The term asymmetric volatility arises from observation that we observe higher volatilities (higher risk) during the market downturn than in the market upturns. The most common mentioned factor that contributes to such risk behavior is increased market leverage that was produced by a negative shock; however, there are also other factors, such as perceived risk/reward balance in different stages of market behavior.

1. Topics
“Volatility”
Helios
Padilla
Mayer
February
22,
2012
1.
(a)
The
term
asymmetric
volatility
arises
from
observation
that
we
observe
higher
volatilities
(higher
risk)
during
the
market
downturn
than
in
the
market
upturns.
The
most
common
mentioned
factor
that
contributes
to
such
risk
behavior
is
increased
market
leverage
that
was
produced
by
a
negative
shock;
however,
there
are
also
other
factors,
such
as
perceived
risk/reward
balance
in
different
stages
of
market
behavior.
As
explained
in
V-­‐‑Lab
documentation,
the
plain
GARCH
model1
cannot
take
into
account
the
stronger
impact
of
negative
shocks
in
time
t-­‐‑1
on
the
variance
in
time
t
than
in
case
of
positive
shocks
–
an
asymmetry
impact
of
negative
shock.
For
that
reason,
the
GARCH
model
has
been
augmented
into
models
such
as
the
Threshold
GARCH
(TGARCH),
the
Asymmetric
GARCH
(AGARCH)
and
the
Exponential
GARCH
(EGARCH).
The
V-­‐‑Lab
documentation
and
the
task
in
this
exam
indicate
that
GJR
GARCH
should
be
used
to
introduce
an
asymmetry
into
the
variance
analysis.
Thus,
my
answers
will
be
based
on
the
use
of
the
GJR
GARCH
model.
2
I
also
assume
that
p=1,
q=1,
so
I
will
be
talking
about
GJR
GARCH
(1,
1)
and
GARCH
(1,
1).
We
assume
that
the
return
time
series
can
take
the
following
form:
,
where
is
expected
return
and
is
the
white
noise,
zero-­‐‑mean
error
term.
The
assumption
is
that
is
serially
uncorrelated,
but
not
necessarily
serially
independent
–
it
can
present
a
conditional
heteroskedasticity,
which
is
taken
into
consideration
by
GJR
GARCH
model.
Error
term
is
then
split
into
a
stochastic
component
( )
and
time-­‐‑dependent
standard
deviation
( ):
,
where
is
assumed
to
be
drawn
from
Gaussian
distribution
and
i.i.d.
(mean=0,
variance
=
1).
The
variance
is
then
formulated
as:
,
(1)
where
So,
GJR
GARCH
models
asymmetry
in
the
ARCH
model
and
as
the
basic
ARCH
includes
(a)
the
error
term,
(b)
a
conditional
variance
(which
brings
ARCH
model
to
GARCH),
plus
(c)
innovation
term,
which
for
asymmetry:
negative
shocks
in
time
t-­‐‑1
have
stronger
impact
on
variance
than
positive
shocks.
Therefore
the
term
takes
value
of
1
GARCH stands for general autoregressive (meaning that past observations are incorporated into the
present situation) conditional (variance is conditioned on time) heteroskedascticity (a time-varying
variance).
2
I am also using a formulation of GJR GARCH consistent with VLAB in order to be able to interpret
results correctly.
ttr eµ += µ te
te
tz ts
ttt zse = tz
2
1
2
11
2
)( --- +++= tttt I bsegaws
þ
ý
ü
î
í
ì
á
³
=-
µ
µ
1-­‐t
1-­‐t
rif
rif
1
0
1tI
1-tI

2. 1,
when
returns
in
time
t-­‐‑1
are
lower
than
expected
return
and
value
0
when
returns
in
time
t-­‐‑1
are
higher
than
expected
return.
As
explained
in
VLAB,
the
effective
coefficient
associated
with
a
negative
shock
is
α+γ.
In
financial
time
series,
we
generally
find
that
γ
is
statistically
significant.
The
estimate
of
the
coefficients
is
performed
through
V-­‐‑Lab
simultaneously
by
maximizing
the
log
likelihood
(MLE).
GJR
GARCH
also
captures
the
volatility
clustering:
if
volatility
was
high
at
time
t-­‐‑1,
it
will
also
be
high
at
time
t.
