econometrics iilipas.uwasa.fi/~sjp/teaching/ecmii/lectures/ecmiic5.pdf · t 1 (7) with !>0and0

Econometrics II

Seppo Pynnonen

Department of Mathematics and Statistics, University of Vaasa, Finland

Spring 2018

Seppo Pynnonen Econometrics II

Volatility Models Application in Risk Management

Part V

Volatility Models

As of Feb 14, 2018Seppo Pynnonen Econometrics II


1 Volatility Models

Background

Conditional Heteroskedasticity

ARCH-models

Properties of ARCH-processes

Estimation of ARCH models

Generalized ARCH models (GARCH)

ARCH-M Model

Asymmetric ARCH: TARCH, EGARCH, PARCH

The TARCH model

The EGARCH model

Power ARCH (PARCH)

Integrated GARCH (IGARCH)

Model Evaluation: Goodness of Fit

Predicting Volatility

Recent developments: Dynamic conditional score approach

Realized volatility

2 Application in Risk Management

Value at Risk (VaR)



Background1 Volatility Models

Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)



Background

Example 1

Consider the following daily close-to-close SP500 values [January 3, 2000

to Feb 3, 2017]

2000 2005 2010 2015

S&P 500 Daily Closing Prices and Returns[Jan 3, 2000 to Feb 3, 2017]

Date

700

1100

1500

1900

2300

Inde

x

−10

−50

510

Retu

rn (%

)



Background

Below are autocorrelations of the log-index.

Obviously the persistence of autocorrelations indicate that the series isintegrated.

The autocorrelations of the return series suggest that the returns might

be stationary



Background

par(mfrow = c(2, 1))

acf(log(spdf$aclose), col = "red", main = "Aucocorrelations of the S&P500 Log Index and Log Returns")

acf(spdf$ret[-1], col = "red", main = "", xlim = c(2, 35), ylim = c(-.1, .1))

0 5 10 15 20 25 30 35

0.0

0.4

0.8

Lag

ACF

Aucocorrelations of the S&P500 Log Index and Log Returns

5 10 15 20 25 30 35

−0.1

00.

000.

10

Lag

ACF



Background

1 2 3 4 5

AC -0.077 -0.053 0.019 -0.010 -0.042

PAC -0.077 -0.059 0.011 -0.011 -0.043

Obviously the (partial) autocorrelations are small.

This implies that there is little time dependence among returns between

different days.



Background

However, autocorrelations of the squared returns strongly suggest that

there is nonlinear time dependence in the returns series.

5 10 15 20 25 30 35

−0.1

0.00.1

0.20.3

0.40.5

Lag

ACF

Autocorrelations of S&P 500 Squared Returns



Background

Also a casual analysis with rolling k = 22-day (≈ month of tradingdays) mean volatility (annualized standard deviation)

mt =1

k

t∑u=t−k+1

ru, (1)

st =

√√√√252

k

t∑u=t−k+1

(ru −mt)2, (2)

t = k , . . .T



Background

2000 2005 2010 2015

S&P 500 Daily Returns and Volatility[Jan 3, 2000 to Feb 3, 2017]

Date

−10

−50

510

Retu

rn (%

)

040

80Vo

latili

ty (%

p.a

)



Background

Because squared observations are the building blocks of thevariance of the series, the results suggest that the variation(volatility) of the series is time dependent.

This leads to the so called ARCH-family of models.3

Note: Volatility not directly observable!!

Methods:

a) Implied volatility

b) Realized volatility

c) Econometric modeling (stochastic volatility, ARCH)

3The inventor of this modeling approach is Robert F. Engle (1982).Autoregressive conditional heteroskedasticity with estimates of the variance ofUnited Kingdom inflation. Econometrica, 50, 987–1008.



Conditional Heteroskedasticity1 Volatility Models

Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)




ARCH-modles

The general setup for ARCH models is

yt = x′tθ + ut (3)

with xt = (x1t , x2t , . . . , xpt)′, θ = (θ1, θ2, . . . , θp)′, t = 1, . . . ,T ,

andut |Ft−1 ∼ N(0, σ2

t ), (4)

where Ft is the information available at time t (usually the pastvalues of ut ; u1, . . . , ut−1), and

σ2t = var[ut |Ft−1] = ω + α1u

2t−1 + α2u

2t−2 + · · ·+ αqu

2t−q. (5)




ARCH-modles

Furthermore, it is assumed that ω > 0, αi ≥ 0 for all i andα1 + · · ·+ αq < 1.

For short it is denoted ut ∼ ARCH(q).

This reminds essentially an AR(q) process for the squaredresiduals, because defining νt = u2

t − σ2t , we can write

u2t = ω + α1u

2t−1 + α2u

2t−2 + · · ·+ αqu

2t−q + νt . (6)

Nevertheless, the error term νt is time heteroskedastic, whichimplies that the conventional estimation procedure used inAR-estimation does not produce optimal results here.




Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)





Consider (for the sake of simplicity) ARCH(1) process

σ2t = ω + αu2

t−1 (7)

with ω > 0 and 0 ≤ α < 1 and ut |ut−1 ∼ N(0, σ2t ).

(a) ut is white noise:(i) Constant mean (zero):

E[ut ] = E[Et−1[ut ]︸︷︷︸=0

] = E[0] = 0. (8)

Note Et−1[ut ] = E[ut |Ft−1], the conditional expectation giveninformation up to time t − 1.4

4The law of iterated expectations: Consider time points t1 < t2 such that Ft1

⊂ Ft2, then for any t > t2

Et1

[Et2

[ut ]]

= E[E[ut |Ft2

]|Ft1

]= E

[ut |Ft1

]= Et1

[ut ]. (9)





(ii) Constant variance: Using again the law of iteratedexpectations, we get

var[ut ] = E[u2t

]= E

[Et−1

[u2t

]]= E

[σ2t

]= E

[ω + αu2

t−1

]= ω + αE

[u2t−1

]... (10)

= ω(1 + α + α2 + · · ·+ αn) + αn+1E[u2t−n−1

]︸︷︷︸→0, as n→∞

= ω

(lim

n→∞

n∑i=0

αi

)

=ω

1− α.





(iii) Autocovariances: Exercise, show that autocovariances are zero,i.e., E [utut+k ] = 0 for all k 6= 0. (Hint: use the law of iteratedexpectations.)

(b) The unconditional distribution of ut is symmetric, butnonnormal:

(i) Skewness: Exercise, show that E[u3t

]= 0.

(ii) Kurtosis: Exercise, show that under the assumptionut |ut−1 ∼ N(0, σ2

t ), and that α <√

1/3, the kurtosis

E[u4t

]= 3

ω2

(1− α)2· 1− α2

1− 3α2. (11)

Hint: If X ∼ N(0, σ2) then E[(X − µ)4

]= 3(σ2)2 = 3σ4.





Because (1− α2)/(1− 3α2) > 1 we have that

E[u4t

]> 3

ω2

(1− α)2= 3 [var[ut ]]

2, (12)

we find that the kurtosis of the unconditional distributionexceed that what it would be, if ut were normally distributed.

