# VOLATILITY MODELS

VOLATILITY MODELS

MEASURING AND FORECASTING VOLATILITY

BEFORE GARCH

Although volatility is viewed as a standard deviation, we will formulate models for the variance and then take square roots.

Moving variance: r t

is the excess return,

 2  t

N

1

1 j

N

0 r t

2

 j

1) all observations from t-N to t are given equal weight

2) all observations before t-N are given no weight

3) the choice of N is left to the trader.

EXPONENTIAL SMOOTHING

 t

2   t

2

1

( 1

) r t

2

1

Once

is known and an initial variance is given such as from the first few observations, then all the variances can be simply computed using something like a spreadsheet.

This model is the same as:

 t

2

 

1

   j

1

 j

1 r t

2

 j a moving variance model with declining weights, no truncation point, and fixed

. This model predicts that all future variances will be the same as today's variance.

GARCH

• GENERALIZED- more general than ARCH model

• AUTOREGRESSIVE-depends on its own past

• CONDITIONAL-variance depends upon past information

• HETEROSKEDASTICITY- fancy word for non-constant variance

HISTORY

• I DEVELOPED THE ARCH MODEL

WHEN I WAS VISITING LSE IN 1979

• IT WAS PUBLISHED IN 1982 WITH

MACRO APPLICATION - THE

VARIANCE OF UK INFLATION

• TIM BOLLERSLEV DEVELOPED THE

GARCH GENERALIZATION AS MY

PHD STUDENT - PUBLISHED IN 1986

THE FIRST FINANCIAL APPLICATIONS:

Engle Lilien and Robins(1987)

French, Schwert and Stambaugh(1987),

Bollerslev(1988) and Bollerslev, Engle, Wooldridge(1988).

SURVEYS:

Bollerslev, Chou and Kroner(1992), Journal of Econometrics,

Special Issue on ARCH Models in Finance

Bollerslev, Engle and Nelson,(1994), ARCH Models, in

Handbook of Econometrics, volume IV, (eds. Engle and

Engle, ARCH: SELECTED READINGS, Oxford University

Press, 1995

The GARCH Model

• r t

 h t

 t h t

    h t

1

 t

2

1

  h t

1

    r t

2

1

• The variance of r t three components

  h t

1 is a weighted average of

– a constant or unconditional variance

– yesterday’s forecast

– yesterday’s news

This model can be rewritten as: h t

( 1

 

)

  j

1

 j

1 r t

2

 j

It can also be rewritten: h t

1

         h t

 

 h t t

2

The forecast of conditional variance one step ahead is given by the first square bracket and the surprise is given by the second. Volatility is predictable but not perfectly.

PARAMETER ESTIMATION

• Historical data reveals when volatilities were large and the process of volatility

• Pick parameters to match the historical volatility episodes

• Maximum Likelihood is a systematic approach:

• Max

L (

)

 

1

2

 

 log h t r t

 m t h t

2

DIAGNOSTIC CHECKING

• Time varying volatility is revealed by volatility clusters

• These are measured by the Ljung Box statistic on squared returns

• The standardized returns no longer r t

/ h should show significant volatilty clustering

• Best models will minimize AIC and

Schwarz criteria

THEOREMS

• GARCH MODELS WITH GAUSSIAN

SHOCKS HAVE EXCESS KURTOSIS

• FORECASTS OF GARCH(1,1) ARE

MONOTONICALLY INCREASING OR

DECREASING IN HORIZON

FORECASTING WITH GARCH r t

2       

) r t

2

1

 

( r t

2

1

 h t

1

)

( r t

2  h t

)

• GARCH(1,1) can be written as ARMA(1,1)

• The autoregressive coefficient is

• The moving average coefficient is

(

 

)

GARCH(1,1) Forecasts h t

   

( r t

2

1

 

)

 

( h t

1

 

)

E t

  t k

  

      t h t

 k

1

  

Monotonic Term Structure of

Volatility

FORECAST PERIOD

FORECASTING AVERAGE

VOLATILITY

E t

 r t

1

...

 r t

 k

 2

E t

E t

( h t

( r t

1

)

2

1

)

...

...

E t

E t

(

( h t

 k r t

)

 k

)

2

• Annualized Vol=square root of 252 times the average daily standard deviation

• Assume that returns are uncorrelated.

TWO YEARS TERM

STRUCTURE OF PORT

0.14

0.13

0.12

0.11

0.10

0.09

0.08

2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500

TERM2000

0.24

0.22

0.20

0.18

0.16

0.14

2900 2950 3000 3050 3100 3150 3200 3250 3300 3350

TERMEND

0.025

0.020

0.015

0.010

0.005

500

0.178

0.176

0.174

0.172

0.170

0.168

0.166

0.164

0.162

2450 2500 2550 2600 2650 2700 2750 2800 2850 2900

TERMMIL_2411

1000 1500

0.22

0.21

0.20

0.19

0.18

1800 1850 1900 1950 2000 2050 2100 2150 2200 2250 2300

TERMMIL_1800

2000

0.180

0.178

0.176

0.174

0.188

0.186

0.184

0.182

2400 2450 2500 2550 2600 2650 2700 2750 2800 2850

TERMMIL_2357

Variance Targeting

• Rewriting the GARCH model h t

   

( h t

1

 t

2

1

 

)

 

( h t

1

 

) the unconditional or long run variance

• this parameter can be constrained to be equal to some number such as the sample variance. MLE only estimates the dynamics

The Component Model h t q t

 q t

 

 

( r t

2

1

 q t

1

)

 

( h t

1

 q t

1

)

 

( q t

1

 

)

 

( r t

2

1

 h t

1

)

• Engle and Lee(1999)

