week 10: var and garch model
DESCRIPTION
Week 10: VaR and GARCH model. Estimation of VaR with Pareto tail. Disadvantages (parametric and nonparametric method): For parametric method: the assumption of normal distribution is always not true. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/1.jpg)
Week 10: VaR and GARWeek 10: VaR and GARCH modelCH model
![Page 2: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/2.jpg)
Estimation of VaR with Pareto tailEstimation 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.
![Page 3: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/3.jpg)
Extreme Value Theory (EVT)Extreme Value Theory (EVT)• Assume {rt} i.i.d with distribution F(x), then the C
DF 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 un
known. 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.
![Page 4: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/4.jpg)
Three types for the above EVT distributionThree 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.
![Page 5: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/5.jpg)
Pareto tailsPareto 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.
![Page 6: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/6.jpg)
How to estimate the tail index How to estimate the tail index ββ??• 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 t
ail to compute the tail index, then β^ =n(c)/∑ri>c log(ri/c) and let H=1/β, then
![Page 7: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/7.jpg)
The properties of Hill estimatorThe properties of Hill estimator
![Page 8: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/8.jpg)
VaR for a derivativeVaR 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-1
asset/ pt-1option),
and Delta=d C(s, T,t, K, σ,r)/dS.
![Page 9: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/9.jpg)
Volatility modelingVolatility 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.
![Page 10: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/10.jpg)
• 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
![Page 11: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/11.jpg)
• 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
![Page 12: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/12.jpg)
![Page 13: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/13.jpg)
![Page 14: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/14.jpg)
![Page 15: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/15.jpg)
The advantages of RiskmetricsThe 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.
![Page 16: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/16.jpg)
Shortcoming of RiskmetricsShortcoming of Riskmetrics
• It ignores the fact that the long-run average variance tends to relative stable over time.
![Page 17: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/17.jpg)
GARCH modelGARCH model
• ARCH(1) model:
![Page 18: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/18.jpg)
The unconditional kurtosis of ARCH(1)The unconditional kurtosis of ARCH(1)
• Suppose the innovations are normal, then E(at
4| Ft-1)=3[ E(at2 | Ft-1) ]2
=3(α0+α1at-12 )2,
it follows that Eat
4 = 3α0 2
( 1 +α1 ) / [ ( 1 -α1
) ( 1 -3α1 2 )]
and Eat
4 /(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.
![Page 19: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/19.jpg)
![Page 20: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/20.jpg)
How to build an ARCH modelHow 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^)p
2)]+2p/n
• Model checking: Ljung-Box statistics. Q=n(n+2) ∑k=1 h (ρ^)
k2/(n-k).
![Page 21: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/21.jpg)
GARCH(1, 1) modelGARCH(1, 1) model
![Page 22: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/22.jpg)
Note that: Note that: σσ22==ωω/(1-/(1-αα--ββ))
![Page 23: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/23.jpg)
![Page 24: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/24.jpg)
![Page 25: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/25.jpg)
The forecast of variance of k-The forecast of variance of k-day cumulative returnday cumulative return
![Page 26: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/26.jpg)
If the returns have zero autocorrelation, then If the returns have zero autocorrelation, then the variance of the K-day returns isthe variance of the K-day returns is
For RiskMetrics model, it is just Kσt+12. But fo
r a GARCH model, we have
![Page 27: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/27.jpg)
If the returns have zero autocorrelation and If the returns have zero autocorrelation and σσt+1t+1<<σσ, then Var forecast of GARCH> RM, then Var forecast of GARCH> RM
![Page 28: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/28.jpg)
GARCH(p, q)GARCH(p, q)
• A process {Rt} is called a GARCH(p, q) model if Rt=σtεt, where
![Page 29: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/29.jpg)
Between GARCH and ARMA Between GARCH and ARMA modelmodel
• Let et=Rt2-σt
2, then Rt2 follows an ARMA mo
dels. This can be seen by Rt
2=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 Rt
2.
![Page 30: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/30.jpg)
Some propertiesSome 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(εt
4)1/2 ∑j=1 pαj /(1- ∑j=1
qβj )<1, then Rt has fourth moment.
![Page 31: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/31.jpg)
Theorem B: Under ∑j=1 pαj +∑j=1
qβj <1, {Rt2} i
s a causal and invertible ARMA(max{p, q}, q) process and exhibits heavier tails than those of εt in the sense of kurtosis.
![Page 32: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/32.jpg)
Some related modelSome 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.
![Page 33: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/33.jpg)
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
![Page 34: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/34.jpg)
Model B: E-GARCH modelModel B: E-GARCH model
![Page 35: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/35.jpg)
Weekend effectWeekend 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+βσt
2+ασt2 Zt
2+γITt+1,
where ITt+1 takes value 1 if day t+1 is a Monday, for example.
![Page 36: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/36.jpg)
More general EGARCH More general EGARCH
![Page 37: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/37.jpg)
IGARCH modelIGARCH model
• A GARCH(p, q) process is called an I-GARCH process if
![Page 38: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/38.jpg)
![Page 39: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/39.jpg)
How to estimate the parameters in a How to estimate the parameters in a GARCH modelGARCH model
• MLE:
![Page 40: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/40.jpg)
• Quasi-Maximum Likelihood Estimation (QMLE):
![Page 41: Week 10: VaR and GARCH model](https://reader034.vdocument.in/reader034/viewer/2022050820/56815db6550346895dcbe473/html5/thumbnails/41.jpg)
Whittle’s estimatorWhittle’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 {Rt
2} is
g(ω)= Var(et) |1- ∑j cj exp(ij ω)|-2 /2π.
And the Whittle’s estimator is given by minimizing ∑j=1
T-1 IT(ωj)/ g(ωj), where
IT(.) is the periodogram of {Rt2} , ωj=2jπ/ T.