Forecasting 1999 - NonLinear Dynamics

Forecasting 1999 - NonLinear Dynamics - Forecasting...

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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 1 Forecasting Financial Markets Nonlinear Dynamics
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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 2 Overview ¾ Fractal distributions ¾ ARFIMA models ¾ Chaotic systems ¾ Phase space ¾ Correlation integrals ¾ Lyapunov exponents
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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 3 Fractal Distributions ¾ Problems with Gaussian theory of financial markets ± Non-normal distribution of returns Fat tails Peaked ¾ Pareto (1897) ± Found that proportion of people owning huge amounts of wealth was far higher than predicted by (log) normal distribution ± “Fat-tails” ± Many examples in nature
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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 4 Pareto-Levy Distributions ¾ Levy (1935) generalized Pareto’s law ± Described family of fat-tailed, high-peak pdf’s ¾ Pareto-Levy density functions ± Ln[f(t)] = i δ t- γ |t| α (1+i β (t/|t|)tan( απ /2) ± Parameters α is measure of peakedness β is measure of skewness (range +/- 1) γ is scale parameter δ is location parameter of the mean
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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 5 Characteristics of Pareto-Levy ¾ α is fractal dimension of probability space ± α = 1 / H ± 0 < α < 2 If α = 2 , ( β = 0, γ = δ = 1 ) distribution is Normal ± EMH: α = 2; FMH: 1 < α < 2 ± Self-similarity If distribution of daily returns has α = a, so will distribution of 5-day returns ± Variance undefined for 1 <= α < 2 ± Mean undefined for α < 1
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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 6 Undefined Variance ¾ Example: Volatility in the S&P 500 Index S&P 500 Index Roling 12 Month Volatility 0% 50% 100% 150% 200% M a r-76 r-78 r-80 r-82 r-84 r-86 r-88 r-90 r-92 r-94 r-96 r-98
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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 7 ARFIMA Models ¾ Generalized ARIMA models ± ARFIMA(p,d,q) Fractional differencing parameter d = H - 0.5 ¾ Models fractal Brownian motion ± Short memory effects ± Long memory effects
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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 8 ARFIMA(0, d, 0) ¾ No short memory effects ¾ Long memory depends on parameter d ± 0 < d < 0.5: black noise ± -0.5 < d < 0: pink noise ± D = 0: white noise ± D = 1: brown noise
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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 9 ARFIMA(0, d, 0) ¾ d < 0.5 ± {y t } is stationary ± Represented as infinite MA process )! 1 ( ! )! 1 ( 0 + = = = d k d k y k k k t k t ψ ε
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Copyright © 1999 – 2006 Investment Analytics Forecasting Financial Markets – Nonlinear Dynamics Slide: 10 ARFIMA(0, d, 0) ¾ d > - 0.5 ± {y t } is invertible ± Represented as infinite AR process )! 1 ( !
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Forecasting 1999 - NonLinear Dynamics - Forecasting...

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