garch

4. Volatility Models∗

4.1 Background

Example 4.1: Consider the following daily close-to-

close SP500 values [January 3, 2000 to Feb 2, 2010]

-.1

-.05

0.0

5.1

log

retu

rn

6.6

6.8

77.

27.

4lo

g cl

ose

2000 2002 2004 2006 2008 2010day

SP500 Daily Prices and Returns

∗Version: Feb 9, 2012

1

Stata commands to generate the above plot from log-

series, lclose (”///” indicates line break)

twoway (tsline lclose, lcolor(blue)) ///(tsline d.lclose, yaxis(2) lcolor(navy)), ///yscale(axis(1) r(5 7)) ///yscale(axis(2) r(-0.1,0.2)) ///ytitle(log close, axis(1)) ///ytitle(log return, axis(2)) ///xtitle(day) ///legend(off) ///tlabel(, format(%tdCCYY)) ///subtitle(SP500 Daily Prices and Returns, margin(b+2.5))

2

Below are autocorrelations of the log-index.

Obviously the persistence of autocorrelations indicate

that the series is integrated.

The autocorrelations of the return series suggest that

the returns might be stationary

ac lclose, lags(20) ytitle(acf) ///xscale(range(0 20)) title(SP500 Index Autocorrelations)

ac d.lclose, lags(20) ytitle(acf) yscale(range(-0.07 0.07)) ///xscale(range(0 20)) title(SP500 Return Autocorrelations)

-0.50

0.000.50

1.00acf

0 5 10 15 20Lag

Bartlett's formula for MA(q) 95% confidence bands

SP500 Index Autocorrelations

-0.06-0

.04-0.0

20.00

0.020.04

acf

0 5 10 15 20Lag

Bartlett's formula for MA(q) 95% confidence bands

SP500 Return Autocorrelations

. corrgram d.close, lags(5) noplot

lag 1 2 3 4 5ac -0.0491 -0.0077 0.0148 -0.0382 0.0191pac -0.0640 -0.0183 0.0613 . .prob 0.0285 0.0856 0.0827 0.1098 0.1650

3

Below are results for ARMA(1,1).

R (stats::arima): AR(1)> arfit <- arima(x = spr, order = c(1,0,1));> print(arfit);> tsdiag(arfit)

Coefficients:ar1 ma1 intercept

0.3772 -0.4822 -0.0246s.e. 0.1436 0.1358 0.0241

sigma^2 estimated as 1.954: log likelihood = -4070.75, aic = 8149.5T = 2320, R2 = 0.013

Residuals:LB p-value

1 0.1393282 4.3843243 16.229995 0.0010172705 18.641046 0.00224155010 24.529902 0.006311485

4

Residual autocorrelations for ARMA(1,1):

Standardized Residuals

Time

0 500 1000 1500 2000

-6-2

26

0 5 10 15 20 25 30

0.00.4

0.8

Lag

ACF

ACF of Residuals

2 4 6 8 10

0.00.4

0.8

p values for Ljung-Box statistic

lag

p valu

e

Lag 1 2 3 4 5 6 7ac 0.008 -0.043 0.071 -0.014 -0.029 -0.002 -0.037p-val 0.354 0.020 0.000 0.246 0.082 0.464 0.039

5

The residual autocorrelations and related Q-statistics

indicate some, although small, autocorrelation left to

the series.

The autocorrelations of the squared residuals strongly

suggest that there is still left time dependency into the

series.

5 10 15 20 25 30

0.00.2

0.4

Lag

ACF

ARMA(1,1) residuals squared

The dependency is nonlinear by nature, however.

6

Also a casual analysis with rolling k = 22-day

(∼ month of trading days) mean volatility

(annualized standard deviation)

(1) mt =1

k

t∑u=t−k+1

ru,

(2) st =

√√√√√252

k

t∑u=t−k+1

(ru −mt)2,

t = k, . . . T

-10-5

05

10

SP500 daily returns and 22-day rolling average

Retur

n (%)

050

100

150

SP500 absolute returns and 22-day rolling volatility

Day

Volat

ility (%

p.a)

2000 2002 2004 2006 2008

sqrt(252) x abs ret22 day volat

7

Because squared observations are the build-

ing blocks of the variance 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.‡

Note: Volatility not directly observable!!

Methods:

a) Implied volatility

b) Realized volatility

c) Econometric modeling (stochastic volatil-

ity, ARCH)

‡The inventor of this modeling approach is RobertF. Engle (1982). Autoregressive conditional het-eroscedasticity with estimates of the variance ofUnited Kingdom inflation. Econometrica, 50, 987–1008.

