1. science
2. mathematics
3. statistics

Quantitative Finance I
Modeling Volatility II
(Lecture 5)
Winter Semester 2014/2015 by Lukas Vacha
* If viewed in .pdf format - for full functionality use Mathematica 7 notebook (.nb) version
of this .pdf
Conditional Heteroscedastic Models - II
Conditional heteroscedastic models are used for modeling σ𝜎t2 as an volatility estimate by
"allowing" heteroscedasticity (time-variation) and capturing that dependence.
Forecasting with GARCH
EWMA
GARCH-t
IGARCH
GARCH-M
TAR-GARCH
Forecasting with GARCH
Forecasts of the GARCH can be obtained recursively
at origin h, the 1-step ahead forecast of σ𝜎h2+1 is:
σ𝜎h2+1 = α𝛼0 + α𝛼1 ah2 + β𝛽1 σ𝜎h2
For multi-forecast, we use at2 = σ𝜎t2 ϵ𝜀t2 and since E ϵ𝜀h2+1 Fh  = 1, we have,
σ𝜎h2+1 = α𝛼0 + (α𝛼1 + β𝛽1 ) σ𝜎h2
σ𝜎h2+2 = α𝛼0 + (α𝛼1 + β𝛽1 ) σ𝜎h2+1
σ𝜎h2+l = α𝛼0 + (α𝛼1 + β𝛽1 ) σ𝜎h2+l , l>1
σ𝜎h2+l →
α𝛼0
1-−α𝛼1 -−β𝛽1
, as l→∞.
Hence, the forecast converges to the unconditional variance.
Exponential moving average (EWMA)
Alternatively, exponential moving average (EWMA) estimator of volatility can be used. It
uses all data points, and recent observations carry larger weights. (weights are exponentially decreasing). It also has the effect of diminishing the problematic “Ghost Features”.
An m-period EWMA with smooth constant λ𝜆 is defined as:
2
QF_I_Lecture5.cdf
Alternatively, exponential moving average (EWMA) estimator of volatility can be used. It
uses all data points, and recent observations carry larger weights. (weights are exponentially decreasing). It also has the effect of diminishing the problematic “Ghost Features”.
An m-period EWMA with smooth constant λ𝜆 is defined as:
σ𝜎t2 =
τ𝜏-−1 2
rt -−1-−τ𝜏
∑m
τ𝜏=1 λ𝜆
where 0<λ𝜆<1
1+λ𝜆+λ𝜆1 +...+λ𝜆m -−1
2
τ𝜏 2
σ𝜎t +1 t = (1 -− λ𝜆) ∑∞
τ𝜏=0 λ𝜆 rt -−τ𝜏
as n→∞,
τ𝜏-−1 2
σ𝜎t2+1 t = (1 -− λ𝜆) rt2 + λ𝜆(1 -− λ𝜆) ∑∞
rt -−τ𝜏 
τ𝜏=1 λ𝜆
using the approximation ∑∞τ=1 λτ-−1 ≃
1
1-−λ
we get:
σ𝜎t2+1 t = (1 -− λ𝜆) rt2 + λ𝜆σ𝜎t2 t -−1 ,
where initial value could be unconditional variance in a historical sample.
In most financial applications, λ𝜆=0.94 is used.
The EWMA model is equivalent to the IGARCH model, that is why the next day forecasts
look very similar to forecasts based on GARCH type models.
The EWMA ignores the long-run variance (unconditional), while the forecasts based on
GARCH family models converge in the long-run to the unconditional variance of the sample.
Example 1 month, 1 year volatility of Apple.
QF_I_Lecture5.cdf
3
Example: Real financial market data -DAX index
Forecasts of variance for the DAX index. Two samples with different unconditional variance.
GARCH with t innovations
Are the GARCH residuals normally distributed ? Usually the financial assets prices are fattailed, i.e. the distribution has more weight in the tails then a normal distribution. In
general, there is a higher probability of large gain (or loss) then indicated by the normal
distribution.
Instead of the Gaussian innovations we can use t-distributed innovations.
Example: with Q-Q plot
Apple (AAPL) daily closing prices GARCH(1,1), Q-Q plots of residuals against the quantiles
of t-distribution (tdof=6.78) and the normal distribution:
4
QF_I_Lecture5.cdf
Apple (AAPL) daily closing prices GARCH(1,1), Q-Q plots of residuals against the quantiles
of t-distribution (tdof=6.78) and the normal distribution:
IGARCH - Integrated GARCH
IGARCH models are unit-root (integrated) GARCH models where α𝛼1 + β𝛽1 = 1, their key
feature is, that past squared shocks are persistent. The unconditional variance of at is
not defined under the IGARCH(1,1) model specification.
The IGARCH effect might be caused by level shifts in volatility, i.e., can be spurious, (for
example long samples. Similar to simple AR processes.) see Mikosch and Stărică (2004).
