Transcript NEW MODELS FOR HIGH AND LOW FREQUENCY VOLATILITY …
NEW MODELS FOR HIGH AND LOW FREQUENCY VOLATILITY Robert Engle NYU Salomon Center Derivatives Research Project
FORECASTING WITH GARCH
DJ RETURNS
.08
.06
.04
.02
.00
-.02
-.04
-.06
-.08
1990 1992 1994 1996 1998 2000 2002 2004 DJRET
DOW JONES SINCE 1990
Dependent Variable: DJRET Method: ML - ARCH (Marquardt) - Normal distribution Date: 01/13/05 Time: 14:30 Sample: 15362 19150 Included observations: 3789 Convergence achieved after 14 iterations Variance backcast: ON GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1) Coefficient Std. Error z-Statistic Prob. C 0.000552
0.000135 4.093478
0.0000
Variance Equation C RESID(-1)^2 GARCH(-1) 9.89E-07 0.066409
0.924912
1.84E-07 5.380913 0.0000
0.004478 14.82844
0.0000
0.005719 161.7365
0.0000
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood -0.000370
-0.001163
0.010200
0.393815
12427.71
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Durbin-Watson stat 0.000356
0.010194
-6.557778
-6.551191
1.985498
.45
.40
.35
.30
.25
.20
.15
.10
.05
1990 1992 1994 1996 1998 2000 2002 2004 2006 HORIZOND
.028
.024
.020
.016
.012
.008
.004
1998 1999 2000 2001 2002 2003 2004 2005 2006 DJSD DJSD1 DJSD2 DJSD3 DJSD4 DJSD5 DJSD0 DJSDEND
.32
.28
.24
.20
.16
.12
1990 1992 1994 1996 1998 2000 2002 2004 2006 HORIZONY
.45
.40
.35
.30
.25
.20
.15
.10
.05
1990 1992 1994 1996 1998 2000 2002 2004 2006 HORIZOND HORIZONM HORIZONQ HORIZONY HORIZON2Y HORIZON5Y
DEFINITIONS
r t is a mean zero random variable measuring the return on a financial asset CONDITIONAL VARIANCE
h t
E t
1
t
UNCONDITIONAL VARIANCE
t
2
t
GARCH(1,1)
h t r t
2 1
h t
1 The unconditional variance is then
t
2 1
t
t
2 2
t
1 2
GARCH(1,1)
h t
t
r t
2 1
h t
1 If omega is slowly varying, then
t
t
t
2
t
t
2 1
t
2
j t
0
j
This is a complicated expression to interpret
SPLINE GARCH
Instead , use a multiplicative form
r t
t g t
t
, where
t
|
t
1
N
(0,1)
g t
)
r t t
2 1 1
g t
1 Tau is a function of time and exogenous variables
UNCONDITIONAL VOLATILTIY
Taking unconditional expectations
E r t
2
t t
t
2 )
t
t
t
Thus we can interpret tau as the unconditional variance.
SPLINE
ASSUME UNCONDITIONAL VARIANCE IS AN EXPONENTIAL QUADRATIC SPLINE OF TIME For K knots equally spaced log 0 1
t
2
t
2
k K
1
k
max
t
t k
, 0 2
ESTIMATION
FOR A GIVEN K, USE GAUSSIAN MLE
L
1 2
t T
1 log
t g t
2
r t t g t
CHOOSE K TO MINIMIZE BIC FOR K LESS THAN OR EQUAL TO 15
EXAMPLES FOR US SP500
DAILY DATA FROM 1963 THROUGH 2004 ESTIMATE WITH 1 TO 15 KNOTS OPTIMAL NUMBER IS 7
RESULTS
LogL: SPGARCH Method: Maximum Likelihood (Marquardt) Date: 08/04/04 Time: 16:32 Sample: 1 12455 Included observations: 12455 Evaluation order: By observation Convergence achieved after 19 iterations C(4) W(1) W(2) W(3) Coefficient Std. Errorz-Statistic Prob. -0.000319 7.52E-05 -4.246643 0.0000
-1.89E-08 2.59E-08 -0.729423 0.4657
2.71E-07 2.88E-08 9.428562 0.0000
-4.35E-07 3.87E-08 -11.24718 0.0000
W(4) W(5) W(6) W(7) C(5) 3.28E-07 5.42E-08 6.058221 0.0000
-3.98E-07 5.40E-08 -7.377487 0.0000
6.00E-07 5.85E-08 10.26339 0.0000
-8.04E-07 9.93E-08 -8.092208 0.0000
1.137277 0.043563 26.10666 0.0000
C(1) 0.089487
C(2) 0.881005
Log likelihood Avg. log likelihood Number of Coefs.
