Week 10: VaR and GARCH model

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Week 10 VaR and GARCH model
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Estimation of VaR with Pareto tail

• Disadvantages (parametric and nonparametric
  method)
• For parametric method the assumption of normal
  distribution is always not true.
• b. For nonparametric estimation, it is usually
  possible only for large a, but not for small a.
  So, one may expect to resort to
• use nonparametric regression for large ato
  estimate small one.

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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-(1kx)1/k, when k?0
• and 1-exp-exp(x), when k0.

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Three types for the above EVT distribution

• Type I k0, the Gumbel family with CDF F(x)
  1-exp-exp(x), x?R.
• b. Type II klt0, the Frechet family with CDF
• F(x) 1-exp-(1kx)1/k, if xlt-1/k and
• 1,
  otherwise.
• c. Type III kgt0, the Weibull family with CDF
• F(x) 1-exp-(1kx)1/k, if xgt-1/k and
• 1,
  otherwise.

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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(r1gtx)x-ßL(x), i.e., rt has
  Pareto tails, then Xt converges to a stable
  distribution with tail index ß. Here the tail
  index ß is always unknown. To evaluate the VaR of
  small a, we estimate the tail index ß first and
  apply the nonparametric method to draw the value
  of VaR for largea0, and then use the VaR(a0) to
  estimate VaR(a) . (how?) This is the so-called
  semi-parametric method.

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How to estimate the tail index ß?

• MLE when the whole distribution of rt is known.
• Linear regression suppose rt have a Pareto
  left tail, for xgt0, P(r1 lt-x) x-ßL(x), then
• log(k/n)log P(r1 ltr(k) ) -ßlog (-r(k))
  log L(-r(k)).
• Hill estimator based on MLE, one can get that
• n/ß log(r1/c) log(r2/c). log(rn/c), when
• P(r1 lt-x) 1-(x/c)-ß, xgtc. When one only uses
  the data in the tail to compute the tail index,
  then
• ß n(c)/?rigtc log(ri/c)
• and let H1/ß, then

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The properties of Hill estimator
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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 Deltad C(s, T,t, K, s,r)/dS.

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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) 1log (S0/K) (r?2/2)T,
• d2 (??T) 1log (S0/K) (r?2/2)T d1
  ??T.

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• In VaR, let Rt be the daily asset log-return and
  St be the daily closing price, then
• Rt1log(St1/ St) .
• Suppose the return is normal distributed with
  mean zero, then one can write it as

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• 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

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The advantages of Riskmetrics

• It is reasonable from the observed return that
  recent returns matter tomorrows variance than
  distance returns.
• It is simple only one parameter is contained in
  the model.
• Relative little data need to be stored to
  calculate tomorrows variance.

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Shortcoming of Riskmetrics

• It ignores the fact that the long-run average
  variance tends to relative stable over time.

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GARCH model

• ARCH(1) model

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The unconditional kurtosis of ARCH(1)

• Suppose the innovations are normal, then
  E(at4 Ft-1)3 E(at2 Ft-1) 2
• 3(a0a1at-12 )2,
• it follows that
• Eat4 3a0 2 ( 1 a1 ) / ( 1 -a1 ) ( 1 -3a1
  2 )
• and
• Eat4 /(Eat2 )2 3 ( 1 -a1 2 )/ ( 1 -3a1 2 )gt3.
• This shows that the tail distribution of at is
  heavier than that of a normal distribution.

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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(s)p2)2p/n
• Model checking Ljung-Box statistics.
• Qn(n2) ?k1 h (?) k2/(n-k).

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GARCH(1, 1) model
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Note that s2?/(1-a-ß)
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The forecast of variance of k-day cumulative
return
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If the returns have zero autocorrelation, then
the variance of the K-day returns is

• For RiskMetrics model, it is just Kst12. But for
  a GARCH model, we have

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If the returns have zero autocorrelation and
st1lts, then Var forecast of GARCHgt RM
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GARCH(p, q)

• A process Rt is called a GARCH(p, q) model if
  Rtstet, where

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Between GARCH and ARMA model

• Let etRt2-st2, then Rt2 follows an ARMA models.
  This can be seen by
• Rt2? ?i1 maxp, q ( aißi) Rt-i2et-
• ?j1 qßj 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.

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Some properties

• Theorem A The necessary and sufficient condition
  for a GARCH(p, q) being a unique strictly
  stationary process with finite variance is ?j1
  paj ?j1 qßj lt1. Further, ERt0, Cov(Rt, Rt-k)0
  for kgt0 and
• Var(Rt)?/(1- ?j1 paj ?j1 qßj ).
• In addition, if
• E(et4)1/2 ?j1 paj /(1- ?j1 qßj )lt1,
• then Rt has fourth moment.

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• Theorem B Under ?j1 paj ?j1 qßj lt1, Rt2
  is a causal and invertible ARMA(maxp, q, q)
  process and exhibits heavier tails than those of
  et in the sense of kurtosis.

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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 st 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.

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• The leverage effect model negative return
  increases variance by more than a positive return
  of the same magnitude.
• Model A let It1, if day ts return is negative
  and zero, otherwise and define

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Model B E-GARCH model
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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
• st12?ßst2ast2 Zt2?ITt1,
• where ITt1 takes value 1 if day t1 is a
  Monday, for example.

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More general EGARCH
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IGARCH model

• A GARCH(p, q) process is called an I-GARCH
  process if

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How to estimate the parameters in a GARCH model

• MLE

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• Quasi-Maximum Likelihood Estimation (QMLE)

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Whittles estimator

• By Theorem B, we see that Rt2 can be written as
• Rt2 c0?j cj Rt-j 2 et, where cjgt0.
• If Var(et) is finite, then the spectral density
  of the process Rt2 is
• g(?) Var(et) 1- ?j cj exp(ij ?)-2 /2p.
• And the Whittles estimator is given by
  minimizing
• ?j1T-1 IT(?j)/ g(?j), where
• IT(.) is the periodogram of Rt2 , ?j2jp/ T.