IGARCH(1,1) is defined as:
at = σ𝜎t ϵ𝜀t ,
σ𝜎t2 = α𝛼0 + β𝛽1 σ𝜎t2-−1 + (1 -− β𝛽1 ) at2-−1 ,
where 0 < β𝛽1 < 1
For α𝛼0 = 0, the IGARCH(1,1) → infinite EWMA.
Example two samples with different unconditional variance (SPX index):
Sample 1: 1/02/1950 12/31/1953:
σ𝜎t2 = 0.000011 + 0.64 σ𝜎t2-−1 + 0.14 at2-−1 , t-dof = 5.079.
Sample 2: 1/02/2007 12/31/2010:
σ𝜎t2 = 0.0000017 + 0.9007 σ𝜎t2-−1 + 0.09989 at2-−1 , t-dof = 5.651.
Sample 1: 1/02/1950 12/31/1953:
σ𝜎t2 = 0.000011 + 0.64 σ𝜎t2-−1 + 0.14 at2-−1 , t-dof = 5.079.
Sample 2: 1/02/2007 12/31/2010:
σ𝜎t2
= 0.0000017 + 0.9007 σ𝜎t2-−1
QF_I_Lecture5.cdf
+ 0.09989 at2-−1 ,
t-dof = 5.651.
Example IGARCH(1,1) artificial processes
IGARCH(1,1)
Simulated series
Simulated Volatility
α𝛼0
0.549
β𝛽1
0.613
New Random Case
Export Simulated Series
10
5
0
-−5
-−10
0
100
200
300
400
500
GARCH-M model
The return of a security may sometimes depend directly on volatility. To model this, we
use GARCH in mean (GARCH-M) model.
GARCH(1,1) - M is formalized as:
rt = μ𝜇 + cσ𝜎t2 + at
at = σ𝜎t ϵ𝜀t ,
σ𝜎t2 = α𝛼0 + α𝛼1 at2-−1 + β𝛽1 σ𝜎t2-−1 ,
where μ𝜇 and 𝒸 are constant. 𝒸 is also called risk premium
5
6
The return of a security may sometimes depend directly on volatility. To model this, we
in mean (GARCH-M) model.
GARCH(1,1) - M is formalized as:
QF_I_Lecture5.cdf
use GARCH
rt = μ𝜇 + cσ𝜎t2 + at
at = σ𝜎t ϵ𝜀t ,
σ𝜎t2 = α𝛼0 + α𝛼1 at2-−1 + β𝛽1 σ𝜎t2-−1 ,
where μ𝜇 and 𝒸 are constant. 𝒸 is also called risk premium
Example GARCH(1,1)-M artificial processes
Note, that with positive risk premium c, returns are positively skewed, as they are positively related to its past volatility
Sample GARCH(1,1)-−M
ACF function of a2t
PACF function of a2t
risk premium c
0.26
α𝛼0
0.39
α𝛼1
0.164
β𝛽1
0.503
New Random Case
Export Simulated Series
4
3
2
1
0
-−1
-−2
0
100
200
300
400
500
Threshold Autoregressive (TAR)-GARCH Model
Threshold Autoregressive (TAR) model can be used to refine the model by allowing for
asymmetric response in the (volatility) equation to the sign of shock. We can observe
asymmetry in declining and rising patterns of examined time series. Model uses simple
threshold to improve linear approximation. We demonstrate the idea on a simple two
regime AR(1) model and then we proceed to threshold model in the variance equation.
QF_I_Lecture5.cdf
7
Examples of Two Regime AR (1) model
Two-Regime AR(1) model is represented by:
xt =
α𝛼1 + β𝛽1 xt -−1 + ϵ𝜀t xt -−1 < k
α𝛼2 + β𝛽2 xt -−1 + ϵ𝜀t xt -−1 ≥ k
Parameters are initially set to: α𝛼1 = α𝛼2 = 0, β𝛽1 = -−1.5 and β𝛽2 = 0.5 to obtain following Two
Regime AR(1) process:
xt =
-−1.5 xt -−1 + ϵ𝜀t xt -−1 < 0
0.5 xt -−1 + ϵ𝜀t xt -−1 ≥ 0
Note, that the process is stationary despite the coefficient -1.5 in the first regime. Series
contains large upward jumps when it becomes negative (due to -1.5 coefficient), and there
are more positive then negative ones. Model also contains no constant term, but E(xt ) is
not zero.
