garch model
DESCRIPTION
general things about garch modelsTRANSCRIPT
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GARCH 1
GARCH Models
Introduction
ARMA models assume a constant volatility In finance, correct specification of volatility isessential
ARMA models are used to model the conditionalexpectation
They write Yt as a linear function of the past plus awhite noise term
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GARCH 2
1970 1975 1980 1985 1990 1995
102
101
100
year
|chan
ge| +
1/2 %
Absolute changes in weekly AAA rate
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GARCH 3
1994 1996 1998 2000 2002 2004 200640
30
20
10
0
10
20
30
year
perc
ent n
et re
turn
Cree Daily Returns
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GARCH 4
1994 1996 1998 2000 2002 2004 20060
5
10
15
20
25
30
35
year
perc
ent n
et re
turn
Cree Daily Returns
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GARCH 5
GARCH models of nonconstant volatility ARCH = AutoRegressive ConditionalHeteroscedasticity
heteroscedasticity = non-constant variance homoscedasticity = constant variance
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GARCH 6
ARMA unconditionally homoscedastic
conditionally homoscedastic
GARCH unconditionally homoscedastic, but
conditionally heteroscedastic
Unconditional or marginal distribution of Rt meansthe distribution when none of the other returns are
known.
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GARCH 7
Modeling conditional means and variances
Idea: If is N(0, 1), and Y = a+ b, then E(Y ) = aand Var(Y ) = b2.
general form for the regression of Yt on X1.t, . . . , Xp,tis
Yt = f(X1,t, . . . , Xp,t) + t (1)
Frequently, f is linear so thatf(X1,t, . . . , Xp,t) = 0 + 1X1,t + + pXp,t.
Principle: To model the conditional mean of Ytgiven X1.t, . . . , Xp,t, write Yt as the conditional mean
plus white noise.
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GARCH 8
Let 2(X1,t, . . . , Xp,t) be the conditional variance of Ytgiven X1,t, . . . , Xp,t. Then the model
Yt = f(X1,t, . . . , Xp,t) + (X1,t, . . . , Xp,t)t (2)
gives the correct conditional mean and variance.
Principle: To allow a nonconstant conditionalvariance in the model, multiply the white noise term
by the conditional standard deviation. This product
is added to the conditional mean as in the previous
principle.
(X1,t, . . . , Xp,t) must be non-negative since it is astandard deviation
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GARCH 9
ARCH(1) processes
Let 1, 2, . . . be Gaussian white noise with unitvariance, that is, let this process be independent
N(0,1).
ThenE(t|t1, . . .) = 0,
and
Var(t|t1, . . .) = 1. (3) Property (3) is called conditionalhomoscedasticity.
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GARCH 10
at = t
0 + 1a2t1. (4)
It is required that 0 0 and 1 0 It is also required that 1 < 1 in order for at to bestationary with a finite variance.
If 1 = 1 then at is stationary, but its variance is Define
2t = Var(at|at1, . . .)
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GARCH 11
From previous slide:
at = t
0 + 1a2t1.
Therefore
a2t = 2t{0 + 1a2t1}.
Since t is independent of at1 and Var(t) = 1E(at|at1, . . .) = 0, (5)
and
2t = 0 + 1a2t1. (6)
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GARCH 12
From previous slide:
2t = 0 + 1a2t1.
If at1 has an unusually large deviation then the conditional variance of at is larger than
usual
at is also expected to have an unusually large
deviation
volatility will propagate since at having a large
deviation makes 2t+1 large so that at+1 will tend to
be large.
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GARCH 13
The conditional variance tends to revert to theunconditional variance provided that 1 < 1 so that
the process is stationary with a finite variance.
The unconditional, i.e., marginal, variance of atdenoted by a(0)
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GARCH 14
The basic ARCH(1) equation is2t = 0 + 1a
2t1. (7)
This gives us
a(0) = 0 + 1a(0).
This equation has a positive solution if 1 < 1:a(0) = 0/(1 1).
