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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
Volatility Models Application in Risk Management
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Background1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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 (%
)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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
)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Properties of ARCH-processes
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Properties of ARCH-processes
(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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Properties of ARCH-processes
(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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Properties of ARCH-processes
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Properties of ARCH-processes
(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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Estimation (skipped material)
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Estimation (skipped material)
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Estimation (skipped material)
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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 .
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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
Estimate Std. Error t value Pr(>|t|)
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
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 509.0074 0
Shapiro-Wilk Test R W 0.985038 0
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
Information Criterion Statistics:
AIC BIC SIC HQIC
2.773163 2.783519 2.773158 2.776820
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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
Estimate Std. Error t value Pr(>|t|)
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
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 573.7686 0
Shapiro-Wilk Test R W 0.9842976 0
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
Information Criterion Statistics:
AIC BIC SIC HQIC
2.777822 2.788177 2.777816 2.781479
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Generalized ARCH models (GARCH)
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:
Estimate Std. Error t value Pr(>|t|)
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
ARCH-M Model
Conditional Variance Dynamics
-----------------------------------
GARCH Model : gjrGARCH(1,1)
Mean Model : ARFIMA(2,0,0)
Distribution : std
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
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
LogLikelihood : -5894.883
AIC : 2.7428 BIC : 2.7561
Because archm estimate is not significance, hence no empirical evidence of volatility
on mean returns.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Asymmetric ARCH: TARCH, EGARCH, PARCH
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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")
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
TGARCH(1,1)
R: rugarch, AR(2)-TGARCH(1, 1) [i.e., GJR] with conditional t-distribution
Conditional Variance Dynamics
-----------------------------------
GARCH Model : gjrGARCH(1,1)
Mean Model : ARFIMA(2,0,0)
Distribution : std
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
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
LogLikelihood : -5895.96
AIC : 2.7428 BIC : 2.7547
gamma1 ≈ .172 is highly significant and positive, thereby indicating leverage.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
TGARCH(1,1)
Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
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
H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
The EGARCH model
Example 6
MA(1)-EGARCH(1,1)-M estimation results.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Power ARCH (PARCH)
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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:
Estimate Std. Error t value Pr(>|t|)
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Power ARCH (PARCH)
The leverage parameter (’gamma1’ ) estimates to unity.
δ = 1.207412 (0.203576) does not deviate significantly from unity
(standard deviation process).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Integrated GARCH (IGARCH)
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].
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
GARCH Model Evaluation
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
GARCH Model Evaluation
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
GARCH Model Evaluation
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Estimation results of TGARCH(1,1) with Student t conditionaldistribution.
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : gjrGARCH(1,1)
Mean Model : ARFIMA(0,0,0)
Distribution : std
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
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
LogLikelihood : -6590.249
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Conditional Heteroskedasticity
Diagnostics for the fitted model:
Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
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
H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility
Predicting Volatility
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, . . ..
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility
Predicting Volatility
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility
Predicting Volatility
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility
Predicting Volatility
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility
Predicting Volatility
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility
Predicting Volatility
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility
Predicting Volatility
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility
Predicting Volatility
Evaluation of predictions unfortunately not that straightforward!(see, Andersen and Bollerslev (1998) International EconomicReview.)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting Volatility
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Predicting 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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach
Dynamic conditional score
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach
Dynamic conditional score
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach
Dynamic conditional score
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach
Dynamic conditional score
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach
Dynamic conditional score
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach
Dynamic conditional score
EGARCH(1,1) with conditional t-distribution results (rugarch):
Conditional Variance Dynamics
-----------------------------------
GARCH Model : eGARCH(1,1)
Mean Model : ARFIMA(0,0,0)
Distribution : std
Optimal Parameters
------------------------------------
Estimate Std. Error t value Pr(>|t|)
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
LogLikelihood : -8533.783
Information Criteria
------------------------------------
Akaike 4.4923
Bayes 4.5021
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Recent developments: Dynamic conditional score approach
Dynamic conditional score
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Realized volatility1 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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Value at Risk (VaR)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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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)].
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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,
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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 .
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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)
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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.
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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:
Estimate Std. Error t value Pr(>|t|)
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
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Value at Risk (VaR)
Standardised Residuals Tests:
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
Information Criterion Statistics:
AIC BIC SIC HQIC
1.835026 1.844104 1.835021 1.838316
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
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).
Seppo Pynnonen Econometrics II
Volatility Models Application in Risk Management
Value at Risk (VaR)
Other Approaches
Empirical Estimation
Quantile Estimation
Extreme value theory [E.g. Longin (2000) JBF]
etc.
Seppo Pynnonen Econometrics II