Two Regime AR(1) model
treshold -− k
0.32
α𝛼1
-−0.49
β𝛽1
-−0.99
α𝛼2
0
β𝛽2
0.66
New Random Case
Export Simulated Series
4
2
0
-−2
-−4
0
100
200
300
400
500
Example
Set parameters of previous example to: α𝛼1 = α𝛼2 = 0, β𝛽1 = 0.2 and β𝛽2 = 0.9 to obtain following Two Regime AR(1) process
0.2 xt -−1 + ϵ𝜀t xt -−1 < k
0.9 xt -−1 + ϵ𝜀t xt -−1 ≥ k
and see the mixture of short-memory and long-memory processes
xt =
8
QF_I_Lecture5.cdf
Set parameters of previous example to: α𝛼1 = α𝛼2 = 0, β𝛽1 = 0.2 and β𝛽2 = 0.9 to obtain following Two Regime AR(1) process
0.2 xt -−1 + ϵ𝜀t xt -−1 < k
0.9 xt -−1 + ϵ𝜀t xt -−1 ≥ k
and see the mixture of short-memory and long-memory processes
xt =
Example of application: AR(1)-TAR-GARCH(1,1)
TAR models can be used for capturing asymmetric response in volatility. If we find the
series to follow AR(1)-GARCH(1,1) process, we can introduce TAR to model asymmetric
response in volatility to shocks. Model is formalized:
rt = ϕ𝜑0 + ϕ𝜑1 rt -−1 + at ,
at = σ𝜎t ϵ𝜀t
α𝛼0 + α𝛼1 at2-−1 + β𝛽1 σ𝜎t2-−1 at -−1 ≤ 0
σ𝜎t2 = 
α𝛼2 + α𝛼3 at2-−1 + β𝛽2 σ𝜎t2-−1 at -−1 > 0
QF_I_Lecture5.cdf
AR(1)-−TAR-−GARCH(1,1)
Simulated series
Simulated Volatility
treshold -− k
1.7
ϕ𝜑0
0.6
ϕ𝜑1
0.612
α𝛼0
0.5
α𝛼1
0.324
β𝛽1
0.2
α𝛼2
0.5
α𝛼3
0.3
β𝛽2
0.5
New Random Case
Export Simulated Series
15
10
5
0
-−5
0
100
200
300
400
500
Empirical example TAR-GARCH(1,1)
Apple (AAPL) daily closing prices sample: 9/1/2009 11/18/2011 (561 obs.)
Estimated volatility eq. (normal innovations):
σ𝜎t2 = 0.00005 + 0.665 σ𝜎t2-−1 + 0.244 at2-−1 I (at -−1 < 0)
Estimated volatility eq. (t innovations): σ𝜎t2 = 0.00004 + 0.71 σ𝜎t2-−1 + 0.240 at2-−1 I (at -−1 < 0)
where I (at -−1 < 0) = 1 if at -−1 < 0.
9
10
QF_I_Lecture5.cdf
EGARCH Model
We have demonstrated that the EGARCH (Nelson, 1991) model be used for capturing
asymmetric response in volatility - the leverage effect. The EGARCH can be defined as:
at = σ𝜎t ϵ𝜀t
lnσ𝜎t2  = α𝛼0 +
1+α𝛼1 L +...+α𝛼s L s
g (ϵ𝜀t -−1 )
1-−β𝛽1 L ...-−β𝛽m L m
g (ϵ𝜀t ) = θ𝜃 ϵ𝜀t + γ𝛾[ϵ𝜀t  -− E ϵ𝜀t ]
where α𝛼0 > 0, θ𝜃, γ𝛾 ∈ ℝ are real constants. For the standard Gaussian random variable ϵ𝜀t ,
E ϵ𝜀t  =
2/∕π𝜋 .
Function g (.) captures the asymmetric relations between stock returns and volatility. It is
both function of magnitude and sign of ϵ𝜀t . Each component of g (ϵ𝜀t ) has mean zero. Thus,
g (ϵ𝜀t ) is a zero mean i.i.d. random sequence.
When ϵ𝜀t is positive (negative) g (ϵ𝜀t ) is linear in ϵ𝜀t with slope θ𝜃+γ𝛾 (θ𝜃-γ𝛾). Thus, g (ϵ𝜀t ) allows
for asymmetrical response of conditional variance (σ𝜎t2 ) to positive or negative returns.
Example EGEARCH(1,0)
(1 -− β𝛽1 L ) lnσ𝜎t2  = α𝛼0 + g (ϵ𝜀t -−1 )
REMARK: Check how your software estimates EGARCH. For Example in Eviews and the
model is defined as:
lnσ𝜎t2  = ω𝜔 + β𝛽 lnσ𝜎t2-−1  + α𝛼
at -−1
σ𝜎t -−1
 + γ𝛾
at -−1 
σ𝜎t -−1

where ω𝜔 > 0, α𝛼, γ𝛾 ∈ ℝ are real constants. When at is negative (positive), then the effect of
the shock on the log of conditional variance is -α𝛼+γ𝛾 (α𝛼+γ𝛾).
Homework #4
Deadline: Tue 11.11.2014
Reading: Starica, C. “Is GARCH (1, 1) as good a model as the Nobel prize accolades would
imply.” Preprint (2003). https://notendur.hi.is/~helgito/NobelGarch.pdf
Homework #5
QF_I_Lecture5.cdf
Homework #5
Deadline: Tue 18.11.2014 3:00 PM
Email the homework to vachal@utia.cas.cz
Estimate any GARCH family model on two non-overlapping time periods (use preferably
stock prices and different dataset than in HW 3) and perform out-of-sample forecast
(arbitrary period). Compare the estimated GARCH family model forecast with the EWMA
forecast using MSE. As a volatility proxy use squared returns. (see paper Startica 2003).
11