8

4.2. Conditional Heteroscedasticity

ARCH-models

The general setup for ARCH models is

(3) yt = x′tθ + ut

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

′,t = 1, . . . , T , and

(4) ut|Ft−1 ∼ N(0, σ2t ),

where Ft is the information available at time

t (usually the past values of ut; u1, . . . , ut−1),

and

(5)σ2t = Var(ut|Ft−1) = ω + α1u

2t−1 + α2u

2t−2 + · · ·+ αqu

2t−q.

9

Furthermore, it is assumed that ω > 0, αi ≥ 0

for all i and α1 + · · ·+ αq < 1.

For short it is denoted ut ∼ ARCH(q).

This reminds essentially an AR(q) process for

the squared residuals, because defining νt =

u2t − σ2

t , we can write

(6)

u2t = ω + α1u

2t−1 + α2u

2t−2 + · · ·+ αqu

2t−q + νt.

Nevertheless, the error term νt is time het-

eroscedastic, which implies that the conven-

tional estimation procedure used in AR-estimation

does not produce optimal results here.

10

Properties of ARCH-processes

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

process

(7) σ2t = ω + αu2

t−1

with ω > 0 and 0 ≤ α < 1 and ut|ut−1 ∼N(0, σ2

t ).

(a) ut is white noise

(i) Constant mean (zero):

(8) E[ut] = E[Et−1[ut]︸︷︷︸=0

] = E[0] = 0.

Note Et−1[ut] = E[ut|Ft−1], the conditional

expectation given information up to time t−1.§

§The law of iterated expectations: Consider time points t1 < t2such that Ft1 ⊂ Ft2, then for any t > t2

(9) Et1 [Et2[ut]] = E [E[ut|Ft2]|Ft1] = E[ut|Ft1] = Et1[ut].

11

(ii) Constant variance: Using again the law

of iterated expectations, we get

(10)

Var[ut] = E[u2t ] = E

[Et−1[u2

t ]]

= E[σ2t ] = E[ω + αu2

t−1]

= ω + αE[u2t−1]

...

= ω(1 + α+ α2 + · · ·+ αn)

+αn+1E[u2t−n−1]︸︷︷︸

→0, as n→∞= ω

(limn→∞

∑ni=0α

i)

= ω1−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 iterated expec-

tations.)

12

(b) The unconditional distribution of ut is

symmetric, but nonnormal:

(i) Skewness: Exercise, show that E[u3t ] = 0.

(ii) Kurtosis: Exercise, show that under the

assumption ut|ut−1 ∼ N(0, σ2t ), and that α <√

1/3, the kurtosis

(11) E[u4t ] = 3

ω2

(1− α)2·

1− α2

1− 3α2.

Hint: If X ∼ N(0, σ2) then E[(X−µ)4] = 3(σ2)2 = 3σ4.

13

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

(12) E[u4t ] > 3

ω2

(1− α)2= 3[Var(ut)]2,

we find that the kurtosis of the unconditional

distribution exceed that what it would be, if

ut were normally distributed.

Thus the unconditional distribution of ut is

nonnormal and has fatter tails than a normal

distribution with variance equal to Var[ut] =

ω/(1− α).

14

(c) Standardized variables:

Write

(13) zt =ut√σ2t

then zt ∼ NID(0,1), i.e., normally and inde-

pendently distributed.

Thus we can always write

(14) ut = zt

√σ2t ,

where zt independent standard normal ran-

dom variables (strict white noise).

This gives us a useful device to check after

fitting an ARCH model the adequacy of the

specification: Check the autocorrelations of

the squared standardized series.

15

Estimation of ARCH models

Given the model

(15) yt = x′tθ + ut

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

N(x′tθ, σ2t ), t = 1, . . . , T .

Then the log-likelihood function becomes

(16) `(θ) =T∑t=1

`t(θ)

with

(17)

`t(θ) = −1

2log(2π)−

1

2logσ2

t −1

2(yt − x′tθ)2/σ2

t ,

where θ = (θ′, ω, α)′.‖

‖In practice more fat tailed (and/or skewed) conditional distri-

butions are assumed.

16

Most popular conditional distributions:

Student t:

(18)

`t = −12

log(π(ν−2)Γ(ν/2)2

Γ((ν+1)/2)2

)− 1

2logσ2

t

−(ν+1)2

log(

1 + (yt−x′tθ)2

σ2t (ν−2)

),

Γ(·) the gamma function, ν > 2 the degrees of free-dom.

Generalized error distribtuion (GED):

(19)

`t = −12

log(

Γ(1/r)3

Γ(3/r)(r/2)2

)− 1

2logσ2

t

−(

Γ(3/r)(yt−x′tθ)2

σ2t Γ(1/r)

)r/2,

r the tail parameter, r = 2 a normal distribution, r < 2

fat-tailed.