Alternatively,
the
shock
at
time
t-­‐‑1
will
also
impact
variance
at
time
t.
There
are
the
following
constraints
on
coefficients:
,
, , .
If
also
is
true,
the
volatility
is
mean
reverting
and
fluctuates
around .
If
this
is
true,
then
we
can
write
down
the
unconditional
variance
as
(2)
In
our
case,
we
observe
a
Carnival
Corp
volatility
data
range
from
02/21/2010
to
02/21/2012.
The
annual
GJR-­‐‑GARCH
volatility
is
plotted
below,
along
with
the
volatility
summary
table
and
estimated
parameters.
1-­‐‑year
volatility
prediction
with
the
GJR-­‐‑GARCH
model
Volatility
summary
table
Price
30.74
Return
-­‐‑0.75
1
Week
Pred:
42.9
Avg
Week
Vol:
43.84
Avg
Month
Vol:
49.89
1
Month
Pred:
43.5
Min
Vol:
16.32
Max
Vol:
189.91
6
Months
Pred:
47.2
Avg
Vol:
39.57
Vol
of
Vol:
45.17
1
Year
Pred:
50.8
0³w 0³a 0³b 0³+ga
1
2
á++ b
g
a s
)var(
2
1
2
tr=
---
=
b
g
a
w
s

3.
Parameter
Estimates
Parameter
t-­‐‑stat
0.03981
8.693
0.01802
6.273
0.94189
393.603
0.07686
19.875
t-­‐‑stat
for
estimated
parameters
shows
that
all
estimates
are
statistically
significant
at
1%;
thus,
all
restrictions
on
coefficients
are
fulfilled
and
the
process
does
follow
GJR-­‐‑
GARCH
model.
The
Corp
Carnival
GJR-­‐‑GARCH
model
can
be
then
written
as:
(3)
A
constant
in
the
model
shows
that
if
the
error
term
and
variance
in
time
t-­‐‑1
were
0
and
returns
in
time
t-­‐‑1
are
higher
than
expected
return,
the
variance
for
Corp
Carnival
would
be
0.03981.
Secondly,
a
1%
change
in
variance
in
time
t-­‐‑1
would
increase
variance
in
time
t
by
0.94189%.
Third,
a
1%
negative
shock
on
returns
in
time
t-­‐‑1
would
impact
variance
in
time
t
by
(0.01802+0.07686=)
0.09488
%.
As ,
volatility
is
mean
reverting
and
unconditional
variance,
=23.98193.
(b)
Below
I
plot
volatilities
over
the
last
3
months
for
GARCH
and
GJR-­‐‑GARCH
models.
Given
the
definition
of
GJR-­‐‑GARCH
model,
asymmetric
impact
of
negative
shock
is
captured
(unlike
in
GARCH
model)
and
therefore
the
volatility
prediction
on
the
basis
of
GJR-­‐‑GARCH
model
is
bigger
than
volatility
prediction
on
the
basis
of
on
GARCH
model.
This
is
specifically
noticed
in
the
case
of
the
biggest
negative
shock
in
returns
on
14
January,
2012.
The
immediate
spike
in
volatility
after
this
date
is
significantly
higher
for
GJR-­‐‑GARCH
than
for
GARCH
model.
3-­‐‑month
volatility
prediction
with
the
GARCH
model
w
a
b
g
2
1
2
11
2
94189.0)07686.001802.0(03981.0 --- +++= tttt I ses
199834.0
2
á=++ b
g
a
)var( tr

4.
