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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