Forecasting 1999 - Chaos Theory

Forecasting 1999 - Chaos Theory - Forecasting Financial...

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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 1 Forecasting Financial Markets Chaos Theory
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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 2 Overview ¾ Fractals ± Self-similarity ¾ Fractal Market Hypothesis ¾ Long Term Memory Processes ± Rescale Range Analysis ± Biased Random walk ± Hurst Exponent ¾ Cycles ¾ Phase Space ¾ Chaos & the Capital Markets
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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 3 The Chaos Game ± Plot point P at random ± Roll dice ± Proceed halfway from P to point labeled with rolled number & plot new point ± Repeat 10,000 times P A (1, 2) B (3, 4) C (5, 6)
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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 4 The Sierpinksi Triangle
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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 5 Chaos ¾ Local randomness, global determinism ± Apparently random process may contain deterministic pattern ¾ Stable, self-similar structure ¾ Sierpinksi Triangle ± Plot order impossible to predict ± But odds of plotting each point are not equal Empty spaces in each triangle have zero probability Local randomness does not equate to independence
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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 6 Characteristics of Fractals ¾ Self-similarity ± The part is similar to the whole Precise similarity in case of Sierpinski triangle ¾ Scale Invariance ± Sub-parts not to same scale as parent ¾ Dimension ± Euclidean space features integer dimensions ± Fractals occupy fractional dimension E.g dimension of Sierpinski triangle is more than a line but less than a plane (1 < d < 2)
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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 7 Fractal Time Series ¾ Dimension measures how “jagged” series is ± Straight line has fractal dimension of 1 ± Random time series has fractal dimension of 1.5 50% chance of rising or falling ± A line can have fractal dimension between 1 and 2 ± At values 1.5 series is less or more jagged than a random series Non-Gaussian
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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 8 Non-Gaussian Properties of Financial Markets ¾ Distribution of Returns ± Higher peak at mean than Normal ± Fatter tails Uniformly fatter – As many observations at 4 σ away from mean as at 2 σ ± Markets tend to stay still or make major moves more often than theory predicts Reflected in option volatility smiles ¾ Term Structure of Volatility ± Scales at faster rate than T 1/2
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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 9 Example: Returns on the DJIA Dow Jones Industrials Returns 0.00 0.10 0.20 0.30 0.40 0.50 0.60 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Standard Deviations Normal 30 Day Returns
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Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory
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Forecasting 1999 - Chaos Theory - Forecasting Financial...

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