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GARCH 15
If 1 = 1 then a(0) is infinite. It turns out that at is stationary nonetheless.
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GARCH 16
For an ARCH(1) process with 1 < 1:
Var(at+k|at, at1, . . .) = (0)+k1{a2t(0)} (0) as k .
In contrast, for any ARMA process:
Var(at+k|at, at1, . . .) = (0).
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GARCH 17
0 5 10 15 200.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
k
Cond
itiona
l var
ianc
e
ARCH(1) case 1
ARCH(1) case 2
AR(1)
Var(at+k|at, . . .) for AR(1) and ARCH(1). (0) = 1in both cases. For ARCH(1), 1 = .9. Case 1:
a2t = 1.5. Case 2: a2t = .5.
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GARCH 18
independence implies zero correlation but not viceversa
GARCH processes are good examples
dependence of the conditional variance on the
past is the reason the process is not independent
independence of the conditional mean on the past
is the reason that the process is uncorrelated
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GARCH 19
Example:
0 = 1, 1 = .95, = .1, and = .8
0 20 40 603
2
1
0
1
2
3White noise
0 20 40 601
2
3
4
5
6Conditional std dev
0 20 40 606
4
2
0
2
4ARCH(1)
0 20 40 6015
10
5
0
5AR(1)/ARCH(1))
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GARCH 20
0 5004
2
0
2
4White noise
0 5000
20
40
60Conditional std dev
0 50050
0
50
100ARCH(1)
0 50040
20
0
20
40
60AR(1)/ARCH(1))
40 20 0 20 400.0010.0030.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99
0.9970.999
Data
Prob
abilit
y
normal plot of ARCH(1)
Parameters: 0 = 1, 1 = .95, = .1, and = .8.
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GARCH 21
Comparison of AR(1) and ARCH(1)
AR(1)
Yt = (Yt1 ) + t.ARCH(1)
at = tt.
t =0 + 1a2t1.
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GARCH 22
Comparison of AR(1) and ARCH(1)
AR(1)
E(Yt) = .
Et(Yt) = + (Yt1 ).ARCH(1)
E(at) = 0.
Et(at) = 0.
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GARCH 23
Comparison of AR(1) and ARCH(1)
AR(1)
2t = 2.
ARCH(1)
2 =0
1 1 .
2t = 0 + 1a2t1.
Recall: 2t = Var(at|at1, . . .).
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GARCH 24
The AR(1)/ARCH(1) model
Let at be an ARCH(1) process Suppose that
ut = (ut1 ) + at.
ut looks like an AR(1) process, except that the noiseterm is not independent white noise but rather an
ARCH(1) process.
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GARCH 25
at is not independent white noise but is uncorrelated Therefore, ut has the same ACF as an AR(1)
process:
u(h) = |h| h.
a2t has the ARCH(1) ACF:
a2(h) = |h|1 h.
need to assume that both || < 1 and 1 < 1 inorder for u to be stationary with a finite variance
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GARCH 26
ARCH(q) models
let t be Gaussian white noise with unit variance at is an ARCH(q) process if
at = tt
and
t =
0 + qi=1
ia2ti
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GARCH 27
GARCH(p, q) models
the GARCH(p, q) model isat = tt
where
t =
0 + qi=1
ia2ti +pi=1
i2ti.
very general time series model: at is GARCH(pG, qG) and
at is the noise term in an ARIMA(pA, d, qA) model
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GARCH 28
Heavy-tailed distributions
stock returns have heavy-tailed oroutlier-prone distributions
reason for the outliers may be that the conditionalvariance is not constant
GARCH processes exhibit heavy-tails Example 90% N(0, 1) and 10% N(0, 25) variance of this distribution is (.9)(1) + (.1)(25) = 3.4 standard deviation is 1.844
distribution is MUCH different that a N(0, 3.4)distribution
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GARCH 29
5 0 5 100
0.1
0.2
0.3
0.4Densities
normalnormal mix
4 6 8 10 120
0.005
0.01
0.015
0.02
0.025Densities detail
normalnormal mix
2 1 0 1 20.0030.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997
Data
Prob
abilit
y
Normal plot normal
10 0 100.0030.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99 0.997
Data
Prob
abilit
y
Normal plot normal mix
Comparison on normal and heavy-tailed distributions.