17

Skewed-Student:

(20)

`t = −12

log(π(ν−2)Γ(ν/2)2(ξ+1/ξ)2

4s2Γ((ν+1)/2)2

)− 1

2logσ2

t

− log(

1 + (m+s(yt−x′tθ))2

σ2t (ν−2)

ξ−2It),

where

(21) It =

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

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

Also skewed normal and skewed ged are available in

some versions.

More details: www.garch.org

18

The maximum likelihood (ML) estimate θ̂ is

the value maximizing the likelihood function,

i.e.,

(22) `(θ̂) = maxθ

`(θ).

The maximization is accomplished by numer-

ical methods.

19

Non-Gaussian series are estimated often by

quasi maximum likelihood (QML), i.e., as if

the error were conditionally normal.

Under fairly general conditions the QML es-

timator of θ̂ is consistent and asymptotically

normal:

(23)√T (θ̂ − θ)→ N(0,J−1

θ IθJ−1θ ),

where

(24) Jθ = −E[∂`t(θ)

∂θ∂θ′

],

and

(25) Iθ = E[∂`t(θ)

∂θ

∂`t(θ)

∂θ′

].

Remark 4.1: If the conditional ditribution is Gaussian,

Jθ = Iθ.

20

Generalized ARCH models

In practice the ARCH needs fairly many lags.

Usually far less lags are needed by modifying

the model to

(26) σ2t = ω + αu2

t−1 + βσ2t−1,

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.

21

Note again that defining νt = u2t −σ2

t , we can

write

(27) u2t = ω + (α+ β)u2

t−1 + νt − βνt−1

a heteroscedastic ARMA(1,1) process.

Applying backward substitution, one easily

gets

(28) σ2t =

ω

1− β+ α

∞∑j=1

βj−1u2t−j

an ARCH(∞) process.

Thus the GARCH term captures all the his-

tory from t− 2 backwards of the shocks ut.

22

Imposing additional lag terms, the model can

be extended to GARCH(r, q) model

(29) σ2t = ω +

r∑j=1

βjσ2t−j +

q∑i=1

αu2t−i

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

Nevertheless, as noted above, in practice

GARCH(1,1) is adequate.

23

Example 4.2: MA(1)-GARCH(1,1) model of SP500returns estimated with conditional normal,ut|Ft−1 ∼ N(0, σ2

t ), and GED, ut|Ft−1 ∼ GED(0, σ2t )

The model is

(30)rt = µ+ ut + θut−1

σ2t = ω + αu2

t−1 + βσ2t−1.

-10-5

05

10

SP500 Returns

2000 2002 2004 2006 2008

24

Using R-program with package fGarch(see http://cran.r-project.org/web/packages/fGarch/index.html)

> fGarch::garchFit(spr ~arma(0,1) + garch(1,1), data = spr,cond.dist = "ged", trace = F)

Conditional Distribution: norm

Coefficient(s):mu ma1 omega alpha1 beta1

0.027604 -0.077629 0.010057 0.076014 0.919177

Std. Errors:based on Hessian

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

mu 0.027604 0.016958 1.628 0.103569ma1 -0.077629 0.022753 -3.412 0.000645 ***omega 0.010057 0.002848 3.532 0.000413 ***alpha1 0.076014 0.009613 7.907 2.66e-15 ***beta1 0.919177 0.009830 93.505 < 2e-16 ***---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Log Likelihood:-3460.251 normalized: -1.490845AIC: 2.985999 BIC: 2.998385 (computed from ’fit@llh’)

25

> fGarch::garchFit(spr ~arma(0,1) + garch(1,1), data = spr,cond.dist = "ged", trace = F)

Conditional Distribution: ged

Coefficient(s):mu ma1 omega alpha1 beta1 shape

0.044617 -0.077417 0.007632 0.074422 0.923302 1.461375



mu 0.044617 0.016108 2.770 0.005608 **ma1 -0.077417 0.021397 -3.618 0.000297 ***omega 0.007632 0.003011 2.535 0.011252 *alpha1 0.074422 0.011051 6.734 1.65e-11 ***beta1 0.923302 0.010954 84.286 < 2e-16 ***shape 1.461375 0.062208 23.492 < 2e-16 ***---Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Log Likelihood:-3432.333 normalized: -1.478816

AIC: 2.962803 BIC: 2.977666 (computed from ’fit@llh’)

Goodness of fit (AIC, BIC) is marginally better for

GED. The shape parameter estimate is < 2 (statisti-

cally significantly) indicated fat tails.

Otherwise the coefficient estimates are about the same.

