Download Fractal Financial Market Analysis

• Seiji Armstrong
• Huy Luong
• Alon Arad
• Kane Hill
•
Seiji
•
•
Introduction , History of Fractal
Huy:
• Failure of the Gaussian hypothesis
•
Alon:
• Fractal Market Analysis
•
Kane:
• Evolution of Mandelbrot’s financial models
1x
8x
Sierpinski Triangle, D = ln3/ln2
Mandelbrot Set, D = 2
• 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 of
frangere: to create irregular fragments.
• Locally random and Globally deterministic
• Underlying Stochastic Process
• Similar system to financial markets !
•
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
Stationary
Gaussian
random walk
Dow Jones Index [Feb 97 - Nov 03]
14000
12000
10000
Stock Value
8000
Brownian motion
14000
6000
12000
4000
10000
2000
0
0
200
Time [day]
400
600
800
Stock Values
8000
100060001200
1400
1600
1800
400
600
4000
2000
0
0
200
Time [day]
800
1000
1200
1400
1600
1800
Dow Jones x(t+9) - x(t) Series
1500
1000
500
0
0
500
1000
1500
2000
-500
-1000
Brownian motion x(t +9) - x(t) Series
1500
-1500
1000
-2000
500
0
0
-500
-1000
-1500
500
1000
1500
2000
Dow Jones Index Price Distribution Frequency [Feb 97 - Nov 03]
450
411
392
400
350
Frequency
300
250
239
224
200
168
140
150
100
50
32
12
0
2
+4SD
+5SD
0
-5SD
-4SD
-3SD
-2SD
-1SD
+1SD
Standard Deviation
+2SD
+3SD
•
Sample Variance of L(t,T) varies in time
•
Tail of histogram fatter than Gaussian
•
Large price fluctuation seen as outliers in Gaussian
• Analyzing fractal characteristics are highly desirable for
non-stationary, irregular signals.
• Standard methods such as Fourier are inappropriate for
stock market data as it changes constantly.
• Fractal based methods .
• Relative dispersional methods ,
• Rescaled range analysis methods do not impose this
assumption
• In 1951, Hurst defined a method to study natural
phenomena such as the flow of the Nile River. Process was
not 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. He
renamed the constant K to H in honor of Hurst. The Hurst
exponent gives a measure of the smoothness of a fractal
object where H varies between 0 and 1.
• It is useful to distinguish between random and non-
random data points.
• If H equals 0.5, then the data is determined to be
random.
• If the H value is less than 0.5, it represents anti-
persistence.
• If the H value varies between 0.5 and 1, this represents
persistence.
• Start with the whole observed data set that covers a total
duration and calculate its mean over the whole of the
available data
• Sum the differences from the mean to get the
cumulative total at each time point, V(N,k), from the
beginning of the period up to any time, the result is a time
series which is normalized and has a mean of zero
• Calculate the range
•
Calculate the standard deviation
• Plot log-log plot that is fit Linear Regression Y on X
where Y=log R/S and X=log n where the exponent H is
the slope of the regression line.
Hurst Exponent
2
y = 0.8489x - 2.1265
2
ln(R/S)
1
1
0
2.00
2.50
3.00
3.50
-1
-1
ln(t)
4.00
4.50
• Gaussian market is a poor model of financial systems.
• A new model which will incorporate the key features of
the financial market is the fractal market model.
•
Paret power law and Levy stability
•
Long tails, skewed distributions
•
Income categories: Skilled workers, unskilled workers,
part time workers and unemployed
• Reality: Temporal dependence of
large and small price
variations, fat tails
• Pr(U > u) ~ uα , 1 < α < 2
Infinite variance: Risk
• The Hurst exponent, H = ½
• Brownian Motion – P(t) = BH[ θ(t)]; ‘suitable subordinator’
is a stable monotone, non decreasing, random processes with
independent increments
• Independence and fat tails : Cotton (1900-1905), Wheat price
in Chicago, Railroad and some financial rates
•Fractional Brownian Motion (FBM)
• Brownian Motion – P(t) = BH[ θ(t)]
• The Hurst exponent, H ≠ ½
• Scale invariance – after suitable renormalization (self -affine
processes are renormalizable (provide fixed points) ) under
appropriate linear changes applied to t and P axes
•Global property of the process’s moments
• Trading time is viewed as θ(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
• L1 = Brownian motion
• L2 = M1963 (mesofractal)
• L3 = M1965 (unifractal)
• L4 = Multifractal models
• L5 = IBM shares
• L6 = Dollar-Deutchmark exchange rate
• L7/8 = Multifractal models
•
Neglecting the big steps
• More clock time - multifractal
model generation.
• Mandelbrot (1960, 1961, 1962, 1963, 1965, 1967, 1972,
1974, 1997, 1999, 2000, 2001, 2003, 2005)
• All papers of Mandelbrot’s were used and analysed from
1960 – 2005 and can be obtained from
www.math.yale.edu/mandelbrot
• Fractal Market Anlysis:
Applying Chaos theory to
Investment and Economcs (Edgar E. Peters) – John Wiley
& Sons Inc. (1994)