Thus the unconditional distribution of ut is nonnormal andhas fatter tails than a normal distribution with variance equalto var[ut ] = ω/(1− α).





(c) Standardized variables:

Writezt =

ut√σ2t

(13)

then zt ∼ NID(0, 1), i.e., normally and independentlydistributed.Thus we can always write

ut = zt

√σ2t , (14)

where zt independent standard normal random variables (strictwhite noise).This gives us a useful device to check after fitting an ARCHmodel the adequacy of the specification: Check theautocorrelations of the squared standardized series.




Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)




Estimation (skipped material, not required for exam)

Given the modelyt = x′tθ + ut (15)

with ut |Ft−1 ∼ N(0, σ2t ), we have yt |{xt ,Ft−1} ∼ N(x′tθ, σ

2t ),

t = 1, . . . ,T .

Then the log-likelihood function becomes

`(η) =T∑t=1

`t(η) (16)

with

`t(η) = −1

2log(2π)− 1

2log σ2

t −1

2(yt − x′tθ)2/σ2

t , (17)

where η = (θ′, ω, α)′.66In practice more fat tailed (and/or skewed) conditional distributions are

assumed.Seppo Pynnonen Econometrics II



Estimation (skipped material)

Most popular conditional distributions:

Student t:

`t = − 12 log

(π(ν−2)Γ(ν/2)2

Γ((ν+1)/2)2

)− 1

2 log σ2t

− (ν+1)2 log

(1 +

(yt−x′tθ)2

σ2t (ν−2)

),

(18)

Γ(·) the gamma function, ν > 2 the degrees of freedom.

Generalized error distribtuion (GED):

`t = − 12 log

(Γ(1/r)3

Γ(3/r)(r/2)2

)− 1

2 log σ2t

−(

Γ(3/r)(yt−x′tθ)2

σ2t Γ(1/r)

)r/2

,

(19)

r the tail parameter, r = 2 a normal distribution, r < 2 fat-tailed.Seppo Pynnonen Econometrics II




Skewed-Student:

`t = − 12 log

(π(ν−2)Γ(ν/2)2(ξ+1/ξ)2

4s2Γ((ν+1)/2)2

)− 1

2 log σ2t

− log(

1 +(m+s(yt−x′tθ))2

σ2t (ν−2)

ξ−2It),

(20)

where

It =

{1, if (yt − x′tθ) ≥ −m/s−1, if (yt − x′tθ) < −m/s

(21)

ξ is the asymmetry parameter (ξ = 1, symmetric).

Also skewed normal and skewed ged are available in some versions.





The maximum likelihood (ML) estimate η is the value maximizingthe likelihood function, i.e.,

`(η) = maxη`(η). (22)

The maximization is accomplished by numerical methods.





Non-Gaussian series are estimated often by quasi maximumlikelihood (QML), i.e., as if the error were conditionally normal.

Under fairly general conditions the QML estimator of η isconsistent and asymptotically normal:

√T (η − η)→ N(0,J −1

η IηJ −1η ), (23)

where

Jη = −E[∂`t(η)

∂η∂η′

], (24)

and

Iη = E[∂`t(η)

∂η

∂`t(η)

∂η′

]. (25)

Remark 5.1: If the conditional distribution is Gaussian, Jη = Iη.




Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)





In practice the ARCH needs fairly many lags.

Usually far less lags are needed by modifying the model to

σ2t = ω + αu2

t−1 + βσ2t−1, (26)

with ω > 0, α > 0, β ≥ 0, and α + β < 1.

The model is called the Generalized ARCH (GARCH) model.

Usually the above GARCH(1,1) is adequate in practice.

Econometric packages call α (coefficient of u2t−1) the ARCH

parameter and β (coefficient of σ2t−1) the GARCH parameter.





Note again that defining νt = u2t − σ2

t , we can write

u2t = ω + (α + β)u2

t−1 + νt − βνt−1 (27)

a heteroskedastic ARMA(1,1) process.

Applying backward substitution, one easily gets

σ2t =

ω

1− β+ α

∞∑j=1

βj−1u2t−j (28)

an ARCH(∞) process.

Thus the GARCH term captures all the history from t − 2backwards of the shocks ut .





Imposing additional lag terms, the model can be extended toGARCH(r , q) model

σ2t = ω +

r∑j=1

βjσ2t−j +

q∑i=1

αu2t−i (29)

[c.f. ARMA(p, q)].

Nevertheless, as noted above, in practiceGARCH(1,1) is adequate.





Example 2

AR(2)-GARCH(1,1) model of SP500 returns estimated with conditionalnormal,ut |Ft−1 ∼ N(0, σ2

t ), and GED, ut |Ft−1 ∼ GED(0, σ2t )

The model is

rt = φ0 + φ1rt−1 + φ2rt−2 + ut

σ2t = ω + αu2

t−1 + βσ2t−1.

(30)




Generalized ARCH models (GARCH): Normal distribution

2000 2005 2010 2015

−10−5

05

10

S&P 500 Returns[Jan 3, 2000 to Feb 10, 2017]

Time





Using R-program with package fGarch

(see http://cran.r-project.org/web/packages/fGarch/index.html)

garchFit(formula = ret ~ arma(2, 0) + garch(1, 1), data = sp, trace = FALSE, cond.dist = "norm")

Conditional Distribution: norm

Estimate Std. Error t value Pr(>|t|)

mu 0.052659 0.012791 4.117 3.84e-05 ***

ar1 -0.059252 0.016360 -3.622 0.000293 ***

ar2 -0.028449 0.016126 -1.764 0.077695 .

omega 0.020125 0.003249 6.195 5.83e-10 ***

alpha1 0.100861 0.009423 10.704 < 2e-16 ***

beta1 0.883553 0.010143 87.105 < 2e-16 ***

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Standardised Residuals Tests:

Statistic p-Value

Jarque-Bera Test R Chi^2 453.3831 0

Shapiro-Wilk Test R W 0.9856473 0

Ljung-Box Test R Q(10) 13.32132 0.2062571

Ljung-Box Test R^2 Q(10) 21.23683 0.01950137

Information Criterion Statistics:

AIC BIC SIC HQIC

2.810593 2.819469 2.810589 2.813728




Generalized ARCH models (GARCH): Generalized error distribution

garchFit(formula = ret ~ arma(2, 0) + garch(1, 1), data = sp, cond.dist = "ged", trace = FALSE)

Conditional Distribution: ged


mu 0.071847 0.012041 5.967 2.42e-09 ***

ar1 -0.060232 0.014108 -4.269 1.96e-05 ***

ar2 -0.032606 0.015640 -2.085 0.0371 *

omega 0.016388 0.003606 4.544 5.51e-06 ***

alpha1 0.101241 0.011273 8.981 < 2e-16 ***

beta1 0.887951 0.011668 76.100 < 2e-16 ***

shape 1.355999 0.041612 32.587 < 2e-16 ***

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1


Statistic p-Value



Ljung-Box Test R Q(10) 14.14198 0.1666134

Ljung-Box Test R^2 Q(10) 19.22586 0.03748597

LM Arch Test R TR^2 18.82164 0.09292479


AIC BIC SIC HQIC

2.773163 2.783519 2.773158 2.776820




Generalized ARCH models (GARCH): Student t-distribution

garchFit(formula = ret ~ arma(2, 0) + garch(1, 1), data = sp, cond.dist = "std", trace = FALSE)