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GARCH 30
For a N(0, 2) random variable X,P{|X| > x} = 2(1 (x/)).
Therefore, for the normal distribution with variance3.4,
P{|X| > 6} = 2(1 (6/3.4)) = .0011.
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GARCH 31
For the normal mixture population which hasvariance 1 with probability .9 and variance 25 with
probability .1 we have that
P{|X| > 6} = 2{.9(1 (6)) + .1(1 (6/5))}= (.9)(0) + (.1)(.23) = .023.
Since .023/.001 21, the normal mixture distributionis 21 times more likely to be in this outlier range than
the normal distribution.
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GARCH 32
Property Gaussian ARMA GARCH ARMA/
WN GARCH
Cond. mean constant non-const 0 non-const
Cond. var constant constant non-const non-const
Cond. distn normal normal normal normal
Marg. mean & var. constant constant constant constant
Marg. distn normal normal heavy-tailed heavy-tailed
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GARCH 33
All of the processes are stationary marginal meansand variances are constant
Gaussian white noise is the baseline process. conditional distribution = marginal distribution
conditional means and variances are constant
conditional and marginal distributions are normal
Gaussian white noise is the source of randomess forthe other processes
therefore, they all have normal conditional
distributions
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GARCH 34
Fitting GARCH models
Fit to 300 observation from a simulated AR(1)/ARCH(1)
Listing of the SAS program for the simulated data
options linesize = 65 ;
data arch ;
infile c:\courses\or473\sas\garch02.dat ;
input y ;
run ;
title Simulated ARCH(1)/AR(1) data ;
proc autoreg ;
model y =/nlag = 1 archtest garch=(q=1);
run ;
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GARCH 35
SAS output
Q and LM Tests for ARCH Disturbances
Order Q Pr > Q LM Pr > LM
1 119.7578
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GARCH 36
Standard Approx
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 0.4810 0.3910 1.23 0.2187
AR1 1 -0.8226 0.0266 -30.92
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GARCH 37
standard errors of the ARCH parameters are ratherlarge
approximate 95% confidence interval for 1 is.70 (2)(0.117) = (.446, .934)
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GARCH 38
Example: S&P 500 returns
60 65 70 75 80 85 90 950.2
0.15
0.1
0.05
0
0.05
0.1
0.15
year
retu
rn re
sidu
al
Residuals when the S&P 500 returns are
regressed against the change in the 3-month
T-bill rates and the rate of inflation.
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GARCH 39
This analysis uses RETURNSP = the return on the S&P 500
DR3 = change in the 3-month T-bill rate
GPW = the rate of wholesale price inflation
RETURNSP is regressed on DR3 and GPW (factormodel)
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GARCH 40
Model
RETURNSP = 0 + 1DR3 + 2GPW+ ut (8)
ut is an AR(1)/GARCH(1,1) process Therefore,
ut = 1ut1 + at,
at is a GARCH(1,1) process:at = tt
wheret =
0 + 1a2t1 + 1
2t1.
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GARCH 41
SAS Program
Key command:
proc autoreg ;
model returnsp = DR3 gpw/nlag = 1 archtest garch=(p=1,q=1);
returnsp = DR3 gpw specifies the regression model nlag = 1 specifies the AR(1) structure. garch=(p=1,q=1) specifies the GARCH(1,1)structure.
archtest specifies that tests of conditionalheteroscedasticity be performed
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GARCH 42
SAS output
The p-values of the Q and LM tests are all very small,less than .0001. Therefore, the errors in the regression
model exhibit conditional heteroscedasticity.