26

Residual Diagnostics:

Autocorrelations of squared standardized residuals:

Normal: GED:Lag LB p-value LB p-value1 4.306651 0.03796362 3.878418 0.048910622 4.823842 0.08964290 4.404404 0.110559433 4.843815 0.18359760 4.436248 0.218047795 6.009314 0.30531370 5.535524 0.35406622

10 16.268283 0.09220573 15.493970 0.11506346

JB = 232.8991, df = 2, p-value = 0.000(dropping the outlier JB = 43.96)

-8-6

-4-2

02

4 SP500 Return residuals [MA(1)-GED-GARCH(1,1)]

2000 2002 2004 2006 2008

2040

6080

Conditional Volatility

Days

Volatility [sq

rt(252*h_t)]

2000 2002 2004 2006 2008

27

The autocorrelations of the squared standardized resid-

uals pass (approximately) the white noise test.

Nevertheless, the normality of the standardized resid-

uals is strongly rejected.

Usually this affects mostly to stantard errors. Com-

mon practice is to use some sort of robust standard

errors (e.g. White)

28

The variance function can be extended by

including regressors (exogenous or predeter-

mined variables), xt, in it

(31) σ2t = ω + αu2

t−1 + βσ2t−1 + πxt.

Note that if xt can assume negative values, it

may be desirable to introduce absolute values

|xt| in place of xt in the conditional variance

function.

For example, with daily data a Monday dummy

could be introduced into the model to cap-

ture the non-trading over the weekends in the

volatility.

29

Example 4.3: Monday effect in SP500 returns and/orvolatility?

”Pulse” (additive) effect on mean and ”innovative”effect on volatility

(32)yt = φ0 + φmMt + φ(yt−1 − φmMt−1) + ut

σ2t = ω + πMt + αu2

t−1 + βσ2t−1,

where Mt = 1 if Monday, zero otherwise.

30

SAS: conditional t-distribution

proc autoreg data = tmp;model sp500 = mon/ nlag = 1 garch = (p=1, q=1) dist = t;hetero mon;

run;...............

The AUTOREG Procedure

GARCH Estimates

SSE 4568.81922 Observations 2321MSE 1.96847 Uncond Var 8.46322547Log Likelihood -2110.3323 Total R-Square 0.0054SBC 4274.91297 AIC 4234.6647MAE 0.93300313 AICC 4234.71312MAPE 140.612286 Normality Test 327.0029

Pr > ChiSq <.0001

Standard ApproxVariable DF Estimate Error t Value Pr > |t| Variable Label

Intercept 1 0.0332 0.0187 1.77 0.0765mon 1 0.0110 0.0485 0.23 0.8202AR1 1 0.0669 0.0230 2.91 0.0036ARCH0 1 0.006221 0.002781 2.24 0.0253ARCH1 1 0.0742 0.0105 7.04 <.0001GARCH1 1 0.9251 0.0102 90.91 <.0001TDFI 1 0.1059 0.0172 6.16 <.0001 Inverse of t DFHET1 1 6.452E-23 1.002E-10 0.00 1.0000

Degrees of freedom estimate: 1/0.1059 ≈ 9.4.

No empirical evidence of a Monday effect in returns

or volatility.

31

ARCH-M Model

The regression equation may be extended

by introducing the variance function into the

equation

(33) yt = x′tθ + λg(σ2t ) + ut,

where ut ∼ GARCH, and g is a suitable func-

tion (usually square root or logarithm).

This is called the ARCH in Mean (ARCH-M)

model [Engle, Lilien and Robbins (1987)¶].

The ARCH-M model is often used in finance,

where the expected return on an asset is re-

lated to the expected asset risk.

The coefficient λ reflects the risk-return trade-

off.

¶Econometrica, 55, 391–407.

32

Example 4.4: Does the daily mean return of SP500depend on the volatility level?

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

(34)yt = φ0 + φ1yt−1 + λ

√σ2t + ut

σ2t = ω + αu2

t−1 + βσ2t−1,

where ut|Ft−1 ∼ tν (t-distribution with ν degrees of

freedom, to be estimated).

proc autoreg data = tmp;model sp500 = / nlag = 1

garch = (p=1, q=1, mean = sqrt)dist = t; /* t-dist */

run;

33

The AUTOREG ProcedureGARCH EstimatesSSE 4566.9544 Observations 2321MSE 1.96767 Uncond Var .Log Likelihood -2110.3437 Total R-Square 0.0058SBC 4274.93573 AIC 4234.68746MAE 0.93278599 AICC 4234.73588MAPE 140.289693 Normality Test 330.0091

Pr > ChiSq <.0001

Standard ApproxVariable DF Estimate Error t Value Pr > |t|

Intercept 1 0.0429 0.0435 0.99 0.3245AR1 1 0.0670 0.0230 2.91 0.0036ARCH0 1 0.006174 0.002776 2.22 0.0262ARCH1 1 0.0740 0.0105 7.03 <.0001GARCH1 1 0.9253 0.0102 90.95 <.0001DELTA 1 -0.009195 0.0483 -0.19 0.8489TDFI 1 0.1064 0.0172 6.17 <.0001 Inverse of t DF

The volatility term in the mean equation (DELTA)

is not statistically significant neither is the constant,

indicating no discernible drift in SP500 index (again

ν̂ = 1/0.1064 ≈ 9.4).

