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Volatility Models

Fin250f: Lecture 11.2 Spring 2010 Reading: Brooks chapter 8. Volatility Models. Outline. Stochastic volatility ARCH(1) GARCH(1,1) GARCH(p,q) GJR and volatility asymmetry High/low volatility time series modeling and long range persistence. Stochastic Volatility. Stochastic Volatility.

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Volatility Models

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  1. Fin250f: Lecture 11.2 Spring 2010 Reading: Brooks chapter 8 Volatility Models

  2. Outline • Stochastic volatility • ARCH(1) • GARCH(1,1) • GARCH(p,q) • GJR and volatility asymmetry • High/low volatility time series modeling and long range persistence

  3. Stochastic Volatility

  4. Stochastic Volatility • Very straightforward • Difficult to estimate • Related to high/low range estimation

  5. ARCH(1)Autoregressive Conditional Heteroskedasticity (Engle)

  6. AR(1)-ARCH(1)Autoregressive Conditional Heteroskedasticity

  7. ARCH(1) • Alpha(1) < 1 • Alpha(0) > 0 • Squared return correlations not persistent enough • Not very useful in finance

  8. GARCH(1,1)(Bollerslev) Generalized Autoregressive Conditional Heteroskedasticity

  9. GARCH(p,q)Generalized Autoregressive Conditional Heteroskedasticity

  10. ARMA(1,1)/GARCH(1,1)Generalized Autoregressive Conditional Heteroskedasticity

  11. Unconditional GARCH(1,1) Variance

  12. GARCH(1,1) Volatility Forecasts

  13. More GARCH(1,1) Forecasts

  14. Keep on Going

  15. Volatility Diagnostics • Squared and absolute returns:Send into usual time series tests • ACF • PACF • Ljung/Box • Engle test (TR^2): Box 8.1

  16. Engle Test • Find residuals of linear model: u(t) • Regress u(t)^2 on lags, u(t-1)^2, u(t-2)^2,…u(t-q)^2 • Get R-squared from this regression • Calculate T*(R-squared) • Chi-squared(q)

  17. GARCH(1,1) standardized residuals

  18. GARCH(1,1) • Most heavily used volatility model on Wall St. • Estimation: • maximum likelihood (not too difficult) • matlab: garcheasy.m

  19. Modifications • EGARCH (log form) • TARCH (threshold nonlinearitiy) • IGARCH (volatility follows random walk) • Asymmetry • For stocks (only) volatility increases by larger amount on market falls • One (of many) models: GJR • Glosten, Jagannathan and Runkle

  20. GJR Model

  21. Volatility Modeling Beyond GARCH • Use other estimates (beyond squared returns) for volatility • VIX • Intraday data (realized volatility) • High/low range information • Build daily estimates of volatility • Apply standard time series tools to volatility or log(volatility)

  22. Modeling Using High/low ranges • voltests.m • Back to the problem of long range correlations and modeling • Special methods • Multi-horizon regressions • Asymmetric volatility impact • One other method • Long memory/fractional integration

  23. Volatility Summary • Lots of predictability (for finance) but, • No perfect model (sort of GARCH(1,1) • Do different objectives matter? • What about out of sample • Some puzzles remain • Long range persistence • Some nonnormal residuals • Sign asymmetry and other nonlinear effects • Connections to trading volume • Basic issue: Why is it changing?

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