0.002418 37.00816 0.0000
0.004612 191.0245 0.0000
-15733.51
-1.263228
11 Akaike info criterion Schwarz criterion Hannan-Quinn criter.
2.528223
2.534785
2.530420
S&P500 1.2
1.0
0.8
0.6
0.4
0.2
0.0
60 65 70 75 80 85 90 95 00 CVOL UVOL
India,5 1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
90 92 CVOL 94 96 UVOL 98 00 02 ANNUAL RV
Argentina, 3 4 3 2 1 0 90 92 CVOL 94 96 UVOL 98 00 02 ANNUAL RV
Japan,4 1.0
0.8
0.6
0.4
0.2
0.0
90 92 CVOL 94 96 UVOL 98 00 02 ANNUAL RV
Brazil,6 3.0
2.5
2.0
1.5
1.0
0.5
0.0
90 92 CVOL 94 96 UVOL 98 00 02 ANNUAL RV
South Africa,3 .9
.8
.7
.6
.5
.4
.3
.2
.1
.0
90 92 CVOL 94 96 UVOL 98 00 02 ANNUAL RV
Poland,1 1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
90 92 CVOL 94 96 UVOL 98 00 02 ANNUAL RV
ESTIMATION
Volatility is regressed against explanatory variables with observations for countries and years.
Within a country residuals are auto correlated due to spline smoothing. Hence use SUR.
Volatility responds to global news so there is a time dummy for each year.
Unbalanced panel
ONE VARIABLE REGRESSIONS
emerging transition log(mc) log(gdp_dll) nlc grgdp gcpi vol_irate vol_forex vol_grgdp vol_gcpi Coefficient 0.0957
-0.0077
-0.0093
0.0015
-1.29E-05 -0.6645
0.6022
0.0089
0.5963
1.1192
0.9364
Table (5) Individual SUR Regressions
Std. Error 0.0176
0.0180
0.0032
0.0055
0.0000
0.1255
0.0418
0.0006
0.0399
0.1008
0.0848
t-Statistic 5.4528
-0.4284
-2.9345
0.2740
-2.3706
-5.2945
14.4181
14.4896
14.9468
11.1056
11.0375
Prob. 0.0000
0.6685
0.0035
0.7842
0.0181
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Det Residual Covariance 6.45E-39 1.53E-38 3.76E-38 2.18E-37 1.23E-37 3.89E-38 1.64E-38 8.59E-39 2.47E-38 8.71E-39 2.84E-38
MULTIPLE REGRESSIONS
emerging transition log(mc) log(gdpus) nlc grgdp gcpi vol_irate vol_gforex vol_grgdp vol_gcpi All Countries 0.0376
( 0.0131 )** -0.0178
( 0.0171 ) -0.0092
( 0.0055 )* 0.0273
( 0.0068 )** -1.8E-05 ( 5.4E-06 )** -0.1603
( 0.1930 ) 0.3976
( 0.1865 )** 0.0020
( 0.0008 )** 0.0222
( 0.0844 ) 0.8635
( 0.1399 )** 0.9981
( 0.3356 )**
Time Effects 0.25
0.2
0.15
0.1
0.05
0 1990 1994 1998 2002
IMPLICATIONS
Unconditional volatility varies over time and can be modeled Volatility mean reverts to the level of unconditional volatility Long run volatility forecasts depend upon macroeconomic and financial fundamentals
HIGH FREQUENCY VOLATILITY
WHERE CAN WE GET IMPROVED ACCURACY?
USING ONLY CLOSING PRICES IGNORES THE PROCESS WITHIN THE DAY.