34

Asymmetric ARCH: TARCH, EGARCH, PARCH

A stylized fact in stock markets is that down-

ward movements are followed by higher volatil-

ity than upward movements.

A rough view of this can be obtained from

the cross-autocorrelations of zt and z2t , where

zt defined in (63).

Example 4.5: Cross-autocorrelations of zt and z22 from

Ex 4.4

-30 -20 -10 0 10 20 30

-0.10-0.05

0.000.05

Lag

ACF

Cross-correlations of z and z^2

Some autocorrelations signify possible leverage effect.

35

The TARCH model

Threshold ARCH, TARCH (Zakoian 1994,

Journal of Economic Dynamics and Control,

931–955 , Glosten, Jagannathan and Run-

kle 1993, Journal of Finance, 1779-1801) is

given by [TARCH(1,1)]

(35) σ2t = ω + αu2

t−1 + γu2t−1dt−1 + βσ2

t−1,

where ω > 0, α, β ≥ 0, α+ 12γ + β < 1, and

dt = 1, if ut < 0 (bad news) and zero other-

wise.

The impact of good news is α and bad news

α+ γ.

Thus, γ 6= 0 implies asymmetry.

Leverage exists if γ > 0.

36

Example 4.6: SP500 returns, MA(1)-TARCH model(EViews, www.eviews.com).

Dependent Variable: SP500Method: ML - ARCH (Marquardt) - Generalized error distribution (GED)Sample (adjusted): 1/04/2000 3/27/2009Included observations: 2321 after adjustmentsConvergence achieved after 28 iterationsMA Backcast: 1/03/2000Presample variance: backcast (parameter = 0.7)GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-1)^2*(RESID(-1)<0) +

C(6)*GARCH(-1)==============================================================

Variable Coefficient Std. Error z-Stat Prob.--------------------------------------------------------------

C 0.012892 0.016635 0.774961 0.4384MA(1) -0.069378 0.022129 -3.135134 0.0017

==============================================================Variance Equation==============================================================

C 0.010407 0.002084 4.993990 0.0000RESID(-1)^2 -0.023073 0.008279 -2.786714 0.0053RES(-1)^2*(RES(-1)<0) 0.144410 0.016134 8.950672 0.0000GARCH(-1) 0.939786 0.009180 102.3715 0.0000==============================================================R-squared 0.007611 Mean dependent var -0.024928Adjusted R-squared 0.007183 S.D. dependent var 1.407094S.E. of regression 1.402032 Akaike info criterion 2.930426Sum squared resid 4558.442 Schwarz criterion 2.947767Log likelihood -3393.759 Hannan-Quinn criter. 2.936745F-statistic 2.964170 Durbin-Watson stat 2.047115Prob(F-statistic) 0.006937==============================================================Inverted MA Roots .07==============================================================

Statistically significant positive asymmetry parameter

estimate indicates presence of leverage.

37

The EGARCH model

Nelson (1991) (Econometrica, 347–370) pro-

posed the Exponential GARCH (EGARCH)

model for the variance function of the form

(EGARCH(1,1))

(36) logσ2t = ω + β logσ2

t−1 + α |zt−1|+ γzt−1,

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

Again, the impact is asymmetric if γ 6= 0,

and leverage is present if γ < 0.

38

Example 4.7: MA(1)-EGARCH(1,1)-M estimation re-sults.

Dependent Variable: SP500Method: ML - ARCH (Marquardt) - Generalized error distribution (GED)Sample (adjusted): 1/04/2000 3/27/2009Included observations: 2321 after adjustmentsConvergence achieved after 25 iterationsMA Backcast: 1/03/2000Presample variance: backcast (parameter = 0.7)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 0.2755MA(1) -0.072900 0.021954 -3.320628 0.0009

=============================================================Variance Equation

=============================================================C(3) -0.066913 0.013105 -5.106018 0.0000C(4) 0.081930 0.016677 4.912873 0.0000C(5) -0.121991 0.012271 -9.941289 0.0000C(6) 0.986926 0.002323 424.9402 0.0000

=============================================================GED PARAM 1.562564 0.051706 30.22032 0.0000=============================================================R-squared 0.007607 Mean dependent var -0.024928Adj R-sq 0.007179 S.D. dependent var 1.407094S.E. of regr 1.402035 Akaike info criterion 2.929546Sum sq resid 4558.461 Schwarz criterion 2.946887Log lik -3392.738 Hannan-Quinn criter. 2.935865F-statistic 2.962535 Durbin-Watson stat 2.040231P(F-stat) 0.006964=============================================================Inverted MA Roots .07=============================================================

Again statistically significant leverage.

