volatility models
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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Volatility Models
Fin250f: Lecture 11.2
Spring 2010
Reading: Brooks chapter 8
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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
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Stochastic Volatility
rt =μ +utut ~N(0,σ t
2 )
log(σ t2 ) =μ + ρ log(σ t−1
2 ) + ztor ARMA(p,q)
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Stochastic Volatility
Very straightforwardDifficult to estimateRelated to high/low range estimation
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ARCH(1)Autoregressive Conditional Heteroskedasticity (Engle)
rt =μ +utut ~N(0,σ t
2 )
σ t2 =α0 +α1ut−1
2
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AR(1)-ARCH(1)Autoregressive Conditional
Heteroskedasticityrt =μ + ρrt−1 +utut ~N(0,σ t
2 )
σ t2 =α0 +α1ut−1
2
ARCH(2) :
σ t2 =α0 +α1ut−1
2 +α2ut−22
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ARCH(1)
Alpha(1) < 1 Alpha(0) > 0Squared return correlations not
persistent enoughNot very useful in finance
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GARCH(1,1)(Bollerslev) Generalized
Autoregressive Conditional Heteroskedasticity
rt =μ +utut ~N(0,σ t
2 )
σ t2 =α0 +α1ut−1
2 +βσ t−12
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GARCH(p,q)Generalized Autoregressive
Conditional Heteroskedasticity
rt =μ +utut ~N(0,σ t
2 )
σ t2 =α0 + α iut−i
2
i=1
q
∑ + βii=1
p
∑ σ t−i2
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ARMA(1,1)/GARCH(1,1)Generalized Autoregressive
Conditional Heteroskedasticityrt =μ + ρrt−1 +θut−1 +utut ~N(0,σ t
2 )
σ t2 =α0 +α1ut−1
2 +βσ t−12
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Unconditional GARCH(1,1) Variance
rt =μ +utut ~N(0,σ t
2 )
σ t2 =α0 +α1ut−1
2 +βσ t−12
E(σ t2 ) =α0 +α1E(ut−1
2 ) + βE(σ t−12 )
E(σ 2 ) =α0 +α1E(σ 2 ) + βE(σ 2 )
E(σ 2 ) =α0
(1−α1 −β)
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GARCH(1,1) Volatility Forecasts
σ t2 = α 0 +α 1ut−1
2 + βσ t−12
σ t−12 = α 0 +α 1ut−2
2 + βσ t−22
σ t2 = α 0 +α 1ut−1
2 + β (α 0 +α 1ut−22 + βσ t−2
2 )
σ t2 = α 0 + βα 0 +α 1ut−1
2 + βα 1ut−12 + β 2σ t−2
2
σ t2 = α 0 β j
j=0
∞
∑ +α 1 β j−1
j=1
∞
∑ ut− j2
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More GARCH(1,1) Forecasts
Et (σ t+12 ) =α0 +α1ut
2 +βσ t2
Et(σ t+22 ) =α0 +α1Et(ut+1
2 ) + βEt(σ t+12 )
Et(σ t+22 ) =α0 +α1Et(σ t+1
2 ) + βEt(σ t+12 )
Et(σ t+22 ) =α0 + (α1 +β)Et(σ t+1
2 )
Et(σ t+32 ) =α0 +α1Et(σ t+2
2 ) + βEt(σ t+22 )
Et(σ t+32 ) =α0 + (α1 +β)Et(σ t+2
2 )
Et(σ t+32 ) =α0 + (α1 +β)(α0 + (α1 +β)Et(σ t+1
2 ))
Et(σ t+32 ) =α0 + (α1 +β)α0 + (α1 +β)2Et(σ t+1
2 )
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Keep on Going
Et (σ t+s2 ) =K + (α1 +β)s−1Et(σ t+1
2 )
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Volatility Diagnostics
Squared and absolute returns:Send into usual time series tests ACF PACF Ljung/Box
Engle test (TR^2): Box 8.1
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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)^2Get R-squared from this regressionCalculate T*(R-squared)Chi-squared(q)
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GARCH(1,1) standardized residuals
zt =rt −μ
σ t2
zt ~N(0,1)
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GARCH(1,1)
Most heavily used volatility model on Wall St.
Estimation: maximum likelihood (not too difficult) matlab: garcheasy.m
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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
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GJR Model
rt =μ +utut ~N(0,σ t
2 )
σ t2 =α0 +α1ut−1
2 +βσ t−12 +γut−1
2 I t−1I t−1 =1 :ut−1 ≤0I t−1 =0 :ut−1 > 0
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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 volatilityApply standard time series tools to
volatility or log(volatility)
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Modeling Using High/low ranges
voltests.mBack to the problem of long range
correlations and modelingSpecial methods
Multi-horizon regressions Asymmetric volatility impact
One other method Long memory/fractional integration
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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?