Volatility Models

Fin250f: Lecture 11.2

Spring 2010

Reading: Brooks chapter 8

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

rt =μ +utut ~N(0,σ t

2 )

log(σ t2 ) =μ + ρ log(σ t−1

2 ) + ztor ARMA(p,q)

Stochastic Volatility

Very straightforwardDifficult to estimateRelated to high/low range estimation

ARCH(1)Autoregressive Conditional Heteroskedasticity (Engle)

rt =μ +utut ~N(0,σ t

2 )

σ t2 =α0 +α1ut−1

2

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

ARCH(1)

Alpha(1) < 1 Alpha(0) > 0Squared return correlations not

persistent enoughNot very useful in finance

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

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

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

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)

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

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 )

Keep on Going

Et (σ t+s2 ) =K + (α1 +β)s−1Et(σ t+1

2 )

Volatility Diagnostics

Squared and absolute returns:Send into usual time series tests ACF PACF Ljung/Box

Engle test (TR^2): Box 8.1

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)

GARCH(1,1) standardized residuals

zt =rt −μ

σ t2

zt ~N(0,1)

GARCH(1,1)

Most heavily used volatility model on Wall St.

Estimation: maximum likelihood (not too difficult) matlab: garcheasy.m

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

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

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)

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

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?