Fractal Markets

8/6/2019 Fractal Markets

1/88

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
http://eleceng.dit.ie/blackledgehttp://eleceng.dit.ie/blackledge


2/88

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


3/88

Financial Risk Management ?


4/88

What is Capitalism?

FearGreed

Balance


5/88

The Price of Greed

Greed Fear

Balance


6/88

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


7/88

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.pdf


8/88

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)


9/88

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


10/88

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


11/88

Euclidean & Fractal Dimensions

co ri ht


12/88

Fundamental Definitionof the Fractal Dimension


13/88

Fractal Types


14/88

Underlying Philosophy

co ri ht

In every way one can see the shape of the sea


15/88

Self-AffineStructures


16/88

Fractals and Texture

co ri ht

Much of Fractal Geometry can be consideredto be an intrinsic study of textureB Mandelbrot


17/88

Fractal Clouds: D=2.1


18/88



19/88


20/88



21/88



22/88



23/88


24/88



25/88



26/88

Random Walks & Hurst Processes

Brownian

Motion


27/88

Random Fractal Walks with aVariable Hurst Exponent

H=0.6 H=0.7

H=0.8 H=0.9


28/88

Random Walks and theFractional Diffusion Equation

q=1: Diffusion Equation

q=2: Wave Equation

1


29/88

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'.


30/88

Fractional Integration


31/88


32/88


33/88

Basic Model for a

Stationary Process

Let n(t) be a white noisesource and


34/88

Transformation Equation


35/88

Greens Function Solution


36/88

Power Spectrum Characteristics

log(Power Spectral Density Function)

= log(constant) + qlog(frequency)


37/88

Fundamental Solution(ignoring scaling constants)

Solution isstatisticallyself-affine


38/88

Characteristic Noise


39/88

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?


40/88


41/88

Evolution Equation for aBrownian Process

Describes the concentration of particles thatmove over a distance xwith probability p(x).

In Fourier space this equation is


42/88

Fractional Diffusion Equation

Consider the characteristic function

We can then write


43/88

Relationship between the LevyIndex and the Fourier Dimension

Greens function is given by

Implies the following relationship:


44/88

Summary

Random walks with a directional bias areHurst process fractional Brownian motion

Hurst processes are a generalisation ofBrownian motion and classical diffusion


45/88

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


46/88

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


47/88

Questions

+Interval (10 Minutes)


48/88

Part II: Contents


Properties of Financial Signals

The Fractal Market Hypothesis

Numerical Algorithms

Example Results

Case Study: ABX indices

Questions


49/88


The Theory ofSpeculation

Louis Bachelier, PhD

Thesis, 1900

Used Brownian motion to

evaluate stock options

Basis for the EMF

Efficient Market Hypothesis


50/88

Example of an EMH Model:The Black-Scholes Equation

Assumes market is astationary Gaussianprocess!


51/88

EMH .v. FMH:Principal Differences


52/88

Is an Economy Based on

Stationary Gaussian Processes?

FTSE: 1984-2007

Log price difference

White Gaussian noise


53/88

Does an Economy have Memory?

Absolute Log price difference

Autocorrelation function


54/88

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


55/88

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


56/88

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


57/88

Does the FMH work?

FMH Simulation

Dow Jones Reality

EMH

Pr[q(t)]=Gauss(x)


58/88

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)


59/88

Computing q(t)

Applying a moving window to the signal

(user defined)

For each window, assume a stationaryprocess where the signal is given by


60/88

Power Spectrum Method


61/88

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


62/88

Principal Algorithm


63/88

Principal Algorithm (continued)


64/88

The Wavelet Transform


65/88

FMH Wavelet


66/88

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.


67/88

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


68/88

Statistical Properties of q(t)


69/88

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)


70/88

E l FTSE (Cl f D )


71/88

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:


72/88

Case Study:ABX Index Analysis

Grades for the ABX Indices from 19 January 2006

to 2 April 2009 based on Close-of-Day prices.


73/88

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.


74/88

AAA: 24-07-2006 - 02-04-2009


75/88

AA: 24-07-2006 - 02-04-2009


76/88

A: 24-07-2006 - 02-04-2009


77/88

BBB: 24-07-2006 - 02-04-2009


78/88

BBB-: 24-07-2006 - 02-04-2009

Statistical Characteristics


79/88

Statistical Characteristicsof q(t) for ABX Indices


80/88

Wh t W t W ?


81/88

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


82/88

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)


83/88

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


84/88

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:


85/88

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/


86/88

Research Project Proposal 2:Currency Exchange Rate Analysis

http://www.augustus.co.uk/
http://www.augustus.co.uk/http://www.augustus.co.uk/


87/88

Q & A


88/88

Documents

Fractal Markets