Conditional Distribution: std


mu 0.068585 0.012185 5.628 1.82e-08 ***

ar1 -0.060299 0.015390 -3.918 8.93e-05 ***

ar2 -0.037943 0.015649 -2.425 0.0153 *

omega 0.013362 0.003325 4.019 5.86e-05 ***

alpha1 0.100479 0.011022 9.116 < 2e-16 ***

beta1 0.893190 0.010926 81.749 < 2e-16 ***

shape 7.009217 0.762147 9.197 < 2e-16 ***

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1


Statistic p-Value



Ljung-Box Test R Q(10) 14.78491 0.1401027

Ljung-Box Test R^2 Q(10) 18.05858 0.05398341

LM Arch Test R TR^2 17.87345 0.1195874


AIC BIC SIC HQIC

2.777822 2.788177 2.777816 2.781479





Goodness of fit (AIC, BIC) is marginally better for GED. The shapeparameter estimate is < 2 (statistically significantly) indicated fat tails.

Otherwise the coefficient estimates are about the same.

LB-tests (Ljung-Box) test indicate that there are no materialautocorrelation left into the standardized residuals or squaredstandardized residuals, i.e., pass the white noise test.

JB-test (Jarque-Bera) test rejects the normality of the standardizedresiduals, which is typical for GARCH applications on returns, [Usuallythis affects mostly to standard errors. Common practice is on use somesort of robust standard errors (e.g. White).]

Thus in all, the AR(2)-GARCH(1,1) indicate satisfactory fit into the

data.





2000 2005 2010 2015

−6−4

−20

24

S&P 500 Return residuals[AR(1)−GED−GARCH(1,1)]

Time

2000 2005 2010 2015

050

100150

S&P 500 Conditional Volatility[Jan 3, 2000 to Feb 10, 2017]

Days

Volatility

(% p.a)

Absolute returnConditional volatility





The variance function can be extended by including regressors(exogenous or predetermined variables), xt , in it

σ2t = ω + αu2

t−1 + βσ2t−1 + πxt . (31)

Note that if xt can assume negative values, it may be desirable tointroduce absolute values |xt | in place of xt in the conditionalvariance function.

For example, with daily data a Monday dummy could beintroduced into the model to capture the non-trading over theweekends in the volatility.





Example 3

Monday effect in SP500 returns and/or volatility?

”Innovation” effect on the mean and volatility

yt = φ0 + φmMt + φyt−1 + ut

σ2t = ω + πMt + αu2

t−1 + βσ2t−1,

(32)

where Mt = 1 if Monday, zero otherwise.





R: rugarch, AR(2)-GARCH(1, 1) with Monday dummy, conditional t-distribution

Conditional Variance Dynamics

-----------------------------------

GARCH Model : sGARCH(1,1)

Mean Model : ARFIMA(2,0,0)

Distribution : std

Robust Standard Errors:


mu 0.064942 0.012173 5.33481 0.000000

ar1 -0.060167 0.012849 -4.68260 0.000003

ar2 -0.037529 0.016085 -2.33314 0.019641

mxreg1 -0.012063 0.031353 -0.38475 0.700425 % Monday effect on returns

omega 0.013391 0.007680 1.74354 0.081240

alpha1 0.100517 0.013682 7.34661 0.000000

beta1 0.893135 0.013430 66.50170 0.000000

vxreg1 0.000000 0.039285 0.00000 1.000000 % Monday effect on variance

shape 6.994795 0.780823 8.95823 0.000000 % degrees of freedom estimate

LogLikelihood : -5976.012

Akaike (AIC): 2.7805, Bayes (BIC): 2.7938

Clearly no evidence of Monday effect in returns or in variance.




GARCH models

Model Diagnostics:

Weighted Ljung-Box Test on Standardized Residuals

------------------------------------

statistic p-value

Lag[1] 0.07741 0.7808

Lag[2*(p+q)+(p+q)-1][5] 2.40611 0.8258

Lag[4*(p+q)+(p+q)-1][9] 6.24829 0.2182

d.o.f=2

H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals

------------------------------------

statistic p-value

Lag[1] 2.331 0.12680

Lag[2*(p+q)+(p+q)-1][5] 8.217 0.02602

Lag[4*(p+q)+(p+q)-1][9] 10.006 0.05001

d.o.f=2




GARCH-models

LB-test indicates yet some potential autocorrelation left in squaredstandardized residuals.

The autocorrelation graphs below show that the significant

autocorrelations, however, are very small (large number of observations,

here, 4,305, tends to make even trivial sample correlations significant).




AR(2)-GARCH(1,1) plots

1970 1974 1978 1982

−10

−50

510

Series with 2 Conditional SD Superimposed

Time

GARC

H mo

del :

sGAR

CH

1970 1974 1978 1982

−10

−50

510

Series with with 1% VaR Limits

Time

Retur

ns

GARC

H mo

del :

sGAR

CH

1970 1974 1978 1982

02

46

810

Conditional SD (vs |returns|)

Time

Volat

ility

GARC

H mo

del :

sGAR

CH

1 5 9 14 20 26 32

ACF of Observations

lag

ACF

−0.05

0.00

0.05

1 5 9 14 20 26 32

ACF of Squared Observations

lag

0.00.1

0.20.3

0.4

GARC

H mo

del :

sGAR

CH

1 5 9 14 20 26 32

ACF of Absolute Observations

lag

ACF

0.00.1

0.20.3

0.4

GARC

H mo

del :

sGAR

CH

−36 −22 −9 1 9 19 30

Cross−Correlations of Squared vs Actual Observations

lag

ACF

−0.10

−0.05

0.00

0.05

GARC

H mo

del :

sGAR

CH

Empirical Density of Standardized Residuals

zseries

Prob

ability

−6 −4 −2 0 2

0.00.1

0.20.3

0.40.5

0.6

Median: 0 | Mean: −0.0623●

●

normal Densitystd (0,1) Fitted Density

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−6−4

−20

2

std − QQ Plot

GARC

H mo

del :

sGAR

CH

ACF of Standardized Residuals

ACF

−0.04

−0.02

0.00

0.02

GARC

H mo

del :

sGAR

CH

ACF of Squared Standardized Residuals

ACF

−0.04

−0.02

0.00

0.02

0.04

GARC

H mo

del :

sGAR

CH

24

68

10

News Impact Curve

σ t2




Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)




ARCH-M Model

The regression equation may be extended by introducing thevariance function into the equation

yt = x′tθ + λg(σ2t ) + ut , (33)

where ut ∼ GARCH, and g is a suitable function (usually squareroot or logarithm).

This is called the ARCH in Mean (ARCH-M) model [Engle, Lilienand Robbins (1987)5].

The ARCH-M model is often used in finance, where the expectedreturn on an asset is related to the expected asset risk.

The coefficient λ reflects the risk-return tradeoff.

