Seiji Armstrong Huy Luong Huy Luong Alon Arad Alon Arad Kane Hill Kane Hill

Seiji Seiji Introduction, History of Fractal Introduction, History of Fractal Huy: Huy: Failure of the Gaussian hypothesis Failure of the Gaussian hypothesis Alon: Alon: Fractal Market Analysis Fractal Market Analysis Kane: Kane: Evolution of Mandelbrot’s financial models Evolution of Mandelbrot’s financial models

Sierpinski Triangle, D = ln3/ln2 1x 8x Mandelbrot Set, D = 2

Fractals are Everywhere: Fractals are Everywhere: Found in Nature and Art Found in Nature and Art Mathematical Formulation: Mathematical Formulation: Leibniz in 17 th century Leibniz in 17 th century Georg Cantor in late 19 th century Georg Cantor in late 19 th century Mandelbrot, Godfather of Fractals: Mandelbrot, Godfather of Fractals: late 20 th century late 20 th century “How long is the coastline of Britain” “How long is the coastline of Britain” Latin adjective Fractus, derivation of frangere: to create irregular fragments. Latin adjective Fractus, derivation of frangere: to create irregular fragments.

Locally random and Globally deterministic Locally random and Globally deterministic Underlying Stochastic Process Underlying Stochastic Process Similar system to financial markets ! Similar system to financial markets !

Louis Bachelier Louis Bachelier Consider a time series of stock price x(t) and designate L (t,T) its natural log relative: 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) L (t,T) = ln x(t, T) – ln x(t) where increment L(t,T) is: where increment L(t,T) is: random random statistically independent statistically independent identically distributed identically distributed Gaussian with zero mean Gaussian with zero mean Stationary Gaussian random walk

Sample Variance of L(t,T) varies in time Sample Variance of L(t,T) varies in time Tail of histogram fatter than Gaussian Tail of histogram fatter than Gaussian Large price fluctuation seen as outliers in Gaussian Large price fluctuation seen as outliers in Gaussian

Analyzing fractal characteristics are highly desirable for non-stationary, irregular signals. 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. Standard methods such as Fourier are inappropriate for stock market data as it changes constantly. Fractal based methods. Fractal based methods. Relative dispersional methods, Relative dispersional methods, Rescaled range analysis methods do not impose this assumption 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 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. 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. 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 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 is less than 0.5, it represents anti- persistence. If the H value varies between 0.5 and 1, this represents 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 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 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 range

Calculate the standard deviation 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. 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.

Gaussian market is a poor model of financial systems. 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. A new model which will incorporate the key features of the financial market is the fractal market model.

Paret power law and Levy stability Paret power law and Levy stability Long tails, skewed distributions Long tails, skewed distributions Income categories: Skilled workers, unskilled workers, part time workers and unemployed Income categories: Skilled workers, unskilled workers, part time workers and unemployed

Reality: Temporal dependence of large and small price variations, fat tails Reality: Temporal dependence of large and small price variations, fat tails Pr(U > u) ~ u α, 1 u) ~ u α, 1 < α < 2 Infinite variance: R isk The Hurst exponent, H = ½ The Hurst exponent, H = ½ Brownian Motion – P(t) = B H [ θ(t)]; ‘suitable subordinator’ is a stable monotone, non decreasing, random processes with independent increments Brownian Motion – P(t) = B H [ θ(t)]; ‘suitable subordinator’ is a stable monotone, non decreasing, random processes with independent increments Independence and fat tails : Cotton ( ), Wheat price in Chicago, Railroad and some financial rates Independence and fat tails : Cotton ( ), Wheat price in Chicago, Railroad and some financial rates

Fractional Brownian Motion (FBM) Fractional Brownian Motion (FBM) Brownian Motion – P(t) = B H [ θ(t)] Brownian Motion – P(t) = B H [ θ(t)] The Hurst exponent, H ≠ ½ 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 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 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 Trading time is viewed as θ(t) - called the cumulative distribution function of a self similar random measure Hurst exponent is fractal variant Hurst exponent is fractal variant Main differences with other models: Main differences with other models: 1. Long Memory in volatility 1. Long Memory in volatility 2. Compatibility with martingale property of returns 2. Compatibility with martingale property of returns 3. Scale consistency 3. Scale consistency 4. Multi-scaling 4. Multi-scaling

L1 = Brownian motion L1 = Brownian motion L2 = M1963 (mesofractal) L2 = M1963 (mesofractal) L3 = M1965 (unifractal) L3 = M1965 (unifractal) L4 = Multifractal models L4 = Multifractal models L5 = IBM shares L5 = IBM shares L6 = Dollar-Deutchmark exchange rate L6 = Dollar-Deutchmark exchange rate L7/8 = Multifractal models L7/8 = Multifractal models

Neglecting the big steps Neglecting the big steps More clock time - multifractal model generation. More clock time - multifractal model generation.

Mandelbrot (1960, 1961, 1962, 1963, 1965, 1967, 1972, 1974, 1997, 1999, 2000, 2001, 2003, 2005) 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 All papers of Mandelbrot’s were used and analysed from 1960 – 2005 and can be obtained from Fractal Market Anlysis: Applying Chaos theory to Investment and Economcs (Edgar E. Peters) – John Wiley & Sons Inc. (1994) Fractal Market Anlysis: Applying Chaos theory to Investment and Economcs (Edgar E. Peters) – John Wiley & Sons Inc. (1994)