Ordinary least squares estimates of the regressionparameters are:
Standard Approx
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 0.0120 0.001755 6.86
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GARCH 43
Using residuals from the OLS estimates, theestimated residual autocorrelations are:
Estimates of Autocorrelations
Lag Covariance Correlation
0 0.00108 1.000000
1 0.000253 0.234934
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GARCH 44
Also, using OLS residuals, the estimate ARparameter is:
Estimates of Autoregressive Parameters
Standard
Lag Coefficient Error t Value
1 -0.234934 0.046929 -5.01
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GARCH 45
Assuming AR(1)/GARCH(1,1) errors, the estimatedparameters of the regression are:
Standard Approx
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 0.0125 0.001875 6.66
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GARCH 46
The estimated GARCH parameters are:AR1 1 -0.2016 0.0603 -3.34 0.0008
ARCH0 1 0.000147 0.0000688 2.14 0.0320
ARCH1 1 0.1337 0.0404 3.31 0.0009
GARCH1 1 0.7254 0.0918 7.91 > ARCH1 (0.1337) reasonably long persistence of volatility.
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GARCH 47
I-GARCH models
I-GARCH or integrated GARCH processes designedto model persistent changes in volatility
A GARCH(p, q) process is stationary with a finitevariance if
qi=1
i +
pi=1
i < 1.
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GARCH 48
A GARCH(p, q) process is called an I-GARCH process if
qi=1
i +
pi=1
i = 1.
I-GARCH processes are either non-stationary or havean infinite variance.
Here are some simulations of ARCH(1) processes:
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GARCH 49
0 0.5 1 1.5 2 2.5 3 3.5 4x 104
150
100
50
0
501 = .9
0 0.5 1 1.5 2 2.5 3 3.5 4x 104
100
0
100
2001 = 1
0 0.5 1 1.5 2 2.5 3 3.5 4x 104
5
0
5 x 104 1 = 1.8
Simulated ARCH(1) processes with 1 = .9, 1, and 1.8.
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GARCH 50
100 80 60 40 20 0 20 40
0.0010.0030.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99
0.9970.999
Prob
abilit
y
1 = .9
50 0 50 100 150
0.0010.0030.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99
0.9970.999
Prob
abilit
y
1 = 1
4 3 2 1 0 1 2 3 4x 104
0.0010.0030.01 0.02 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.98 0.99
0.9970.999
Prob
abilit
y
1 = 1.8
Normal plots of ARCH(1) processes with 1 = .9, 1, and 1.8.
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GARCH 51
Comments on the figures
all three processes do revert to their mean, 0 larger the value of 1 the more the volatility comes insharp bursts
processes with 1 = .9 and 1 = 1 looks similar none of the processes in the figure show muchpersistence of higher volatility
to model persistence of higher volatility, one needs anI-GARCH(p, q) process with q 1
Next figure shows simulations from I-GARCH(1,1)processes
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GARCH 52
0 1000 20000
5
10
15
20
25
301 = 0.95, Cond. std dev
0 1000 200020
10
0
10
20
301 = 0.95, GARCH(1,1)
0 1000 20000
5
10
15
20
25
301 = 0.4, Cond. std dev
0 1000 200040
20
0
20
401 = 0.4, GARCH(1,1)
0 1000 20000
50
100
150
2001 = 0.2, Cond. std dev
0 1000 2000300
200
100
0
100
200
3001 = 0.2, GARCH(1,1)
0 1000 20005
10
15
20
25
30
35
401 = 0.05, Cond. std dev
0 1000 2000100
50
0
50
1001 = 0.05, GARCH(1,1)
Simulations of I-GARCH(1,1) processes. 1 + 1 = 1
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GARCH 53
To fit I-GARCH in SAS:
proc autoreg ;
model returnsp =/nlag = 1 garch=(p=1,q=1,type=integrated);
run ;
The default value of type is nonneg which onlyconstrains the GARCH coefficients to be
non-negative.
type=integrated in addition imposes thesum-to-one constraint of the I-GARCH model
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GARCH 54
What does infinite variance mean?
let X be a random variable with density fX the expectation of X is
xfX(x)dx
provided that this integral is defined.