5Econometrica, 55, 391–407.Seppo Pynnonen Econometrics II



ARCH-M Model

Example 4

Does the daily mean return of SP500 depend on the volatility level?

Model AR(2)-GARCH(1, 1)-M with conditional t-distribution

yt = φ0 + φ1yt−1 + φ2yt−2 + λ√σ2t + ut

σ2t = ω + αu2

t−1 + γu2t−1dt−1 + βσ2

t−1,

(34)

where dt−1 = 1 if ut−1 < 0 and dt−1 = 0 if ut−1 ≥ 0, ut |Ft−1 ∼ tν (t-distribution

with ν degrees of freedom, to be estimated).




ARCH-M Model


-----------------------------------

GARCH Model : gjrGARCH(1,1)


Distribution : std



mu -0.004074 0.026120 -0.155985 0.876045

ar1 -0.054971 0.012766 -4.306086 0.000017

ar2 -0.028803 0.016263 -1.771137 0.076538

archm 0.052837 0.034168 1.546393 0.122010

omega 0.017516 0.003684 4.754470 0.000002

alpha1 0.000000 0.011675 0.000011 0.999991

beta1 0.895287 0.016684 53.662160 0.000000

gamma1 0.171338 0.020950 8.178302 0.000000

shape 8.268432 1.135133 7.284108 0.000000


AIC : 2.7428 BIC : 2.7561

Because archm estimate is not significance, hence no empirical evidence of volatility

on mean returns.




Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)





A stylized fact in stock markets is that downward movements arefollowed by higher volatility than upward movements.

A rough view of this can be obtained from thecross-autocorrelations of zt and z2

t , where zt defined in equation(13).

The third panel in the middle of the plots of Example 4 showsthese in terms of returns cross-autocorrelations.

The significant correlations shown by the graph suggest presence ofleverage effect.




Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)




The TARCH model

Threshold ARCH, TARCH (Zakoian 1994, Journal of EconomicDynamics and Control, 931–955 , Glosten, Jagannathan andRunkle 1993, Journal of Finance, 1779-1801) is given by[TARCH(1,1)]

σ2t = ω + αu2

t−1 + γu2t−1dt−1 + βσ2

t−1, (35)

where ω > 0, α, β ≥ 0, α + 12γ + β < 1, and dt = 1, if ut < 0

(bad news) and zero otherwise.

The impact of good news is α and bad news α + γ.

Thus, γ 6= 0 implies asymmetry.

Leverage exists if γ > 0.




The TARCH model

Example 5

SP500 returns, AR(2)-TGARCH(1,1) model (rugarch).

# Specify

spec.3 <- ugarchspec( # TGARCH (gjrGARCH)

variance.model = list( # Variance model

model = "gjrGARCH", # TGARCH specidication,

garchOrder = c(1, 1) # GARCH(1,1)

), # end of variance model

mean.model = list( # Mean model

armaOrder = c(2, 0) # AR(2)

), # end of mean model

distribution.model = "std" # conditional distribution (Student t)

) # end of ugarchspec

spec.3 # check out the specification

## Fit the specification

fit.3 <- ugarchfit(spec = spec.3, data = sp[, "ret"])

fit.3

plot(fit.3, which = "all")




TGARCH(1,1)

R: rugarch, AR(2)-TGARCH(1, 1) [i.e., GJR] with conditional t-distribution


-----------------------------------



Distribution : std



mu 0.036026 0.010926 3.2974 0.000976

ar1 -0.056725 0.012685 -4.4717 0.000008

ar2 -0.030416 0.016205 -1.8769 0.060527

omega 0.015530 0.003292 4.7176 0.000002

alpha1 0.000000 0.011562 0.0000 1.000000

beta1 0.898983 0.016589 54.1914 0.000000

gamma1 0.171677 0.021600 7.9479 0.000000

shape 8.296701 1.132325 7.3271 0.000000


AIC : 2.7428 BIC : 2.7547

gamma1 ≈ .172 is highly significant and positive, thereby indicating leverage.




TGARCH(1,1)


------------------------------------

statistic p-value

Lag[1] 0.005089 0.9431

Lag[2*(p+q)+(p+q)-1][5] 1.723152 0.9902

Lag[4*(p+q)+(p+q)-1][9] 5.158234 0.4184

d.o.f=2



------------------------------------

statistic p-value

Lag[1] 9.11 0.002542

Lag[2*(p+q)+(p+q)-1][5] 10.44 0.007110

Lag[4*(p+q)+(p+q)-1][9] 12.40 0.014720

d.o.f=2

Diagnostics results are similar to vanilla GARCH(1,1) in Example 3 above.




AR(2)-TGARCH(1,1) plots

1970 1972 1974 1976 1978 1980 1982

−10−5

05

10

Series with 2 Conditional SD Superimposed

Time

GARC

H mode

l : gjrG

ARCH

1970 1972 1974 1976 1978 1980 1982

−10−5

05

10

Series with with 1% VaR Limits

Time

Retur

ns

GARC

H mode

l : gjrG

ARCH

1970 1972 1974 1976 1978 1980 1982

02

46

810

Conditional SD (vs |returns|)

Time

Volati

lity

GARC

H mode

l : gjrG

ARCH

1 4 7 11 15 19 23 27 31 35

ACF of Observations

lag

ACF

−0.05

0.00

0.05

1 4 7 11 15 19 23 27 31 35

ACF of Squared Observations

lag

0.00.1

0.20.3

0.4

GARC

H mode

l : gjrG

ARCH

1 4 7 11 15 19 23 27 31 35

ACF of Absolute Observations

lag

ACF

0.00.1

0.20.3

0.4

GARC

H mode

l : gjrG

ARCH

−36 −26 −16 −7 0 6 13 21 29

Cross−Correlations of Squared vs Actual Observations

lag

ACF

−0.10

−0.05

0.00

0.05

GARC

H mode

l : gjrG

ARCH

Empirical Density of Standardized Residuals

zseries

Probab

ility

−6 −4 −2 0 2

0.00.1

0.20.3

0.40.5

0.6

Median: 0.03 | Mean: −0.0331●

●

normal Densitystd (0,1) Fitted Density

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−6−4

−20

2

std − QQ Plot

GARC

H mode

l : gjrG

ARCH

ACF of Standardized Residuals

ACF

−0.04

−0.02

0.00

0.02

GARC

H mode

l : gjrG

ARCH

ACF of Squared Standardized Residuals

ACF

−0.04

−0.02

0.00

0.02

GARC

H mode

l : gjrG

ARCH

24

68

1012

14

News Impact Curve

σ t2




Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)




The EGARCH model

Nelson (1991) (Econometrica, 347–370) proposed the ExponentialGARCH (EGARCH) model for the variance function of the form(EGARCH(1,1))

log σ2t = ω + β log σ2

t−1 + α |zt−1|+ γzt−1, (36)

where zt = ut/√σ2t is the standardized shock.

Again, the impact is asymmetric if γ 6= 0, and leverage is present ifγ < 0.




The EGARCH model

Example 6

MA(1)-EGARCH(1,1)-M estimation results.




