t)
Applying a moving window to the signal
(user defined)
For each window, assume a stationaryprocess where the signal is given by
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Power Spectrum Method
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Data Fitting Methods
Least Squares Method: Minimise
Orthogonal Linear Regression (OLR)
Note: DC level is omitted from input as itmeasures the scale of the signal and notits fractal dimension
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Principal Algorithm
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Principal Algorithm (continued)
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The Wavelet Transform
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FMH Wavelet
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Example Result
Dow Jones Close-of-Day data from 02-11-1932 to 25-03-2009.
Dow Jones data (blue after normalization) and q(t) (red)computed using a window of size 1024.
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FTSE . V . DJ
DJ Close-of-Day data from25-04-1988 to 20-03-2009.
FTSE Close-of-Day data from25-04-1988 to 20-03-2009.
100 bin Histogram of q
100 bin Histogram of q
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Statistical Properties of q(t)
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Post-processing: Filtering q(t)
By low pass filtering q(t) macrotrends can bedetected.
Imperative that a filter is used that:
- guarantees smoothness
- shape preserving
- locally data consistent
Requires application of a
Variation Diminishing Smoothing Kernel (VDSK)
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E l FTSE (Cl f D )
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Example: FTSE (Close-of-Day)19 March 2004 26 November 2008
q(t)
FTSE (Normalised)
Min on 10 May 07
Max on 19 June 07
Macrotrends
Normalised
Gradients
Case Study:
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Case Study:ABX Index Analysis
Grades for the ABX Indices from 19 January 2006
to 2 April 2009 based on Close-of-Day prices.
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What is an ABX Index ?
The index is based on a basket of Credit Default Swap(CDS) contracts for the sub-prime housing equity sector.
Credit Default Swaps operate as a type of insurance policyfor banks or other holders of bad mortgages.
If the mortgage goes bad, then the seller of the CDS mustpay the bank for the lost mortgage payments.
Alternatively, if the mortgage stays good then the seller
makes a lot of money.
The riskier the bundle of mortgages the lower the rating.
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AAA: 24-07-2006 - 02-04-2009
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AA: 24-07-2006 - 02-04-2009
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A: 24-07-2006 - 02-04-2009
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BBB: 24-07-2006 - 02-04-2009
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BBB-: 24-07-2006 - 02-04-2009
Statistical Characteristics
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Statistical Characteristicsof q(t) for ABX Indices
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Wh t W t W ?
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What Went Wrong?The Human Factor
I am in blood,
steppd in so far that,
should I wade no more,
returning were as tedious as go oer.
Replace the word blood for debt and thetrend was set irrespective of the Hypothesis
S
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Summary
Market dynamics are fractional dynamics
Many economic signals are non-stationary randomscaling fractals
FMH operator:
FMH Hypothesis:
A change in q(t) precedes a change in amacroeconomic index
S (C ti d)
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Summary (Continued)
q(t) is computed using the OLR methodand a moving window
q(t) is filtered using a Gaussian VDSK toreveal macrotrends
q(t) provides a measure of
Bear .v. Bull
O P bl
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Open Problems
What is the best way to interpret q(t), i.e. how
should q(t) be post-processed to best reflect the FMH:
- statistical interpretation of Pr[q(t)] ?
- Bayesian analysis?
What other fractal measures could be used ?
What is the effect of considering a multi-dimensional
model with FMH operator:
Research Project Proposal 1:
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Research Project Proposal 1:Multi-Fractal Analysis
Fuzzy logic
KnowledgeData Base
DSP
Decision
Feature vector to include multi-fractal parameters q=1,2,
Feature
vector
Economic signal
http://www.tradingsolutions.com
Research Project Proposal 2:
http://www.tradingsolutions.com/http://www.tradingsolutions.com/8/6/2019 Fractal Markets
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Research Project Proposal 2:Currency Exchange Rate Analysis
http://www.augustus.co.uk/
http://www.augustus.co.uk/http://www.augustus.co.uk/8/6/2019 Fractal Markets
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Q & A
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