39

Power ARCH (PARCH)

Ding, Granger, and Engle (1993). A long

memory property of stock market returns and

a new model. Journal of Empirical Finance.

PARC(1,1)

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

where γ is the leverage parameter. Again

γ > 0 implies leverage.

40

Example 4.8: R: fGarch::garchFit MA(1)-APARCH(1,1)

results for SP500 returns.

R parametrization: MA(1), mu and thetagfa <- 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)[data = sp500r]

Conditional Distribution:ged


mu 0.014249 0.016825 0.847 0.397049ma1 -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


41

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

δ̂ = 1.207412 (0.203576) does not deviate significantly

from unity (standard deviation process).

42

Integrated GARCH (IGARCH)

Often in GARCH α̂+ β̂ ≈ 1. Engle and Boller-

slev (19896). Modelling the persistence of

conditional variances, Econometrics Reviews

5, 1–50, introduce integrated GARCH with

α+ β = 1.

(38) σ2t = ω + αu2

t−1 + (1− α)σ2t−1.

Close to the EWMA (Exponentially Weighted

Moving Average) specification

(39) σ2t = αu2

t−1 + (1− α)σ2t−1

favored often by practitioners (e.g. RiskMet-

rics).

Unconditional variance does not exist [more

details, see Nelson (1990). Stationarity and

persistence in in the GARCH(1,1) model. Econo-

metric Theory 6, 318–334].

43

4.3 Predicting Volatility

Predicting with the GARCH models is straight-

forward.

Generally a k-period forward prediction is of

the form

(40) σ2t|k = Et

[u2t+k

]= E

[u2t+k|Ft

]k = 1,2, . . ..

44

Because

(41) ut = σtzt,

where generally zt ∼ i.i.d(0,1).

Thus in (40)

(42)Et[u2

t+k] = Et[σ2t+kz

2t+k]

= Et[σ2t+k]Et[z2

t+k] (zt are i.i.d(0,1))

= Et[σ2t+k].

This can be utilize to derive explicit predic-

tion formulas in most cases.

45

ARCH(1):

(43) σ2t+1 = ω + αu2

t

(44)σ2t|k = ω(1−αk−1)

1−α + αk−1σ2t+1

= σ2 + αk−1(σ2t+1 − σ

2),

where

(45) σ2 = Var[ut] =ω

1− αRecursive fromula:

(46) σ2t|k =

σ2t+1 for k = 1

ω + ασ2t|k−1 for k > 1

46

GARCH(1,1):

(47) σ2t+1 = ω + αu2

t + βσ2t .

(48)

σ2t|k = ω(1−(α+β)k−1)

1−(α+β) + (α+ β)k−1σ2t+1

= σ2 + (α+ β)k−1(σ2t+1 − σ

2),

where

(49) σ2 =ω

1− α− β.

Recursive fromula:

(50) σ2t|k =

σ2t+1 for k = 1

ω + (α+ β)σ2t|k−1 for k > 1

47

IGARCH:

(51) σ2t+1 = ω + αu2

t + (1− α)σ2t .

(52) σ2t|k = (k − 1)ω + σ2

t+1.

Recursive fromula:

(53) σ2t|k =

σ2t+1 for k = 1

ω + σ2t|k−1 for k > 1

48

TGARCH:

(54) σ2t+1 = ω + αu2

t + γu2t dt + βσ2

t .

(55)

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)

with

(56) σ2 =ω

1− α− 12γ − β

.

Recursive fromula:

(57)

σ2t|k =

σ2t+1 for k = 1

ω + (α+ 12γ + β)σ2

t|k−1 for k > 1

49

EGARCH and APARCH prediction equations

are a bit more involved.

Recursive formulas are more appropriate in

these cases.

The volatility forecasts are applied for exam-

ple in Value At Risk computations.

50

Evaluation of predictions unfortunately not

that straightforward! (see, Andersen and Boller-

slev (1998) International Economic Review.)

51

4.4 Realized volatility

Under certain assumptions, volatility during

a period of time can be estimated more and

more precisely as the frequency of the returns

increases.

Daily (log) returns rt are sums of intraday re-

turns (e.g. returns calculated at 30 minutes

interval)

(58) rt =m∑h=1

rt(h)

where rt(h) = logPt(h)− logPt(h− 1) is the

day’s t intraday return in time interval [h− 1, h],

h = 1, . . . ,m, Pt(h) is the price at time point

h within the day t, Pt(0) is the opening price

and Pt(m) is the closing price.

