FractalFinancial FINAL

7/28/2019 FractalFinancial FINAL

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Seiji Armstrong

Huy Luong

Alon Arad

Kane Hill


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Seiji Introduction , History of Fractal

Huy: Failure of the Gaussian hypothesis

Alon:

Fractal Market Analysis

Kane: Evolution of Mandelbrots financial models


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Sierpinski Triangle, D = ln3/ln2

1x

8x

Mandelbrot Set, D = 2


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Fractals are Everywhere:

Found in Nature and Art

Mathematical Formulation: Leibniz in 17th century

Georg Cantor in late 19th century

Mandelbrot, Godfather of Fractals: late 20th century

How long is the coastline of Britain

Latin adjective Fractus, derivation offrangere: to create irregular fragments.


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Locally random and Globally deterministic

Underlying Stochastic Process

Similar system to financial markets !


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Louis Bachelier - 1900 Consider a time series of stock price x(t) and designate L (t,T)

its natural log relative:

L (t,T) = ln x(t, T) ln x(t)

where increment L(t,T) is:

random statistically independent

identically distributed

Gaussian with zero mean

StationaryGaussianrandom walk


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0

2000

4000

6000

8000

10000

12000

14000

0 200 400 600 800 1000 1200 1400 1600 1800

Stock

Value

Time [day]

Dow Jones Index [Feb 97 - Nov 03]

0

2000

4000

6000

8000

10000

12000

14000

0 200 400 600 800 1000 1200 1400 1600 1800

Stock

Values

Time [day]

Brownian motion


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-2000

-1500

-1000

-500

0

500

1000

1500

0 500 1000 1500 2000

Dow Jones x(t+9) - x(t) Series

-1500

-1000

-500

0

500

1000

1500

0 500 1000 1500 2000

Brownian motion x(t +9) - x(t) Series


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1232

168

224

392411

239

140

0 20

50

100

150

200

250

300

350

400

450

-5SD -4SD -3SD -2SD -1SD +1SD +2SD +3SD +4SD +5SD

Frequency

Standard Deviation

Dow Jones Index Price Distribution Frequency [Feb 97 - Nov 03]


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Sample Variance of L(t,T) varies in time

Tail of histogram fatter than Gaussian

Large price fluctuation seen as outliers in Gaussian


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Analyzing fractal characteristics are highly desirable fornon-stationary, irregular signals.

Standard methods such as Fourier are inappropriate forstock market data as it changes constantly.

Fractal based methods .

Relative dispersional methods ,

Rescaled range analysis methods do not impose thisassumption


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In 1951, Hurst defined a method to study naturalphenomena such as the flow of the Nile River. Process wasnot random, but patterned. He defined a constant, K,

which measures the bias of the fractional Brownian

motion.

In 1968 Mandelbrot defined this pattern as fractal. Herenamed the constant K to H in honor of Hurst. The Hurst

exponent gives a measure of the smoothness of a fractalobject where H varies between 0 and 1.


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It is useful to distinguish between random and non-random data points.

If H equals 0.5, then the data is determined to berandom.

If the H value is less than 0.5, it represents anti-

persistence.

If the H value varies between 0.5 and 1, this representspersistence.


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Start with the whole observed data set that covers a totalduration and calculate its mean over the whole of theavailable data


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Sum the differences from the mean to get thecumulative total at each time point, V(N,k), from thebeginning of the period up to any time, the result is a time

series which is normalized and has a mean of zero


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Calculate the range


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Calculate the standard deviation


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Plot log-log plot that is fit Linear Regression Y on Xwhere Y=log R/S and X=log n where the exponent H isthe slope of the regression line.


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y = 0.8489x - 2.1265

-1

-1

0

1

1

2

2

2.00 2.50 3.00 3.50 4.00 4.50

ln(R/S)

ln(t)

Hurst Exponent


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Gaussian market is a poor model of financial systems.

A new model which will incorporate the key features ofthe financial market is the fractal market model.


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Paret power law and Levy stability

Long tails, skewed distributions

Income categories: Skilled workers, unskilled workers,part time workers and unemployed


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Reality: Temporal dependence of large and small pricevariations, fat tails

Pr(U > u) ~ u , 1 < < 2 Infinite variance: Risk

The Hurst exponent, H =

Brownian Motion P(t) = BH[(t)]; suitable subordinator

is a stable monotone, non decreasing, random processes withindependent increments

Independence and fat tails : Cotton (1900-1905), Wheat price

in Chicago, Railroad and some financial rates


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Fractional Brownian Motion (FBM)

Brownian Motion P(t) = BH[(t)]

The Hurst exponent, H

Scale invariance after suitable renormalization (self -affineprocesses are renormalizable (provide fixed points) ) underappropriate linear changes applied to t and P axes


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Global property of the processs moments

Trading time is viewed as (t) - called the cumulative

distribution function of a self similar random measure

Hurst exponent is fractal variant

Main differences with other models: 1. Long Memory in volatility

2. Compatibility with martingale property of returns

3. Scale consistency

4. Multi-scaling


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L1 = Brownian motion

L2 = M1963 (mesofractal)

L3 = M1965 (unifractal)

L4 = Multifractal models

L5 = IBM shares

L6 = Dollar-Deutchmark exchange rate

L7/8 = Multifractal models


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Neglecting the big steps

More clock time - multifractal

model generation.


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Mandelbrot (1960, 1961, 1962, 1963, 1965, 1967, 1972,1974, 1997, 1999, 2000, 2001, 2003, 2005)

All papers ofMandelbrots were used and analysed from1960 2005 and can be obtained from

www.math.yale.edu/mandelbrot

Fractal Market Anlysis: Applying Chaos theory toInvestment and Economcs (Edgar E. Peters) John Wiley& Sons Inc. (1994)
http://www.math.yale.edu/mandelbrothttp://www.math.yale.edu/mandelbrothttp://www.math.yale.edu/mandelbrothttp://www.math.yale.edu/mandelbrothttp://www.math.yale.edu/mandelbrothttp://www.math.yale.edu/mandelbrothttp://www.math.yale.edu/mandelbrothttp://www.math.yale.edu/mandelbrot

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FractalFinancial FINAL