Volatility
summary
table
Price
30.74
Return
-­‐‑0.75
1
Week
Pred:
40.19
Avg
Week
Vol:
40.66
Avg
Month
Vol:
43.23
1
Month
Pred:
40.38
Min
Vol:
17.33
Max
Vol:
107.53
6
Months
Pred:
41.68
Avg
Vol:
38.77
Vol
of
Vol:
36.61
1
Year
Pred:
43.04
Parameter
Estimates
Parameter
t-­‐‑stat
0.01141
6.306
0.02821
18.816
0.97128
906.887
3-­‐‑month
volatility
prediction
with
the
GJR-­‐‑GARCH
Model
Volatility
summary
table
Price
30.74
Return
-­‐‑0.75
1
Week
Pred:
42.9
Avg
Week
Vol:
43.84
Avg
Month
Vol:
49.89
1
Month
Pred:
43.5
Min
Vol:
16.32
Max
Vol:
189.91
6
Months
Pred:
47.2
Avg
Vol:
39.57
Vol
of
Vol:
45.17
1
Year
Pred:
50.8
Parameter
Estimates
Parameter
t-­‐‑stat
0.03981
8.693
0.01802
6.273
0.94189
393.603
0.07686
19.875
(c)
In
both
cases,
when
volatility
is
estimated
with
GARCH
of
with
GJR-­‐‑GARCH
models,
volatility
predications
are
higher
over
the
long-­‐‑term
horizon
(1
year)
than
over
the
w
a
b
w
a
b
g

5. short-­‐‑term
horizon
(1
week)
–
see
plots
from
V-­‐‑Lab
below.
When
we
take
into
consideration
longer
horizons,
we
implicitly
“tag
along”
t-­‐‑k
period
impacts
for
a
larger
number
of
k.
For
example,
if
we
observe
1
week
data,
we
would
be
considering
5
lagged
period
effects.
When
we
observe
1
year
data,
we
would
be
considering
250
lagged
period
effects.
The
volatility
impact
is
therefore
multiplying
through
periods
and
therefore
longer-­‐‑term
forecasts
result
in
higher
volatilities
than
short-­‐‑term
forecasts.
Annualized
Volatility
Predictions
with
the
GARCH
Model
Annualized
Volatility
Predictions
with
the
GJR-­‐‑GARCH
Model
2.
The
VIX
index
measures
the
short-­‐‑term
implied
volatility
of
the
S&P
500
index
and
has
become
a
benchmark
for
volatility
in
equity
markets.
Volatility
is
calculated
as
s
standard
deviation
of
returns.
However,
if
we
want
to
talk
about
the
implied
or
realized
volatility
of
VIX,
we
actually
talk
about
the
volatility
of
volatility.
Volatility
of
volatility
calculated
as
the
standard
deviation
of
the
percentage
change
in
VIX
volatility
and
it
tells
us
how
fast
volatility
changes.
Prior
and
during
the
financial
crisis,
the
standard
risk
measures,
such
as
VaR,
were
based
on
short-­‐‑term
risk
measurements
–
1
to
10
day
horizons.
However,
investors
normally
hold
their
positions
for
a
longer
period
of
time
than
10
days.
As
short-­‐‑term
risk
measures
during
the
low
risk
environment
did
not
alert
to
any
possible
increase
in
risk,
everyone
was
confidents
that
their
long-­‐‑term
positions
are
safe
by
accepting
short-­‐‑term
risk
assessment.
However,
none
actually
considered
the
possibility
(and
velocity)
of
change
in
risk
and
volatility
of
markets.
During
the
low
risk,
low
volatility
and
low
interest
period,
everyone
tried
to
increase
their
leverage
on
the
back
of
cheap
assets,
attractive
structured
products
offered
and
a
variety
of
insurance
instruments
available
to
insure
against
any
possible
risk
event.
The
problem
arose
when
market
volatility
started
increasing,
investors
were
holding
now
highly
risky
positions
that
they
were
able
to
sell
only
at
deep
discounts,
insurers
were
not
capitalized
enough
to
provide
payouts.
Thus,
the
major
problem
was
that
future
volatility
(which
was
observed
through
the
VoV
chart)
was
not
incorporated
at
all
in
risk
assessment
models
and
none
was
prepared
for
what
came
forward.
Short-­‐‑term
risk
assessment
models
produce
short-­‐‑term
results
and
observations.
If
one
compares
10-­‐‑day,
30-­‐‑day,
60-­‐‑day
or
1-­‐‑year
volatility
index
(VIX)
and
its
implied

6. volatility,
short-­‐‑term
volatilities
will
be
lower
than
long-­‐‑term
volatilities.
The
reason
behind
this
is
that
long-­‐‑term
volatility
depends
on
macroeconomic
policies
(monetary,
fiscal,
balance
of
payments)
and
success
of
their
implementation.