52

The realized variance for day t is defined as

(59) σ̂2t,m =

m∑h=1

r2t (h)

and the realized volatility is σ̂t,m =√σ̂2t,m which

is typically presented in percentages per an-

num (i.e., scaled by the square root of the

number of trading days and presented in per-

centages).

Under certain conditions it can be shown that

σ̂t,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). Re-

alized volatility. In Handbook of Finan-

cial Time Series, T.G. Andersen, R.A.

Davis, J-P. Kreiss and T. Mikosh (eds),

Springer, New York, pp. 555–575.

53

4.4 An Application in Risk Management

Several types of risk categories: Credit risk,

operational risk, and market risk

Value at Risk (VaR)

Value at Risk (VaR) can be utilized to quan-

tify market (risk due to general market de-

cline)

For a given probability p and time horizon h VaR in-

dicates the risk for a portfolio to loose the amount of

VaR or more.

or

VaR indicates the maximum loss with prob-

ability 1− p in time horizon h.

54

In probability terms VaR is simply the pth

quantile of the distribution.

Thus, given probability p and time horizon h,

for a long position VaR > 0 is the threshold

(quantile) loss defined by

(60) p = P[∆Vt(h) ≤ −VaR] = Fh(−VaR)

where Vt is the value of the long position

(investment) at time point t,

(61) ∆Vt(h) = Vt+h − Vt

is the change in the value, and Fh(·) is the

distribution function of ∆Vt(h).

55

For a short position

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

Important: VaR deals with the tail-probabilities.

Fh a key problem.

56

In practice VaR is convenient to compute in

terms of (one period) returns.

rt =∆Vt

Vt−1.

Thus, assuming that rt ∼ (µt, σ2t ) and stan-

dardizing

(63) zt =rt − µtσt

.

57

We can write generally:

Long position:

(64) −VaR = Vt (σtzp + µt)

Short position:

(65) VaR = Vt(σtz1−p + µt

)

where Vt is the amount of investment and

zp is the percentile of the distribution of the

standardized return zt defined in (63).

58

Remark 4.4.1: If standardized returns are t-distributedwith ν > 2 degrees of freedom then, e.g. for longposition:

(66) −VaR = V ×

(σt√

ν/(2− ν)tp(ν) + µt

),

where σt is the scale parameter (estimated as a stan-

dard deviation) of the t-distribution and tp(ν) is the

p-percentile obtained from p = P[t ≤ tp(ν)].

59

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 days

Daily observations

60

Example 4.4.1: Portfolio manager has 10Meur posi-tion stocks with volatility 25% (p.a) and changes areassumed 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,

Generally for normal distribution, given one

day predicted variance σ2t+1, p% VaR is

(67) VaR = Value× z1−p σt+1,

and h-day

(68) VaR(h) = Value× z1−p√hσt+1 =

√h VaR,

61

Remark 4.4.2: 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.

62

RiskMetrics:

EWMA volatility

(69) µt = 0, σ2t+1 = ασ2

t + (1− α)r2t ,

α ∈ (0,1), typically α ∈ (0.90,1).

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

plies

(70) σ2t|h = σ2

t+1 for all h ≥ 1.

63

Example 4.4.2: 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/

USD

-4-2

02

Chan

ge (%

) (log

)

Time

EUR-USD Daily Exchange Rates

1999 2001 2003 2005 2007 2009

64

EWMA volatility, α = 0.94

> sigma2.t <- var(deur) # starting value

> for (t in 1:length(deur)) {sigma2.t[t+1] <- 0.94*sigma2.t[t] + 0.06*deur[t]^2

}

-4-2

02

Chang

e (%)

(log)

510

1520

25

Volat

[p.a %

]

Time

EUR-USD Daily Changes and Volatility

1999 2001 2003 2005 2007 2009

(p.a volatility√

252 sigma2.t)

65

h = 10 day p = 0.05 VaR:

> sqrt(sigma2.t[length(sigma2.t)]) # sigma(t+1) (%)

[1] 1.216485

Thus σt+1 = 1.216485% = 0.01216485

VaR:

> V <- 10 ## Million EUR

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

> VaR

[1] 0.200094

VaR(10):

> VaR10 <- sqrt(10)*VaR; VaR10

[1] 0.632753

Thus, h = 10 days p = 0.05 VaR for the 10Meur po-

sition is 632 753 Eur

66

Remark 4.4.3: Returns are typically log returns,

rt = 100× log(Pt/Pt−1).

Thus, more exactly

(71) VaR = V× (exp(z1−p σt+1)− 1)

and

(72) VaR(h) = V×(

exp(z1−p√hσt+1)− 1

).

> 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.

67

Non-zero mean return:

If

(73)rt = µ+ ut, ut = σtzt, µ 6= 0

σ2t = λσ2

t−1 + (1− λ)u2t−1,

Note: The mean evolves at rate hµ and the

volatility at rate√hσ.

