Week 10: VaR and
GARCH model
Estimation of VaR with Pareto tail
•
Disadvantages (parametric and nonparametric
method):
a. For parametric method: the assumption of
normal distribution is always not true.
b. For nonparametric estimation, it is usually
possible only for large α, but not for small α.
So, one may expect to resort to
use nonparametric regression for large αto
estimate small one.
Extreme Value Theory (EVT)
• Assume {rt} i.i.d with distribution F(x), then the
CDF of r(1) , denoted by Fn,1(x) is given by
Fn,1(x)=1-[1-F(x)]n.
In practice F(x) is unknown and then Fn,1(x) is
unknown. The EVT is concerned with finding two
sequence {an) and {bn} such that
(r(1) – an)/ bn
converges to a non-degenerated distribution as n
goes to infinity. Under the independent
assumption, the limit distribution is given by
F*(x)=1-exp[-(1+kx)1/k], when k≠0
and
1-exp[-exp(x)], when k=0.
Three types for the above EVT distribution
a. Type I: k=0, the Gumbel family with CDF:
F*(x)= 1-exp[-exp(x)], x∈R.
b. Type II: k<0, the Frechet family with CDF
F*(x)= 1-exp[-(1+kx)1/k], if x<-1/k and
=1,
otherwise.
c. Type III: k>0, the Weibull family with CDF
F*(x)= 1-exp[-(1+kx)1/k], if x>-1/k and
=1,
otherwise.
Pareto tails
• In risk management, we are mainly interested in
the Frechet family, which includes stable and
student-t distribution.
• We know that when {rt} i.i.d with tail distribution
P(|r1|>x)=x-βL(x), i.e., rt has Pareto tails, then {Xt}
converges to a stable distribution with tail index
β. Here the tail index β is always unknown. To
evaluate the VaR of small α, we estimate the tail
index β first and apply the nonparametric
method to draw the value of VaR for largeα0, and
then use the VaR(α0) to estimate VaR(α) . (how?)
This is the so-called semi-parametric method.
How to estimate the tail index β?
• MLE: when the whole distribution of rt is known.
• Linear regression: suppose {rt} have a Pareto left tail, for
x>0, P(r1 <-x)= x-βL(x), then
log(k/n)=log P(r1 <r(k) )= -βlog (-r(k)) + log L(-r(k)).
• Hill estimator: based on MLE, one can get that
n/β= log(r1/c)+ log(r2/c)+….+ log(rn/c), when
P(r1 <-x)= 1-(x/c)-β, x>c. When one only uses the data in
the tail to compute the tail index, then
β^ =n(c)/∑ri>c log(ri/c)
and let H=1/β, then
The properties of Hill estimator
VaR for a derivative
• Suppose that instead of a stock, one owns
a derivative whose value depends on the
stock. One can estimate a VaR for this
derivative by
VaR for derivative =LX VaR for asset,
where L=(Delta pt-1asset/ pt-1option),
and Delta=d C(s, T,t, K, σ,r)/dS.
Volatility modeling
• Volatility is important in options trading: for
example the price of a European call
option, the well known Black-Scholes
option pricing formula states that the price
is C(S0) = S0(d1) – Ke –rT (d2),
where
d1= (T) –1[log (S0/K) + (r+2/2)T],
d2= (T) –1[log (S0/K) + (r–2/2)T] = d1 –
T.
• In VaR, let Rt be the daily asset log-return
and St be the daily closing price, then
Rt+1=log(St+1/ St) .
Suppose the return is normal distributed with
mean zero, then one can write it as
• The variance as measure by square return,
exhibit strong autocorrelation, so that if the
recent period was one of high variance,
then tomorrow is tend to have high
variance. To capture this phenomenon, the
easiest way is to use
The advantages of Riskmetrics
• It is reasonable from the observed return
that recent returns matter tomorrow’s
variance than distance returns.
• It is simple: only one parameter is
contained in the model.
• Relative little data need to be stored to
calculate tomorrow’s variance.
Shortcoming of Riskmetrics
• It ignores the fact that the long-run
average variance tends to relative stable
over time.