If
investors
are
aware
that
long-­‐‑term
volatility
is
higher
than
the
short-­‐‑term,
they
can
decide
to
either
engage
in
short-­‐‑term
investments
that
are
less
volatile
and
accept
lower
returns,
or
try
to
incorporate
higher
long-­‐‑term
volatility
in
their
expected
returns.
Alternatively,
long-­‐‑
term
investments
with
higher
volatility
can
be
also
hedged
with
proper
instruments
that
would
sufficiently
account
for
higher
volatility
in
the
future.
Long-­‐‑term
investments
represent
higher
volatility
because
at
the
time
of
investment
decision
it
is
not
clear
whether
policymakers
will
be
successful
at
mitigating
future
risks
(for
example,
a
probability
of
Greek
default,
global
economic
slowdown,
Iran’s
nuclear
policy
impact
on
oil
supply,
etc..).
Furthermore,
new
macroeconomic
risks
will
arise
during
the
investment
period
and
more
uncertainties
will
have
to
be
taken
into
consideration.
3.
Correlation
is
a
measure
of
relation
between
two
variables
or
series.
If
variables
are
moving
in
the
same
direction,
correlation
is
positive
(up
to
+1,
which
indicates
perfect
positive
correlation).
When
variables
are
moving
in
opposite
direction,
correlation
is
negative
(up
to
-­‐‑1,
which
means
perfect
negative
correlation).
Engel
(2000)
defines
unconditional
correlation
between
two
variables
r1
and
r2,
each
with
mean
zero
as
,
(1)
Where
is
covariance
between
variables
r1
and
r2,
and
and
is
variance
of
r1
and
r2,
respectively.
This
formula
does
not
include
time
component
and
therefore
it
is
assumed
that
correlation
is
not
based
on
information
known
from
previous
periods.
However,
we
know
that
correlations
are
sensitive
to
time.
This
time
sensitivity
is
taken
into
consideration
in
conditional
correlation,
where
both
covariances
and
variances
are
based
on
information
known
the
previous
period:
(2)
In
his
work,
Engle
(2000)
also
shows
that
conditional
correlation
can
be
interpreted
as
conditional
covariance
between
standardized
disturbances.
For
that
reason,
he
writes
the
returns
( )
as
the
conditional
standard
deviation
times
the
standardized
disturbance
( ),
with
mean
zero
and
variance
1:
and
(3)
If
we
substitute
(3)
into
(2),
we
get
(4)
There
are
many
ways
to
estimate
conditional
correlations
and
I
will
describe
two
based
on
multivariate
GARCH
models.
GARCH
models
assume
that
volatilities
are
correlations
are
functions
of
lagged
returns.
Bollerslev
(1990)
specified
the
constant
conditional
)()(
)(
2
2
2
1
21
2,1
rErE
rrE
=r
)( 21rrE )( 2
1rE )( 2
2rE
)()(
)(
2
,2
2
,11
,2,11
,2,1
ttt
ttt
t
rErE
rrE
-
-
=r
tih ,
ti,e
)( 2
,1, titti rEh -= 2,1,,,, == ihr tititi e
)(
)()(
)(
,3,11
2
,2
2
,11
,2,11
,2,1 ttt
ttt
ttt
t E
EE
E
ee
ee
ee
r -
-
-
==

7. correlation
(CCC)
multivariate
GARCH
specification,
where
conditional
covariances
and
variances
are
time-­‐‑varying,
but
conditional
correlations
are
constant.
Conditional
variances
are
modelled
by
univariate
GARCH
models
and
correlation
matrix
is
estimated
by
MLE.
The
assumption
of
constant
correlations
allows
for
comparison
between
periods.
The
model
is
described
as
follows.
The
multivariate
GARCH
assumes
that
distribution
of
returns
(rt)
from
n
assets
has
mean
zero
and
covariance
matrix
Ht
(Engle,
Sheppard,
2001):
,
(5)
where
,
and
Conditional
covariance
matrix
Ht
is
then
decomposed
into
nxn
Rt
matrix
m
where
(6)
and
D
is
conditional
correlation
matrix
with
constant
correlations,
= ;
,
where
,
i=1,
…N.
Thus,
the
time
variation
in
covariance
matrix
Ht
is
then
explained
only
by
time-­‐‑varying
conditional
variances
for
all
rt.
