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
McFadden), Elsevier.
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?
• CAPM GIVES AN ANSWER
• MULTI-BETA GIVES ANOTHER
• PRICING KERNEL COVERS ALL