BUT THERE ARE MANY COMPLICATIONS. HOW CAN WE USE THIS?
ONE MONTH OF DAILY RETURNS
21 20 19 18 17 16 9000 9100 9200 9300 9400 9500 9600 9700 9800 PRICEDAY
INTRA-DAILY RETURNS
21 20 19 18 17 16 9000 9100 9200 9300 9400 9500 9600 9700 9800 PRICEDAY PRICE10
ARE THESE DAYS THE SAME?
CAN WE USE THIS INFORMATION TO MEASURE VOLATILITY BETTER?
DAILY HIGH AND LOW
hl t
range
log(
high t
low t
) DAILY REALIZED VOLATILITY
dv t
i n
1 log(
p
/
p
)
PARKINSON(1980)
HIGH LOW ESTIMATOR IF RETURNS ARE CONTINUOUS AND NORMAL WITH CONSTANT VARIANCE, ln( 2 )
range
2
daily
TARCH MODEL WITH RANGE
C RESID(-1)^2 RESID(-1)^2*(RESID(-1)<0) GARCH(-1) RANGE(-1)^2
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood 1.07E-06 2.03E-07 5.268049 0.0000
-0.100917 0.011398 -8.853549 0.0000
0.096744 0.010951 8.834209 0.0000
0.879976 0.010518 83.65995 0.0000
0.075963 0.008281
9.172690
0.0000
-0.001360
0.010330
0.404010
12550.46
S.D. dependent var Akaike info criterion Schwarz criterion Durbin-Watson stat 0.010323
-6.616277
-6.606403
2.001541
A MULTIPLE INDICATOR MODEL FOR VOLATILITY USING INTRA-DAILY DATA
Robert F. Engle Giampiero M. Gallo
Forthcoming, Journal of Econometrics
Absolute returns
|
r t ha t
|
ha t
t r t
1 2
ha t
1 • Insert asymmetric effects (sign of returns) ha t | r t 1 | 2 ha t 1 • Insert other lagged indicators ha t 2 1 r ha t 1 | r t 1 1 r t 1 d t 1 t 1 r hl t 2 1 r t 1 t 1 r dv t 2 1 r t 1 t 1 ,
Repeat for daily range, hl t :
hh t
hl t
h
h
hl t
2 1
hh t
t
h hh t
1
r h t
1
h hl d t
2 1
t
1
r h t
2 1
r d h t
2 1
t
1
h dv t
2 1
h
2
dv d t
1
t
1 .
And for realized daily volatility, dv t :
hd t
d
d dv t
2 1
dv t
d hd t hd
1
t
t
r d t
1
d dv d t
2 1
t
1
r d t
2 1
r d d t
2 1
t
1
d hl t
2 1
d
2
hl d t
1
t
1 .
Smallest BIC-based selection
ha t
5.026 - 0.030
2.805
1.068
r t
2 1 0.901
43.432
ha t
-1 - 0.745
3.293
r t
1 0.101
2.328
hl t
2 1
hh t
7.622
4.885
0.109
5.407
hl t
2 1 0.850
32.713
hh t
1 - 0.878
3.608
r t
1
hd t
8.061
2.366
dv t
1 2 0.736
91.479
hd t
-1 -1.183
28.350
r t
-1 0.122
6.688
dv d t
-1
t
-1 0.123
23.911
r t
2 1
Forecasting
• one step-ahead
h T
1|
T
ha hh T T
1|
T
1|
T hd T
1|
T
d a h
A *
T
, 2 , 2 , 2 , 2 , 2 ,
T T T T T T T T
,
T
,
T
• multi-step-ahead
h
ha hh hd
d a h
A
ha hh hd
1|
T
1|
T
1|
T
Ah
1|
T
Term Structure of Volatility 1
20 18 16 14 12 10 01/08/98 03/09/98 05/05/98 07/01/98 08/27/98 Absolute returns
IMPLICATIONS
Intradaily data can be used to improve volatility forecasts Both long and short run forecasts can be implemented if all the volatility indicators are modeled Daily high/low range is a particularly valuable input These methods could be combined with the spline garch approach.