The
off-­‐‑diagonal
elements
of
the
conditional
covariance
matrix
can
be
then
written
as
(Silvennoinen,
Terasvirta,
2008)
,
and
(7)
While
the
advantage
of
this
model
is
that
is
easy
to
estimate
(we
only
need
non
linear
n
estimates
of
univariate
GARCH
models),
the
problem
is
that
correlations
are
not
always
constant.
Engle
(2000,
2001)
therefore
suggested
an
alternative
estimation
of
conditional
correlations
with
dynamic
conditional
correlation
(DCC)
multivariate
GARCH
specification,
which
is
derived
from
Bollerslev’s
CCC
model,
but
allows
for
conditional
correlations
to
be
time-­‐‑varying.
The
conditional
covariance
matrix
is
then
written
as
(8)
Conditional
correlations
are
then
estimated
on
the
basis
of
exponential
smoothing:
,
(9)
where
smoothing
is
introduced
through
),0( tt HNr »
ttt DRRH =
[ ]ijtt hH =
{ }tit hdiagR ,=
ijtr ijr [ ]ijD r=
1=iir
[ ] ijjtitijt hhH r= ji ¹ Nji ££ ,1
tttt RDRH =
[ ] jit
stj
s
s
sti
s
s
s
stjsti
s
tji R ,
2
,
1
2
,
1
1
,,
,, =
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
=
-
¥
=
-
¥
=
¥
=
--
åå
å
elel
eel
r

8.
(10)
and
the
conditional
correlation
estimator
is
then
(11)
This
process
can
be
estimated
by
using
GARCH
(1,
1)
model:
(12)
The
unconditional
expectation
for
the
cross
product
is
and
for
variances, .
The
conditional
correlation
estimator
is
written
as
in
(11).
The
model
is
mean-­‐‑reverting
as
long
as ,
however,
when
the
sum
equals
to
1,
the
model
is
as
explained
in
(10)
and
(11).
Reference:
Asai
Manabu
and
Michael
McAleer,
2005,
“Dynamic
Correlations
in
Symmetric
Multivariate
SV
Models”,
in:
MODSIM
2005
International
Congress
on
Modelling
and
Simulation.
Modelling
and
Simulation
Society
of
Australia
and
New
Zealand,
Zerger
A.
and
Argent
R.,
eds.
Bollerslev,
Tim,
1990,
“Modelling
the
Coherence
in
Short-­‐‑Run
Nominal
Exchange
Rates:
A
Multivariate
Generalized
ARCH
Model,”
The
Review
of
Economics
and
Statistics
Vol.
72,
No.
3
(Aug.,
1990),
pp.
498-­‐‑505.
Engle,
Robert
F
,
2000,
“Dynamic
Conditional
Correlation
–A
Simple
Class
Of
Multivariate
Garch
Models,”
July
1999
(revised
May
2000),
a
research
supported
by
NSF
grant
SBR-­‐‑9730062
and
NBER,
27
pp.
Engle,
Robert,
2001,
“GARCH
101:
The
Use
of
ARCH/GARCH
Models
in
Applied
Econometrics,”
Journal
of
Economic
Perspectives,
Vol.
15,
No.
4
(Fall
2001),
pp.
157–168.
Engle,
Robert
F.
and
Kevin
Sheppard,
2001,
“Theoretical
and
Empirical
properties
of
Dynamic
Conditional
Correlation
Multivariate
GARCH”,
2001,
46
pp.
Hafner,
Christian
M.
and
Philip
Hans
Franses,
2009,
“A
Generalized
Dynamic
Conditional
Correlation
Model:
Simulation
And
Application
To
Many
Assets”,
Institute
de
Statistique,
Universite
Catholique
de
Louvain,
Workin
Paper
0904,
January
14,
2009,
25
pp.
Silvennoinen,
Annasvinna,
and
Timo
Terasvirta,
2008,
“Multivariate
GARCH
Models,
”
SSE/EFI
Working
Paper
Series
in
Economic
and
Finance,
January
2008,
No.
669,
25
pp.
( )( ) ( )1,,1,1,,, 1 --- +-= tjitjtitji qq leel
tjjtii
tji
tji
qq
q
,,
,,
,, =r
( ) stjsti
s
s
jijitjijitijitji qq --
¥
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ø
ö
çç
è
æ
-
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