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GARCH 55
If 0
xfX(x)dx = (9)
and 0
xfX(x)dx = (10)then the expectation is, formally, + not defined if both integrals are finite, then the expectation is thesum of these two integrals
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GARCH 56
Exercise: fX(x) = 1/4 if |x| < 1 andfX(x) = 1/(4x
2) if |x| 1 fX is a density since
fX(x)dx = 1
Then expectation does not exist since 0
xfX(x)dx =
and 0
xfX(x)dx =
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GARCH 57
What are the implications of having no
expectation?
assume sample of iid sample from fX law of large numbers sample mean will converge tothe expectation
law of large numbers doesnt apply if expectation isnot defined
there is no point to which the sample mean canconverge
it will just wander without converging
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GARCH 58
0 0.5 1 1.5 2 2.5 3 3.5 4x 104
1
0.5
0
0.51 = .9
0 0.5 1 1.5 2 2.5 3 3.5 4x 104
6
4
2
0
2
41 = 1
0 0.5 1 1.5 2 2.5 3 3.5 4x 104
5
0
5
10
151 = 1.8
Sample means of ARCH(1) processes with 1 = .9, 1, and 1.8.
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GARCH 59
What are the implications of having infinite
variance
now suppose that the expectation of X exists andequals X
the variance
(x X)2fX(x)dx
if this integral is + then the variance is infinite law of large numbers sample variance will convergeto the variance
variance of X is infinity the sample variance willconverge to infinity
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GARCH 60
0 0.5 1 1.5 2 2.5 3 3.5 4x 104
0
2
4
6
81 = .9
0 0.5 1 1.5 2 2.5 3 3.5 4x 104
0
50
100
1501 = 1
0 0.5 1 1.5 2 2.5 3 3.5 4x 104
0
1
2
3
4 x 105 1 = 1.8
Sample variances: ARCH(1) with 1 = .9, 1, and 1.8.
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GARCH 61
GARCH-M processes
if we fit a regression model with GARCH errors could use the conditional standard deviation (t)
as one of the regression variables
when the dependent variable is a return the market demands a higher risk premium for
higher risk
so higher conditional variability could cause higher
returns
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GARCH 62
GARCH-M models in SAS add keyword mean,e.g.,
proc autoreg ;
model returnsp =/nlag = 1 garch=(p=1,q=1,mean);
run ;
or for I-GARCH-Mproc autoreg ;
model returnsp =/nlag = 1 garch=(p=1,q=1,mean,type=integrated);
run ;
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GARCH 63
GARCH-M example: S&P 500
GARCH(1,1)-M was fit in SAS is the regression coefficient for t = .5150 standard error = .3695
t-value = 1.39 p-value = .1633
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GARCH 64
since p-value = .1633 could accept the null hypothesis that = 0
no strong evidence that there are higher returns
during times of higher volatility.
volatility of S&P 500 is market risk so this issomewhat surprising (think of CAPM)
may be that the effect is small but not 0 ( ispositive, after all)
AIC criterion does select the GARCH-M model
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GARCH 65
E-GARCH
E-GARCH models are used to model the leverageeffect
prices become more volatile as prices decrease
E-GARCH, model is
log(t) = 0 +
qi=1
1g(ti) +pi=1
i log(ti),
where
g(t) = t + {|t| E(|t|)} log(t) can be negative no constraints onparameters
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GARCH 66
From the previous page:g(t) = t + {|t| E(|t|)}
To understand g note that
g(t) = E(|t|) + ( + )|t| if t > 0,and
g(t) = E(|t|) + ( )|t| if t < 0, typically, 1 < < 0 so that 0 < + < = .7 in the S&P 500 example E(|t|) =
2/pi = .7979 (good calculus exercise)
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GARCH 67
4 2 0 2 41
01
2
34
56
t
g( t
)
= 0.7
4 2 0 2 41
01
2
34
56
t
g( t
)
= 0
4 2 0 2 41
01
2
34
56
t
g( t
)
= 0.7
4 2 0 2 41
01
2
34
56
t
g( t
)
= 1
The g function for the S&P 500 data (top left
panel) and several other values of .