The EGARCH model

Dependent Variable: SP500

Method: ML - ARCH (Marquardt) - Generalized error distribution (GED)

Sample (adjusted): 1/04/2000 3/27/2009

LOG(GARCH) = C(3) + C(4)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(5)

*RESID(-1)/@SQRT(GARCH(-1)) + C(6)*LOG(GARCH(-1))

=============================================================

Variable Coefficient Std. Error z-Statistic Prob.

-------------------------------------------------------------

C 0.017476 0.016027 1.090422 0.2755

MA(1) -0.072900 0.021954 -3.320628 0.0009

=============================================================

Variance Equation

=============================================================

C(3) -0.066913 0.013105 -5.106018 0.0000

C(4) 0.081930 0.016677 4.912873 0.0000

C(5) -0.121991 0.012271 -9.941289 0.0000

C(6) 0.986926 0.002323 424.9402 0.0000

=============================================================

GED PARAM 1.562564 0.051706 30.22032 0.0000

=============================================================

...

=============================================================

Inverted MA Roots .07

=============================================================

Again statistically significant leverage.




Power ARCH (PARCH)

1 Volatility Models

Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)




Ding, Granger, and Engle (1993). A long memory property ofstock market returns and a new model. Journal of EmpiricalFinance. PARC(1,1)

σδt = ω + βσδt−1 + α(|ut−1| − γut−1)δ, (37)

where γ is the leverage parameter. Again γ > 0 implies leverage.




Power ARCH (PARCH)

Example 7

R: fGarch::garchFit MA(1)-APARCH(1,1) results for SP500 returns.

R parametrization: MA(1), mu and theta

gfa <- fGarch::garchFit(sp500r~arma(0,1) + aparch(1,1), data = sp500r, cond.dist = "ged", trace=F)

Mean and Variance Equation:

data ~ arma(0, 1) + aparch(1, 1)

Conditional Distribution: ged

Error Analysis:


mu 0.014249 0.016825 0.847 0.397049

ma1 -0.076112 0.021454 -3.548 0.000389 ***

omega 0.013396 0.003474 3.856 0.000115 ***

alpha1 0.055441 0.008217 6.747 1.51e-11 ***

gamma1 1.000000 0.016309 61.316 < 2e-16 ***

beta1 0.935567 0.007620 122.775 < 2e-16 ***

delta 1.207412 0.203576 5.931 3.01e-09 ***

shape 1.579328 0.068530 23.046 < 2e-16 ***

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Log Likelihood: -3388.234 normalized: -1.459816




Power ARCH (PARCH)

The leverage parameter (’gamma1’ ) estimates to unity.

δ = 1.207412 (0.203576) does not deviate significantly from unity

(standard deviation process).




Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)





Often in GARCH α + β ≈ 1. Engle and Bollerslev (1989).Modelling the persistence of conditional variances, EconometricsReviews 5, 1–50, introduce integrated GARCH with α + β = 1.

σ2t = ω + αu2

t−1 + (1− α)σ2t−1. (38)

Close to the EWMA (Exponentially Weighted Moving Average)specification

σ2t = αu2

t−1 + (1− α)σ2t−1 (39)

favored often by practitioners (e.g. RiskMetrics).

Unconditional variance does not exist [more details, see Nelson(1990). Stationarity and persistence in in the GARCH(1,1) model.Econometric Theory 6, 318–334].




Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)




GARCH Model Evaluation

The success of the specified model to capture the empiricalcharacteristics of observed data is evaluated by variousgoodness-of-fit measures and tests.

If ut follows a GARCH-model, the focus in the goodness-of-fitevaluation is in the standardized series

zt =utσt

that are iid (independent and identically distributed) if σt is thecorrect volatility model.





Accordingly, is the estimated GARCH model is correct, thestandardized estimated values

zt =utσt

(40)

should be (empirically) iid, where σt is the square root of theestimated GARCH variance σ2.

Comparison between different GARCH models can be based incriterion functions (AIC, BIC).

For the selected model autocorrelations of z2t (as well as zt) should

be empirically zero.





Testing for zero autocorrelations are based as in ordinary timeseries analysis on individual autocorrelations and Ljung-BoxQ-statistic (see Sec. 4)

Q(m) = T (T + 2)m∑

k=1

1

T − kr2k (41)

where rk is the sample autocorrelation of squared zts in GARCHanalysis and m is the number of tested autocorrelations.

If there is no autocorrelation, i.e., the null hypothesis

H0 : ρ1 = ρ2 = · · · = ρm = 0,

where ρk = corr[z2t , z

2t+k

], k = 1, . . . ,m, holds, Q(m) ∼ χ2

m

(approximately).





Example 8 (Apple returns Jan 2005 – Feb 2018)

open high low close vol aclose ret-%

2005-01-04 4.56 4.68 4.50 4.57 274202600 3.09 1.02

2005-01-05 4.60 4.66 4.58 4.61 170108400 3.12 0.87

2005-01-06 4.62 4.64 4.52 4.61 176388800 3.12 0.08

2005-01-07 4.64 4.97 4.62 4.95 556862600 3.35 7.03

2005-01-10 4.99 5.05 4.85 4.93 431327400 3.33 -0.42

2005-01-11 4.88 4.94 4.58 4.61 652906800 3.12 -6.59

.

.

2018-02-06 154.83 163.72 154.00 163.03 68243800 162.37 4.09

2018-02-07 163.09 163.40 159.07 159.54 51608600 158.89 -2.16

2018-02-08 160.29 161.00 155.03 155.15 54390500 154.52 -2.79

2018-02-09 157.07 157.89 150.24 156.41 70672600 156.41 1.22

2018-02-12 158.50 163.89 157.51 162.71 60819500 162.71 3.95

2018-02-13 161.95 164.75 161.65 164.34 32282900 164.34 1.00




Specify first the best fitting conditional distribution for TGARCH(1,1)model in terms of AIC and BIC:

==================

AIC BIC

------------------

norm 4.077 4.086

snorm 4.078 4.089

std 3.997* 4.008* best

sstd 3.997 4.010

ged 4.005 4.016

sged 4.005 4.018

nig 3.998 4.011

ghyp 3.998 4.012

jsu 3.997 4.010

==================

Both criteria end up with Student t-distribution.




Estimation results of TGARCH(1,1) with Student t conditionaldistribution.

*---------------------------------*

* GARCH Model Fit *

*---------------------------------*


-----------------------------------



Distribution : std



mu 0.138839 0.026818 5.1770 0.000000

omega 0.050989 0.032663 1.5611 0.118511

alpha1 0.034519 0.014135 2.4421 0.014602

beta1 0.914287 0.031020 29.4742 0.000000

gamma1 0.091469 0.035459 2.5795 0.009893

shape 4.917627 0.434076 11.3290 0.000000





Diagnostics for the fitted model:


------------------------------------

statistic p-value

Lag[1] 3.593 0.05803

Lag[2*(p+q)+(p+q)-1][2] 3.650 0.09367

Lag[4*(p+q)+(p+q)-1][5] 5.627 0.10998

d.o.f=0



------------------------------------

statistic p-value

Lag[1] 0.07995 0.7774

Lag[2*(p+q)+(p+q)-1][5] 0.83804 0.8951

Lag[4*(p+q)+(p+q)-1][9] 1.84842 0.9223

d.o.f=2

Weighted ARCH LM Tests

------------------------------------

Statistic Shape Scale P-Value

ARCH Lag[3] 0.1313 0.500 2.000 0.7171

ARCH Lag[5] 1.4713 1.440 1.667 0.5999

ARCH Lag[7] 1.7169 2.315 1.543 0.7770

The results indicate that there are no autocorrelations in standarized

values (returns) or their squares. Also the ARCH LM test shows no

further dependencies in the standardized series.



