![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
=
-
¥
=
-
¥
=
¥
=
--
åå
å
elel
eel
r](https://image.slidesharecdn.com/topicsvolatility-181129134313/85/Topics-Volatility-7-320.jpg)
Topics Volatility
- 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 eµ += µ te te tz ts ttt zse = tz 2 1 2 11 2 )( --- +++= tttt I bsegaws 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 0³w 0³a 0³b 0³+ga 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 ses 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 ee ee ee 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 = - ¥ = - ¥ = ¥ = -- åå å elel eel 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 leel tjjtii tji tji qq q ,, ,, ,, =r ( ) stjsti s s jijitjijitijitji qq -- ¥ = -- å+÷÷ ø ö çç è æ - -- =-+-+= ,, 1 ,,1,,,1,,,, 1 1 () eeba b ba rrbrear r 1, =jir 1á+ ba