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GARCH 68
SAS fits the E-GARCH model fixed as 1
estimated
E-GARCH model is specified by using type=exp asin:
proc autoreg ;
model returnsp =/nlag = 1 garch=(p=1,q=1,mean,type=exp);
run ;
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GARCH 69
Back to the S&P 500 example
SAS can fit six different AR(1)/GARCH(1,1) models type = integrated, exp, or nonneg
GARCH-in-mean effect can be included or not
following table contains the AIC statistics models ordered by AIC (best fitting to worse)
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GARCH 70
Model AIC AIC
E-GARCH-M 1783.9 0E-GARCH 1783.1 0.8GARCH-M 1764.6 19.3GARCH 1764.1 19.8I-GARCH-M 1758.0 25.9I-GARCH 1756.4 27.5
AIC statistics for six AR(1)/GARCH(1,1) models
fit to the S&P 500 returns data. AIC is change
in AIC between a given model and E-GARCH-M.
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GARCH 71
AR(2) and E-GARCH(1,2)-M, E-GARCH(2,1)-M,and E-GARCH(2,2)-M models were tried
none of these lowered AIC
none had all parameters significant at p = .1
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GARCH 72
Listing of SAS output for the E-GARCH-Mmodel:
The AUTOREG Procedure
Estimates of Autoregressive Parameters
Standard
Lag Coefficient Error t Value
1 -0.234934 0.046929 -5.01
Algorithm converged.
Exponential GARCH Estimates
SSE 0.44211939 Observations 433
MSE 0.00102 Uncond Var .
Log Likelihood 900.962569 Total R-Square 0.1050
SBC -1747.2885 AIC -1783.9251
Normality Test 24.9607 Pr > ChiSq
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GARCH 73
Standard Approx
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 -0.003791 0.0102 -0.37 0.7095
DR3 1 -1.2062 0.3044 -3.96
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GARCH 74
The GARCH zoo
Heres a sample of other GARCH models mentioned in
Bollerslev, Engle, and Nelson (1994):
QARCH = quadratic ARCH TARCH = threshold ARCH STARCH = structural ARCH SWARCH = switching ARCH QTARCH = quantitative threshold ARCH vector ARCH diagonal ARCH factor ARCH
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GARCH 75
GARCH Models in Finance
Remember the problem of implied volatility it
depended on K and T !
Black-Scholes assumes a constant variance But GARCH effects are common so the Black-Scholesmodel is not adequate
Having a volatility function (smile) is a quick fix but not logical
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GARCH 76
3540
4550
55
0
50
100
1500.015
0.02
0.025
0.03
0.035
0.04
exercise pricematurity
impl
ied
vola
tility
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GARCH 77
Options can be priced assuming the log-returns are a
GARCH process (rather than a random walk)
Multinomial (not binomial) tree to have differentlevels of volatility
Need to keep track of price and conditional variance
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GARCH 78
Ritchken and Trevor use an NGARCH (nonlinearasymmetric GARCH) model:
log(St+1/St) = r + ht ht/2 +
htt+1
ht+1 = 0 + 1ht + 2ht(t+1 c)2where t is WhiteNoise(0, 1).
c = 0 is an ordinary GARCH model
is a risk premium
Under the risk-neutral (martingale) measure:log(St+1/St) = (r ht/2) +
htt+1,
ht+1 = 0 + 1ht + 2ht(t+1 c)2,where c = c+ and t is WhiteNoise(0, 1).