Predicting Volatility1 Volatility Models

Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)





Predicting with the GARCH models is straightforward.

Generally a k-period forward prediction is of the form

σ2t|k = Et

[u2t+k

]= E

[u2t+k |Ft

](42)

k = 1, 2, . . ..





Becauseut = σtzt , (43)

where generally zt ∼ i.i.d(0, 1).

Thus in (42)

Et

[u2t+k

]= Et

[σ2t+kz

2t+k

]= Et

[σ2t+k

]Et

[z2t+k

](zt are i.i.d(0, 1))

= Et

[σ2t+k

].

(44)

This can be utilized to derive explicit prediction formulas in mostcases.





ARCH(1):σ2t+1 = ω + αu2

t (45)

σ2t|k = ω(1−αk−1)

1−α + αk−1σ2t+1

= σ2 + αk−1(σ2t+1 − σ2),

(46)

whereσ2 = var[ut ] =

ω

1− α(47)

Recursive fromula:

σ2t|k =

σ2t+1 for k = 1

ω + ασ2t|k−1 for k > 1

(48)





GARCH(1,1):σ2t+1 = ω + αu2

t + βσ2t . (49)

σ2t|k = ω(1−(α+β)k−1)

1−(α+β) + (α + β)k−1σ2t+1

= σ2 + (α + β)k−1(σ2t+1 − σ2),

(50)

whereσ2 =

ω

1− α− β. (51)

Recursive fromula:

σ2t|k =

σ2t+1 for k = 1

ω + (α + β)σ2t|k−1 for k > 1

(52)





IGARCH:

σ2t+1 = ω + αu2

t + (1− α)σ2t . (53)

σ2t|k = (k − 1)ω + σ2

t+1. (54)

Recursive fromula:

σ2t|k =

σ2t+1 for k = 1

ω + σ2t|k−1 for k > 1

(55)





TGARCH:σ2t+1 = ω + αu2

t + γu2t dt + βσ2

t . (56)

σ2t|k =

ω(1−(α+ 12γ+β)k−1)

1−(α+ 12γ+β)

+ (α + 12γ + β)k−1σ2

t+1

= σ2 + (α + 12γ + β)k−1(σ2

t+1 − σ2)

(57)

withσ2 =

ω

1− α− 12γ − β

. (58)

Recursive fromula:

σ2t|k =

σ2t+1 for k = 1

ω + (α + 12γ + β)σ2

t|k−1 for k > 1(59)





EGARCH and APARCH prediction equations are a bit moreinvolved.

Recursive formulas are more appropriate in these cases.

The volatility forecasts are applied for example in Value At Riskcomputations.





Evaluation of predictions unfortunately not that straightforward!(see, Andersen and Bollerslev (1998) International EconomicReview.)




TGARCH prediction example

Example 9

Using the above recursive formula in equation (59) the graph below

shows the estimated TGARCH volatility of Example 5 for the last two

years (500 trading days) together with absolute returns (annualized) and

predicted volatility for 66 days forwards, starting from February 16, 2017

(notice the convergence prediction towards the unconditional, or long run

volatility).




TGARCH estimated and predicted volatilities

2016 2017

010

2030

4050

60

S&P 500 Estimated and Predicted Return Volatility

Day

Volat

ility (

%, p

.a)

Estimated volatPredicted volatAbsolute returnsLong run volat



Recent developments: Dynamic conditional score approach1 Volatility Models

Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)




Dynamic conditional score (DCS)

Creal, Koopman, and Lucas (2008) and Harvey and Chakravarty(2008)2 Generally, assume the conditional distribution of yt+1,given information up to time point t is ft(yt+1).

2Harvey, A.C. and T. Chajravarty (2008), Beta-t-(E)GARCH. WorkingPaper in Economics, Faculty of Economics, Cambridge Univesity, UK.

Creal, D., S.J. Koopman, and A. Lucas 2008, A general framework forobservation driven time-varying parameter models, Tinnbergen IntituteDiscussion Papers (TI 2008-108).

Creal, D., S.J. Koopman, and A. Lucas 2013, Generalized autoregressive scoremodels with applications, Journal of Applier Econometrics 28, 777-795.




Dynamic conditional score

Then applied to conditional ARCH-framework, the DCSrepresentation for GARCH(1,1) is

σ2t+1 = ω + φut + κσ2

t , (60)

where

ut = σ2t

∂ log ft(yt)

∂σ2t

, (61)

where ∂ log ft(yt+1)/∂σ2t is the conditional score of the

distribution.





If ft is the normal distribution, then (60) reduces to

σ2t+1 = ω + φ(y2

t − σ2t ) + κσ2

t , (62)

such that ut = y2t − σ2

t , φ = α, and κ = β − α.

If ft(yt) is t-distribution with ν degrees of freedom,

ut =(ν + 1)y2

t

(ν − 2)σ2t + y2

t

− 1, −1 ≤ ut ≤ ν, ν > 2. (63)





In particular EGARCH-specification of DCS has proven to beuseful. As such, in the Gaussian (normal distribution) case thedynamic equaiton applies to log σ2

t+1.





For the t-distributon with ν > 2 degrees of freedom, it turns outbetter work with ψt+1 = (ν − 2)1/2σt+1 and defineλt+1 = logψt+1, such that the dynamics to be dealt with is

λt+1 = δ + φλt + κut (64)

where

ut =(ν + 1)y2

t

ν exp(2λt) + y2t

− 1. (65)





The leverage effect is incorporated by specification

λt+1 = δ + φλt + κut + κ∗sign(−yt)(ut + 1) (66)

where sign(−yt) = 1 for yt < 0 and −1 for yt ≥ 0, such thatκ∗ > 0 indicates leverage.

R-package betategarch can be used for estimation (see theR-example on the course web-site).

Web-site for CDS (or GAS): http://gasmodel.com/index.htm





Example 10

Consider Apple inc. Daily returns.

On Thursday September 28, 2000 the company issued a profit warning,implying a one day plunge of 52% (78% in log-difference) in the stockprice.

Below are estimation results for usual EGARCH with conditionalt-distribution using R rugarch package and CDS using R betategarch

package.