68

Equations in (64) and (65) give one period

VaRs.

h-period VaRs are obtained straightforwardly

Long (h-period):

(74) −VaR(h) = V × (√hσt+1zp + hµ).

Short (h-period):

(75) VaR(h) = V × (√hσt+1z1−p + hµ).

69

ARMA-GARCH(1,1) returns

(76)Φ(L)rt = φ0 + Θ(L)ut+1

σ2t+1 = ω + αu2

t + βσ2t ,

|φ1| < 1.

Notations:

h-period log-return

(77) rt(h) = 100 log

(Pt

Pt−h

)or

(78) rt(h) = rt + rt−1 + · · ·+ rt−h+1.

Note: rt = rt(1).

70

Denote a h-period predicted return from t as

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

or

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

where

(81) µt|j = Et[rt+j

]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 long run mean,

(82) µt[h] = µh+ (rt − µ)φ11− φh11− φ1

.

71

MA-representation of an ARMA-process

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

where µ = φ0/Φ(L), ut = σtzt with

zt ∼ i.i.d(0,1).

For an AR(1), ψj = φj1.

72

Prediction errors:

(84) et[h] = rt(h)− µt(h) =h∑

j=1

(rt+j − µt|j).

From the MA-representation

(85) et|j ≡ rt+j − µt|j =j−1∑k=0

ψkut+j−k,

with ψ0 = 1 .

73

After collecting terms

(86) et[h] =h−1∑j=0

ψ̃jut+h−j,

where

(87) ψ̃j =j∑

k=0

ψk

with ψ̃0 = ψ0 = 1.

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

independent of σt+j,

(88) σ2t [h] ≡ Vart[et[h]] =

h−1∑j=0

ψ̃2j σ

2t|(h−j).

74

Example 4.4.3: MA(1)-GARCH(1,1)

(89)rt+1 = µ+ θut + ut+1

σ2t+1 = ω + αu2

t + βσ2t

(90) ψ̃0 = 1, ψ̃j = 1 + θ, j > 0.

(91) µt|j = Et[rt+j

]=

{µ+ θut, for j = 1

µ, for j > 1.

Then

(92) µt[h] = hµ+ θut

and

(93)

σ2t [h] = σ2(1 + (h− 1)(1 + θ)2)

+(

[1−(α+β)h](1+θ)2

1−(α+β)

)(σ2

t+1 − σ2),

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

run”) variance.

75

Example 4.4.4: Consider the EUR/USD example.

GARCH(1,1) with conditional normal distribution yields:

> gf <- garchFit(deur~garch(1,1), data = deur, trace = F)> summary(gf)

Title:GARCH Modelling

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

Mean and Variance Equation:data ~ garch(1, 1)[data = deur]

Conditional Distribution:norm

Coefficient(s):mu omega alpha1 beta1

-0.01745817 0.00055545 0.02718111 0.97254904



mu -0.0174582 0.0110655 -1.578 0.115omega 0.0005555 0.0004541 1.223 0.221alpha1 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


76

Description:Tue Apr 14 10:23:07 2009 by user: sp

Standardised Residuals Tests:Statistic p-Value

Jarque-Bera Test R Chi^2 57.70481 2.948752e-13Shapiro-Wilk Test R W 0.9946075 4.230857e-08Ljung-Box Test R Q(10) 15.03485 0.1307965Ljung-Box Test R Q(15) 19.76627 0.1810835Ljung-Box Test R Q(20) 22.66414 0.3055854Ljung-Box Test R^2 Q(10) 13.51373 0.1963489Ljung-Box Test R^2 Q(15) 16.70982 0.3365053Ljung-Box Test R^2 Q(20) 29.01204 0.08752258LM Arch Test R TR^2 14.72602 0.2567627

Information Criterion Statistics:AIC BIC SIC HQIC

1.835026 1.844104 1.835021 1.838316

77

Estimate of ω is insignificant and

α̂+ β̂ = 0.0271811 + 0.9725490 = 0.9997302,

which, however, differs at 5% from 1 (Wald test,

EViews).

Empirically, however, ≈ EWMA.

GARCH(1,1)-VaR(10):

s.t.h <- function(gfit, 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"]T <- length(gfit@residuals)s2 <- omega / (1 - alpha - beta)s2.tp1 <- (omega + alpha*gfit@residuals[T]^2 + beta*[email protected][T])

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

names(sth) <- NULLreturn(sth)

}V <- 10gVaR10 <- V*(0.01*s.t.h(gf,10)*qnorm(.95)); gVaR10[1] 0.6297266

I.e., 629,727 eur, which is about the same as in EWMA

(Example 4.4.3).

78

Other Approaches:

Empirical Estimation

Quantile Estimation

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

JBF]

etc.

79

garch

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