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GARCH 79
Now there are five unknown parameters:
h0 1, 2, and 3 c
These parameters are estimated by nonlinear
least-squares
Implied GARCH parameters
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GARCH 80
Volatility smile is explained as due to GARCH
effects:
nonconstant variance nonnormal marginal distribution
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GARCH 81
From Chapter 7, Bank of Volatility, of When
Genius Failed by Roger Lowenstein:
Early in 1998, Long-Term began to short large amounts
of equity volatility.
This simple trade, second nature to Rosenfeld and David
Modest, would be indecipherable to 999 out of 1,000
Americans.
Equity vol comes straight from the Black-Scholes model.
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GARCH 82
The stock market, for instance, typically varies
by about 15 percent to 20 percent a year.
And when the model told them that the markets were
mispricing equity vol, they were willing to bet the firm
on it.
The options market was anticipating volatility in
the stock market of roughly 20 percent. Long-term
viewed this as incorrect ... Thus, it figured that options
prices would sooner or later fall.
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GARCH 83
Long-Term began to short options on the S&P 500 are
similar European options. They were selling
volatility.
In fact, it sold insurance (options) both
waysagainst a sharp downturn and against a sharp
rise.
-
GARCH 84
1994 1996 1998 2000 20020
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08S&
P 50
0 ab
solu
te lo
g re
turn
-
GARCH 85
1997 1998 1999 2000 20010.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
S&P5
00, c
ondi
tiona
l SD,
ann
ualiz
ed
AR(1)/EGARCH(1,1) model
Annualized SD =253 daily variance
-
GARCH 86
data capm ;
set Sasuser.capm ;
logR_sp500 = dif(log(close_sp500)) ;
logR_msft = dif(log(close_msft)) ;
run ;
proc gplot ;
plot logR_sp500*date ;
run ;
proc autoreg ;
model logR_sp500 = /nlag=1 garch=(p=1,q=1,type=exp) ;
output out=outdata cev=cev ;
run ;
proc gplot ;
plot cev*date ;
run ;
-
GARCH 87
nlag p q E-GARCH M-GARCH AIC
1 2 2 no no 14,3661 1 2 no no 15,0021 1 2 yes no 15,1051 1 2 yes yes 15,1031 2 1 yes no 15,1051 1 1 yes no 15,1070 1 1 yes no 14,3662 2 2 yes no 15,1121 2 2 yes no 15,1042 2 3 yes no 15,1002 3 2 yes no 15,1001 3 2 yes no 15,101
-
GARCH 88
In fact, it sold insurance (options) both
waysagainst a sharp downturn and against a sharp
rise.
0 0.2 0.4 0.6 0.8 190
100
110st
ock
price
0 0.2 0.4 0.6 0.8 10
5
10
call
price
0 0.2 0.4 0.6 0.8 10
2
4
time
put p
rice
-
GARCH 89
The Greeks
C(S, T, t,K, , r) = price of an option
=
SC(S, T, t,K, , r) Delta
=
tC(S, T, t,K, , r) Theta
R = rC(S, T, t,K, , r) Rho
V =
C(S, T, t,K, , r) Vega
-
GARCH 90
Put-call parity
Put and call prices with same K and T are related:
P (S, T, t,K, , r) = C(S, T, t,K, , r) + er(Tt)K S.Therefore, the call and put have the same vegas
P (S, T, t,K, , r) =
C(S, T, t,K, , r)
but difference deltas
SP (S, T, t,K, , r) =
SC(S, T, t,K, , r) 1
0 < (call) = (d1) < 1
1 < (put) = (d1) 1 < 0
-
GARCH 91
From previous slide:
1 < (put) = (d1) 1 < 0
Suppose we buy N1 call options and N2 put options.
Delta of the portfolio is
N1(d1) +N2((d1) 1)which is zero if
N1N2
=1 (d1)(d1)
Hedging is possible with a put and call with different K
and T each then has its own value of d1
-
GARCH 92
Selling equity vol was very clever, but as Lowenstein
remarks:
This wasso unlike the partners credorank
speculation.