The Fractal Market Hypothesis

J M Blackledge

Stokes Professor of DSP (ICT) Dublin Institute of Technologyhttp://eleceng.dit.ie/blackledge

What is Capitalism ?

FearGreed

Balance

The Price of Greed

Greed Fear

Balance

Contents of Presentation• What are Fractals?

• Random Fractal Walkers

• The Origins of Mathematical Finance

• The Fractal Market Hypothesis (FMH)

• Some Example Results?

• Questions

Fractal: Where does the word come from?

“The term fractal is 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 preserved in fragment”.

B Mandelbrot

What are Fractals?

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Euclidean objects Fractal objects

Example: The Sierpinski Triangle

• N=3, r=1/2 at each step• D=log(3)/log(2) =1.584962501...NrD = 1

Euclidean & Fractal Dimensions

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Self-Similarity:

Underlying Philosophy

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In every way one can see the shape of the sea

The Fractal Geometry of Nature

Fractals and Texture

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“Much of Fractal Geometry can be considered to be an intrinsic study of texture” B Mandelbrot

Fractal Clouds: D=2.1

Fractal Clouds: D=2.2

Fractal Clouds: D=2.3

Fractal Clouds: D=2.4

Fractal Clouds: D=2.5

Fractal Clouds: D=2.6

Fractal Clouds: D=2.7

Fractal Clouds: D=2.8

Fractal Clouds: D=2.9

Fractal Art: The CAD of Natural Objects

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Random Fractal Walkers

Brownian Motion

Random Fractal Walks with a Variable Hurst Exponent

H=0.6 H=0.7

H=0.8 H=0.9

Random Fractal Walks and Partial Differential Equations

• q=1: Diffusion Equation• q=2: Wave Equation• 1<q<2: Fractional (Fractal) Diffusion Equation

Hurst exponent H: Fourier dimension: 0<H<1 0<q<2; D=(5-2q)/2, 0.5<q<1.5

H>0.5 (q>1): random walk with persistenceand directional bias

Example: Diffusion of Light

Source imaged Source imaged Source convolvedthrough air through steam with a Gaussian PSF

Fractional Calculus

• L’hospital to Leibnitz (1695):‘Given that dnf/dtn exists for all integer n, what if n be 1/2'.

• Leibnitz to L’hospital:‘It will lead to a paradox ... From this paradox, one day useful consequences will be drawn'.

Fractional Integration

Origins of Mathematical Finance

The Theory of Speculation

Louis Bachelier, PhD Thesis, 1900• Used Brownian motion to

evaluate stock options.

• Basis for the EMF

Efficient Market Hypothesis

EMH .v. FMH:Principal Differences

Example of EMH:Black-Scholes Analysis

Assumes market is astationary Gaussian process!

Is an economy based on Stationary Gaussian Processes?FTSE: 1984-2007

Log price difference

White Gaussian noise

Does an economy have memory?

Absolute Log price difference

Autocorrelation function

Does an economy have repeating 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

Is an economy a random fractal walk?

0 200 400 600 800 1000 12001700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

Hilbert Transform

Unification of the Observations: The FMH with Black Swans

The FMH operator

q(t)=1.1q(t)=1.8

q(t)=1.1q(t)

Does the FMH work?FMH Simulation

Dow Jones Reality

EMH

Non-stationary Signal Processing

• Requires application of ‘moving windows’ to compute q(t) at different scales: Time-frequency methods in DSP

• Hypothesis:A change in q(t) precedes a change in a macroeconomic index (e.g. FTSE)

Equivalence with Wavelet Analysis

• Wavelet Transform

• FMH Transform

Example: FTSE (COD)19 March 2004 – 26 November 2008

q(t)

FTSE (Normalised)

Min on 10 May ‘07

Max on 19 June ‘07

Macrotrends

NormalisedGradients

Does it work over smaller times scales?

FTSE (Normalised): Jan-May, 2007

q(t)

Macrotrends

Gradients

Systems Development

Fuzzy logic

Knowledge Data Base

DSP

q[i+1]-q[i]<0: good marketq[i+1]-q[i]>0: poor market

q[i] Decision

Feature vector includes:• Information Dimension - ‘Fractal Entropy’• Correlation Dimension - ‘Fractal Matched Filter’• Multi-fractals - c.f. Statistical Moments

Example system: http://www.tradingsolutions.com

Featurevector

Economic signal

The Fractal Market Hypothesis

• Market dynamics are fractional dynamics

• Economic signals are random fractals with non-stationary characteristics

• FMH operator:

• Methods of solution given inApplication of the Fractal Market Hypothesis for ModellingMacroeconomic Time SeriesBlackledge J M, ISAST Transactions on Electronics and Signal Processing, No.1 Vol. 2, 89-110, 2008 (ISSN:1797-2329) http://www.isastorganization.org/ES2ready.pdf

Questions