EGARCH(1,1) with conditional t-distribution results (rugarch):


-----------------------------------

GARCH Model : eGARCH(1,1)


Distribution : std

Optimal Parameters

------------------------------------


mu 0.153169 0.029941 5.1158 0.000000

omega 0.012111 0.001378 8.7873 0.000000

alpha1 -0.022861 0.008203 -2.7868 0.005323

beta1 0.992399 0.000112 8871.8528 0.000000

gamma1 0.106220 0.012831 8.2785 0.000000

shape 5.364418 0.430779 12.4528 0.000000


Information Criteria

------------------------------------

Akaike 4.4923

Bayes 4.5021





CDS beta-t-EGARCH estimation results (betategarch):

Coefficients:

omega phi1 kappa1 kappastar df

Estimate: 0.8534224 0.992201606 0.033855484 0.013010095 5.3635420

Std. Error: 0.1222253 0.003348272 0.005721808 0.003547565 0.4217289

Log-likelihood: -8537.771862

BIC: 4.502041

Both estimation results indicate leverage, otherwise there are no material

differences in goodness of fit. However, the plot of the conditional

standard deviations (Run the R-code on the web-site) show the difference

of the sensitivity of conditional standard deviation paths to a single

outlier.



Realized volatility1 Volatility Models

Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)



Realized volatility

Realized volatility

Under certain assumptions, volatility during a period of time canbe estimated more and more precisely as the frequency of thereturns increases.

Daily (log) returns rt are sums of intraday returns (e.g. returnscalculated at 30 minutes interval)

rt =m∑

h=1

rt(h) (67)

where rt(h) = logPt(h)− logPt(h − 1) is the day’s t intradayreturn in time interval [h − 1, h], h = 1, . . . ,m, Pt(h) is the priceat time point h within the day t, Pt(0) is the opening price andPt(m) is the closing price.



Realized volatility

Realized volatility

The realized variance for day t is defined as

σ2t,m =

m∑h=1

r2t (h) (68)

and the realized volatility is σt,m =√σ2t,m which is typically presented in

percentages per annum (i.e., scaled by the square root of the number oftrading days and presented in percentages).

Under certain conditions it can be shown that σt,m → σt as m→∞, i.e.,

when the intraday return interval → 0.

For a resent survey on RV, see: Andersen, T.G. and L. Benzoni (2009). Realized

volatility. In Handbook of Financial Time Series, T.G. Andersen, R.A. Davis, J-P.

Kreiss and T. Mikosh (eds), Springer, New York, pp. 555–575. SSRN version is

available at:

http://papers.ssrn.com/sol3/papers.cfm?abstract id=1092203&rec=1&srcabs=903659



1 Volatility Models

Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)



Financial industry has several types of risk categories, like: Creditrisk, operational risk, and market risk.

We consider here quantifying market risk by VaR (Value at Risk).



Value at Risk (VaR)1 Volatility Models

Background


ARCH-models




ARCH-M Model


The TARCH model

The EGARCH model

Power ARCH (PARCH)





Realized volatility


Value at Risk (VaR)



Value at Risk (VaR)

Value at Risk (VaR) can be utilized to quantify market (risk due togeneral market decline)

For a given probability p and time horizon h VaR indicates the risk for a

portfolio to loose the amount of VaR or more.

or

VaR indicates the maximum loss with probability 1− p in timehorizon h.



Value at Risk (VaR)

In probability terms VaR is simply the pth quantile of thedistribution.

Thus, given probability p and time horizon h, for a long positionVaR > 0 is the threshold (quantile) loss defined by

p = P[∆Vt(h) ≤ −VaR] = Fh(−VaR) (69)

where Vt is the value of the long position (investment) at timepoint t,

∆Vt(h) = Vt+h − Vt (70)

is the change in the value, and Fh(·) is the distribution function of∆Vt(h).



Value at Risk (VaR)

For a short position

p = P[∆Vt(h) ≥ VaR] = 1− Fh(VaR). (71)

Important: VaR deals with the tail-probabilities.

Fh a key problem.



Value at Risk (VaR)

In practice VaR is convenient to compute in terms of (one period)returns.

rt =∆Vt

Vt−1.

Thus, assuming that rt ∼ (µt , σ2t ) and standardizing

zt =rt − µtσt

. (72)



Value at Risk (VaR)

We can write generally:

Long position:

−VaR = Vt × (σtzp + µt) (73)

Short position:

VaR = Vt × (σtz1−p + µt) (74)

where Vt is the amount of investment and zp is the percentile ofthe distribution of the standardized return zt defined in (72).



Value at Risk (VaR)

Remark 5.2: If standardized returns are t-distributed with ν > 2 degreesof freedom then, e.g. for long position:

− VaR = V ×

(σt√

ν/(2− ν)tp(ν) + µt

), (75)

where σt is the scale parameter (estimated as a standard deviation) of

the t-distribution and tp(ν) is the p-percentile obtained from

p = P[t ≤ tp(ν)].



Value at Risk (VaR)

Typically:

p is 0.01 or 0.05 ”5 or 1 percent VaR”h is set by a regulation committee, e.g. from 1 or 10 daysDaily observations



Value at Risk (VaR)

Zero Mean Return

Example 11

Portfolio manager has 10Meur position stocks with volatility 25% (p.a)and changes are assumed normally distributed with zero mean.

One day 5% VaR?

Assuming 252 trading days then ∆Vt(1) ∼ N(0, 0.252/252),−z.05 = z.95 = 1.645

VaR = 10× (0.25/√

252)× 1.645 ≈ 0.259Meur,



Value at Risk (VaR)

Generally for normal distribution, given one day predicted varianceσ2t+1, p% VaR is

VaR = Value× z1−p σt+1, (76)

and h-day

VaR(h) = Value× z1−p√h σt+1 =

√h VaR, (77)



Value at Risk (VaR)

Remark 5.3: If the return distribution is non-symmetric then of course

zp 6= −z1−p, implying that VaRs for long and short positions must be

evaluated separately.



Value at Risk (VaR)

RiskMetrics:

EWMA volatility

µt = 0, σ2t+1 = ασ2

t + (1− α)r2t , (78)

α ∈ (0, 1), typically α ∈ (0.90, 1).

That is IGARCH(1, 1) with ω = 0. This implies

σ2t|h = σ2

t+1. (79)



Value at Risk (VaR)

Example 12

10,000,000 position in USD (Euro-area portfolio manager) EUR/USD

(price of one USD in euros, ↓ dollar depreciates)

0.70.8

0.91.0

1.11.2

EUR/U

SD

-4-2

02

Chang

e (%) (lo

g)

Time

EUR-USD Daily Exchange Rates

1999 2001 2003 2005 2007 2009



Value at Risk (VaR)

EWMA volatility, α = 0.94

-4-2

02

Chang

e (%) (lo

g)

510

1520

25

Volat

[p.a %]

Time

EUR-USD Daily Changes and Volatility

1999 2001 2003 2005 2007 2009

(p.a volatility√

252σt , where σt is daily standard deviation)



Value at Risk (VaR)

h = 10 day p = 0.05 VaR:

From data σt+1 = 1.216485% = 0.01216485VaR:V = 10 Meur,

VaR = 10×√h × σt+1 × z.95,

i.e VaR = 10×√

10× 0.01216485× 1.64 = .632753 Meur.