GARCH model
• ARCH(1) model:
The unconditional kurtosis of ARCH(1)
• Suppose the innovations are normal, then
E(at4| Ft-1)=3[ E(at2 | Ft-1) ]2
=3(α0+α1at-12 )2,
it follows that
Eat4 = 3α0 2 ( 1 +α1 ) / [ ( 1 -α1 ) ( 1 -3α1 2 )]
and
Eat4 /(Eat2 )2 =3 ( 1 -α1 2 )/ ( 1 -3α1 2 )>3.
This shows that the tail distribution of at is
heavier than that of a normal distribution.
How to build an ARCH model
• Build an econometric model (e.g. an ARMA
model) for the return series to remove any linear
dependence in data abd use the residual series
of the model to test for ARCH effects
• Specify the ARCH order and perform estimation:
AIC(p)=log[(σ^)p2)]+2p/n
• Model checking: Ljung-Box statistics.
Q=n(n+2) ∑k=1 h (ρ^) k2/(n-k).
GARCH(1, 1) model
Note that: σ2=ω/(1-α-β)
The forecast of variance of kday cumulative return
If the returns have zero autocorrelation, then
the variance of the K-day returns is
For RiskMetrics model, it is just Kσt+12. But
for a GARCH model, we have
If the returns have zero autocorrelation and
σt+1<σ, then Var forecast of GARCH> RM
GARCH(p, q)
• A process {Rt} is called a GARCH(p, q)
model if Rt=σtεt, where
Between GARCH and ARMA model
• Let et=Rt2-σt2, then Rt2 follows an ARMA
models. This can be seen by
Rt2=ω+ ∑i=1 max{p, q} ( αi+βi) Rt-i2+et∑j=1 qβj et-j.
This also explains why simple GARCH
models, such as GARCH(1, 1) may
provide a parsimonious representation for
some complex autodependence structure
of Rt2.
Some properties
• Theorem A: The necessary and sufficient
condition for a GARCH(p, q) being a
unique strictly stationary process with finite
variance is ∑j=1 pαj +∑j=1 qβj <1. Further,
ERt=0, Cov(Rt, Rt-k)=0 for k>0 and
Var(Rt)=ω/(1- ∑j=1 pαj +∑j=1 qβj ).
In addition, if
E(εt4)1/2 ∑j=1 pαj /(1- ∑j=1 qβj )<1,
then Rt has fourth moment.
Theorem B: Under ∑j=1 pαj +∑j=1 qβj <1, {Rt2}
is a causal and invertible ARMA(max{p, q},
q) process and exhibits heavier tails than
those of εt in the sense of kurtosis.
Some related model
• GARCH-M model: when the conditional standard
deviation is a regressive variable, we called this
model as GARCH-in-mean (GARCH-M) model,
i.e.,
Yt =aXt +b σt+ at,
Where at is a GARCH model.
For example, when Y is a return, it may depends
on the variability, higher variability will lead to
higher returns.
The leverage effect model: negative
return increases variance by more
than a positive return of the same
magnitude.
Model A: let It=1, if day t’s return is negative
and zero, otherwise and define
Model B: E-GARCH model
Weekend effect
It is always known that days that followed
a weekend or a holiday have higher
variance than average day. We can try the
following model:
σt+12=ω+βσt2+ασt2 Zt2+γITt+1,
where ITt+1 takes value 1 if day t+1 is a
Monday, for example.
More general EGARCH
IGARCH model
• A GARCH(p, q) process is called an IGARCH process if
How to estimate the parameters in a
GARCH model
• MLE:
• Quasi-Maximum Likelihood Estimation
(QMLE):
Whittle’s estimator
By Theorem B, we see that Rt2 can be written as
Rt2 =c0+∑j cj Rt-j 2 +et, where cj>0.
If Var(et) is finite, then the spectral density of the
process {Rt2} is
g(ω)= Var(et) |1- ∑j cj exp(ij ω)|-2 /2π.
And the Whittle’s estimator is given by minimizing
∑j=1T-1 IT(ωj)/ g(ωj), where
IT(.) is the periodogram of {Rt2} , ωj=2jπ/ T.