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• 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)