Thus, h = 10 days p = 0.05 VaR for the 10Meur position is 632 753 Eur



Value at Risk (VaR)

Remark 5.4: Returns are typically log returns,

rt = 100× log(Pt/Pt − 1).

Thus, more exactly

VaR = V× (exp(z1−p σt+1)− 1) (80)

andVaR(h) = V×

(exp(z1−p

√h σt+1)− 1

). (81)

> V <- 10 ## million EUR

> VaR <- V*(exp(0.01*sqrt(sigma2.t[length(sigma2.t)])*qnorm(0.95))-1)

[1] 0.2021093

and> VaR10 <- V*(exp(0.01*sqrt(10)*sqrt(sigma2.t)*qnorm(.95))-1); VaR10

[1] 0.6532005

i.e., VaR(10) is 653 200 eur, about 20 teur bigger than above.



Value at Risk (VaR)

Non-zero mean return

Ifrt = µ+ ut , ut = σtzt , µ 6= 0

σ2t = λσ2

t−1 + (1− λ)u2t−1,

(82)

Note: The mean evolves at rate hµ and the volatility at rate√h σ.



Value at Risk (VaR)

Equations in (73) and (74) give one period VaRs.

h-period VaRs are obtained straightforwardly

Long (h-period):

−VaR(h) = V × (√h σt+1 zp + hµ). (83)

Short (h-period):

VaR(h) = V × (√h σt+1 z1−p + hµ). (84)



Value at Risk (VaR)

ARMA-GARCH(1,1) returns

Φ(L)rt = φ0 + Θ(L)ut+1

σ2t+1 = ω + αu2

t + βσ2t ,

(85)

|φ1| < 1.

Notations:

h-period log-return

rt(h) = 100 log

(Pt

Pt−h

)(86)

orrt(h) = rt + rt−1 + · · ·+ rt−h+1. (87)

Note: rt = rt(1).



Value at Risk (VaR)

Denote a h-period predicted return from t as

µt [h] = Et [100× log(Pt+h/Pt)] = Et [rt+h(h)] (88)

orµt [h] = µt|1 + µt|2 + · · ·+ µt|h, (89)

whereµt|j = Et [rt+j ] (90)

are one period returns predicted j periods ahead.

For an AR(1)-process rt|j = µ+ φj1(rt − µ) with µ = φ0/(1− φ1)the ling run mean,

µt [h] = µ h + (rt − µ)φ11− φh11− φ1

. (91)



Value at Risk (VaR)

MA-representation of an ARMA-process

rt = µ+ ut + ψ1ut−1 + ψ2ut−1 + · · ·, (92)

where µ = φ0/Φ(L), ut = σtzt withzt ∼ i.i.d(0, 1).

For an AR(1), ψj = φj1.



Value at Risk (VaR)

Prediction errors:

et [h] = rt(h)− µt(h) =h∑

j=1

(rt+j − µt|j). (93)

From the MA-representation

et|j ≡ rt+j − µt|j =

j−1∑k=0

ψkut+j−k , (94)

with ψ0 = 1 .



Value at Risk (VaR)

After collecting terms

et [h] =h−1∑j=0

ψjut+h−j , (95)

where

ψj =

j∑k=0

ψk (96)

with ψj = ψj = 1.

Using ut+j = σt+jzt+j with zt+j i.i.d, and thus independent ofσt+j ,

σ2t [h] ≡ vart [et [h]] =

h−1∑j=0

ψ2j σ

2t|(h−j). (97)



Value at Risk (VaR)

Example 13

MA(1)-GARCH(1,1)rt+1 = µ+ θut + ut+1

σ2t+1 = ω + αu2

t + βσ2t

(98)

ψ0 = 1, ψj = 1 + θ, j > 0. (99)

µt|j = Et[rt+j

]=

{µ+ θut , for j = 1µ, for j > 1.

(100)

Thenµt [h] = hµ+ θut (101)

andσ2t [h] = σ2(1 + (h − 1)(1 + θ)2)

+

( [1−(α+β)h

](1+θ)2

1−(α+β)

)(σ2

t+1 − σ2),(102)

where σ2 = ω/(1− α− β) is the unconditional (”long run”) variance.



Value at Risk (VaR)

Example 14

Consider the EUR/USD example. GARCH(1,1) with conditional normal

distribution yields:

garchFit(formula = deur ~ garch(1, 1), data = deur)

Conditional Distribution: norm

Error Analysis:


mu -0.0174582 0.0110655 -1.578 0.115

omega 0.0005555 0.0004541 1.223 0.221

alpha1 0.0271811 0.0037161 7.314 2.58e-13 ***

beta1 0.9725490 0.0037294 260.779 < 2e-16 ***

---

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Log Likelihood:

-2363.183 normalized: -0.9159624



Value at Risk (VaR)


Statistic p-Value

Jarque-Bera Test R Chi^2 57.70481 2.948752e-13

Shapiro-Wilk Test R W 0.9946075 4.230857e-08

Ljung-Box Test R Q(10) 15.03485 0.1307965

Ljung-Box Test R Q(15) 19.76627 0.1810835

Ljung-Box Test R Q(20) 22.66414 0.3055854

Ljung-Box Test R^2 Q(10) 13.51373 0.1963489

Ljung-Box Test R^2 Q(15) 16.70982 0.3365053

Ljung-Box Test R^2 Q(20) 29.01204 0.08752258

LM Arch Test R TR^2 14.72602 0.2567627


AIC BIC SIC HQIC

1.835026 1.844104 1.835021 1.838316



Value at Risk (VaR)

Estimate of ω is insignificant and

α+ β = 0.0271811 + 0.9725490 = 0.9997302,

which, however, differs at 5% from 1 (Wald test). Empirically ≈ EWMA, however.GARCH(1,1)-VaR(10):Using R with the fGarch results in gf object and the following small R function:

s.th <- function(gf, h){

## VaR(h) return standard deviation, sigma2.t[h]

## gf: fGarch object with GARCH(1,1)

## h: VaR periods

## REMARK: rescale with 0.01 if input in percentages!

omega <- gfit@fit$coef["omega"]

alpha <- gfit@fit$coef["alpha1"]

beta <- gfit@fit$coef["beta1"]

n <- length(gf@residuals)

s2 <- omega / (1 - alpha - beta) # unconditional variance sigma2

s2.tp1 <- (omega + alpha*gf@residuals[n]^2 + beta*[email protected][n]) # sigma2(t+1)

sth <- sqrt(s2*h + ((1- (alpha + beta)^h)/(1 - (alpha + beta)))*(s2.tp1 - s2))

names(sth) <- NULL

return(sth)

}

V <- 10

gVaR10 <- V * (0.01 * s.th(gf,10) * qnorm(.95)); gVaR10

[1] 0.6297266

I.e., 629,727 eur, which is about the same as in EWMA (Example 12).



Value at Risk (VaR)

Other Approaches

Empirical Estimation

Quantile Estimation

Extreme value theory [E.g. Longin (2000) JBF]

etc.


econometrics iilipas.uwasa.fi/~sjp/teaching/ecmii/lectures/ecmiic5.pdf · t 1 (7) with !>0and0

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