Transcript Risk - college of business administration

Lecture 8: Conditional Heteroscdastic Models
• The following topics will be covered:
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•
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ARCH
GARCH
GARCH extensions
Alternative Approach developed in French et al (1987)
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Basic Idea
ut  E(rt Ft 1 ) ,
 t2  Var(rt Ft 1 )  E[(rt  t ) 2 Ft 1 ]
Assume that rt follows a simple time series model such as a stationary ARMA(p,q) model:
rt  t  at , -- mean equation
p
q
i 1
i 1
where  t  0  i rt i   i at i
where p and q are non-negative integers.
at/σt is known as a standardized shock.
Combining (3.2) and (3.3), we have
 t2  Var(rt Ft 1 )  Var(at Ft 1 ) -- volatility equation
In other words, we attempt to model the non-conventional residual terms here.
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ARCH
Basic idea: (a) mean-corrected asset return at is serially uncorrelated, but dependent; (b)
the dependence of at can be described by a simple quadratic function of its lagged values.
One way to model this is to use an ARCH(m) model which assumes:
at   t  t ,
 t2   0  1at21     m at2m
where εt is a sequence of independent and identically distributed random variables with mean zero and
variance 1. – page 83 – this is to model the volatility equation.
Large past squared shock a t21 etc imply a large conditional variance for the mean-corrected
return at. Large shocks tend to be followed by another large shock. – see Figure 3.2 and 3.3
on page 84 and 85.
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Properties and Weakness of ARCH Model
For an ARCH(1) model,
E(at)=0
Var(at)=α0/(1-α1)
Kurtosis of at is greater than 3 – the shock at of a conditional Gaussian ARCH(1) model is
more likely than a Gaussian white noise series to produce “outliers”.
Weakness
(1) Symmetric effect of positive versus negative
(2) Overpredict the volatility because they respond slowly to large isolated shock to
the return series.
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Procedures in Building an ARCH Model
(1) Build an ARMA model to remove any linear dependence in the data
at = rt - μt
(2) Use the residual to test ARCH effects –
( SSR0  SSR1 ) / m
where SSR0 and SSR1 are defined in page 87.
F
SSR1 /(T  2m  1)
(3) Use PACF of at2 to determine the ARCH order
(4) Estimation – using MLE
a. Under normality assumption – page 88
b. Under a Student-t distribution – page 88-89
(5) Model checking

a
a. This is to examine at  t , either follows a normal or a standardized
t
Student-t distribution
(6) Forecasting
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Example
• The monthly log stock returns of Intel Corp (page 90-91)
proc autoreg data=intl; model lret=/garch=(q=3); output out=a1; run;
proc autoreg data=intl; model lret=/garch=(q=1); output out=a1; run;
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GARCH
Sometimes, it requires many parameters to adequately describe the volatility process of an asset return.
An alternative model is GARCH. For a log return series rt, assuming that the mean equation of the
process can be adequately described by an ARMA model. Let at = rt - µt be the mean-corrected log
return. at would follow a GARCH(m,s) model if
at   t  t ,
m
   0   i a
2
t
i 1
s
2
t i
   j t2 j
(3.13)
j 1
An ARCH model is a special case GARCH.
See page 93 – 95.
The modeling approach for ARCH can be used to build a GARCH model.
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Example
Example 3.3: the monthly excess returns of S&P 500 index starting from 1926 to 1992
SAS Program
/* joint estimate AR(3)-GARCH(1,1) Model */
proc autoreg data=msi;
model ret = / nlag=3 garch=(p=1,q=1);
run;
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IGARCH, GARCH-M, EGARCH
IGARCH
If the AR polynomial of the GARCH representation has a unit root, then we apply an
IGARCH model. An IGARCH(1,1) model can be written as
at   t  t ,  t2   0  1 t21  (1  1 )at21 , 1>β1>0.
GARCH-M:
A simple GARCH(1,1)-M model can be written as
at   t  t ,  t2   0  1at21  1 t21 , c is called the risk premium
rt    c t2  at ,
parameter. A positive c indicates that the return is positively related to its past volatility.
The formulation of GARCH-m model implies that there are serial correlations in the
return series rt.
EGARCH -- nonlinear
g( t )  t   [  t  E(  t )], -- this is to allow for asymmetric effects between positive
(   ) t  E (|  t |) if  t  0
and negative asset returns since g ( t )  
(   ) t  E (|  t |) if  t  0
An EGARCH(m, s) model can be written as
1  1 B     s B s
2
at   t  t , ln( t )   0 
g ( t 1 )
m
1  1 B     m B
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French, Schwert and Stambaugh (1987)’s Approach
n
n
rt   rt ,i , thus Var(rt | Ft 1 )  Var(rt ,i | Ft 1 )  2 Cov[(rt ,i , rt , j ) | Ft 1 ]
m
m
i 1
i 1
i j
Case 1: if rt ,i is a white noise series, then Var(rtm | Ft 1 )  nVar(rt ,1 ) ,
where Var(rt,1) can be estimated from the daily returns, specifically,

n
Var (rt ,1 ) 
 (r
i
n 1
. Then estimated monthly volatility is

n
 m2 
 rt ) 2
t ,i
n (rt ,i  rt ) 2
i
n 1
Case 2: If rt ,i follows an MA(1) model, then
Var(rtm | Ft 1 )  nVar(rt ,1 )  2(n 1)Cov(rt ,1, rt ,2 )

n
 
2
m
n (rt ,i  rt ) 2
i
n 1
n 1


 2 2(n  1)Cov(rt ,i  rt )(rt ,i 1  rt )
i 1
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Example
Example 3.4: consider monthly volatility of the log returns of S&P 500 index from January
1980 to December 1999. Calculate the volatility by three methods: (1) French, Schwert
and Stambaugh (1987) assuming rt,I is a white noise; (2) French, Schwert and Stambaugh
(1987) assuming rt,I follows a MA(1); (3) GARCH(1,1).
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Exercises
(1)
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