• q is long run component and (h-q) is transitory

• volatility mean reverts to a slowly moving long run component

GARCH(p,q) r t

 h t

 t h t

   j p

1

 j h t

 j

 t

2

 j

 j q

1

 j h t

 j

The Leverage Effect -

Asymmetric Models

• Engle and Ng(1993) following Nelson(1989)

• News Impact Curve relates today’s returns to tomorrows volatility

• Define d as a dummy variable which is 1 for down days h t

    r t

2

1

  r t

2

1 d t

1

  h t

1

NEWS IMPACT CURVE

VOLATILITY

NEWS

Other Asymmetric Models

EGARCH: NELSON(1989) log( h t

)

    log( h t

1

)

  r t

1 h t

1

  r t

1 h t

1

NGARCH: ENGLE(1990) h t

   

( r t

1

 

) 2

  h t

1

PARTIALLY NON-PARAMETRIC

ENGLE AND NG(1993)

VOLATILITY

NEWS

EXOGENOUS VARIABLES IN

A GARCH MODEL

• Include predetermined variables into the variance equation

• Easy to estimate and forecast one step

• Multi-step forecasting is difficult

• Timing may not be right h t

    r t

2

1

  h t

1

  z t

1

EXAMPLES

• Non-linear effects

• Deterministic Effects

• News from other markets

– Heat waves vs. Meteor Showers

– Other assets

– Implied Volatilities

– Index volatility

• MacroVariables or Events

WHAT IS THE BEST MODEL?

• The most reliable and robust is

GARCH(1,1)

• For short term forecasts, this is good enough.

• For long term forecasts, a component model with leverage is often needed.

• A model with economic causal variables is the ideal

Component with Leverage h t

 q t

 

( r t

2

1

 q t

1

)

 

( h t

1

 q t

1

)

 

( r t

2

1 d t

1

.

5 q t

1

) q t d t

 

1 if

 r t

( q t

1

0

 

)

 

( r t

2

1

 h t

1

)

Procter and Gamble Daily

Returns 2/89-3/99

Variance Equation Garch(1,1)

C 5.11E-06

ARCH(1) 0.047402

GARCH(1) 0.927946

1.24E-06

0.006681

0.011114

4.112268

7.094764

83.49235

AIC -5.7086 , SCHWARZ CRITERION -5.699617

CORRELOGRAM OF

SQUARED RESIDUALS

AC PAC Q-Stat Prob

1 0.030

0.030 2.3720 0.124

2 -0.018

3 -0.010

-0.019 3.1992 0.202

-0.009 3.4859 0.323

4 0.018

5 -0.011

6 -0.002

7 0.013

8 -0.016

9 0.010

10 0.003

0.018 4.3016 0.367

-0.013 4.6317 0.462

-0.001 4.6408 0.591

0.013 5.0865 0.649

-0.017 5.7164 0.679

0.012 5.9973 0.740

0.002 6.0183 0.814

P&G TARCH

C

ARCH(1)

0.0000 0.0000 4.6121 0.0000

0.0269 0.0093 2.9062 0.0037

(RESID<0)*ARCH(1)0.0520 0.0141 3.6976 0.0002

GARCH(1) 0.9123 0.0139 65.8060 0.0000

AIC -5.7114

SCHWARZ CRITERION -5.7002

P&G EGARCH

C

|RES|/SQR[GARCH](1)

RES/SQR[GARCH](1)

EGARCH(1)

-0.3836

0.1186

-0.0392

0.9656

0.0672 -5.7052

0.0153 7.7645

0.0103 -3.8195

0.0070 137.9063

AIC -5.7114

SCHWARZ CRITERION -5.7002

P&G Asymmetric Component

Perm: C

Perm: [Q-C]

0.0002 0.0000 14.1047

0.9835 0.0049 201.5877

Perm: [ARCH-GARCH] 0.0335 0.0079 4.2577

Tran: [ARCH-Q] -0.0361 0.018 -2.0045

Tran: (RES<0)*[ARCH-Q] 0.0910 0.0213 4.2838

Tran: [GARCH-Q] 0.8063 0.0819 9.8403

AIC -5.7132, SCHWARZ CRITERION -5.6974

P&G VOLATILITIES

0.7

0.6

0.5

0.4

0.3

0.2

0.1

2/22/89 1/23/91 12/23/92 11/23/94 10/23/96 9/23/98

V_PG_T ARCH V_PG_GARCH V_PG_ACOMP

DISCUSS GAUSSIAN

ASSUMPTION

• EMPIRICAL EVIDENCE INDICATES THAT

INNOVATIONS ARE LEPTOKURTIC

• GAUSSIAN GARCH IS QMLE

– CONSISTENT BUT NEEDS

BOLLERSLEV

WOOLDRIDGE STANDARD ERRORS

– T-DISTRIBUTION

MAY BE MORE EFFICIENT

– CAN DO

SEMI-PARAMETRIC ESTIMATOR OF

ENGLE AND GONZALES-RIVERA

Bollerslev Wooldridge

Standard Errors

ROBUST TO NON-NORMAL ERRORS

Perm: C

Perm: [Q-C]

0.0002 0.0000 8.3857

0.9835 0.0081 121.9762

Perm: [ARCH-GARCH] 0.0335 0.0107 3.1264

Tran: [ARCH-Q] -0.036 0.0242 -1.4898

Tran: (RES<0)*[ARCH-Q] 0.0910 0.0429 2.1220

Tran: [GARCH-Q] 0.8063 0.1223 6.5919

RISK PREMIA

• WHEN RISK IS GREATER, EXPECTED

RETURNS SHOULD BE GREATER

– HOW MUCH?

– WHAT COUNTS AS RISK?