Quantitative Finance I Modeling Volatility II (Lecture 5)

Transcription

Quantitative Finance I Modeling Volatility II (Lecture 5)
Quantitative Finance I
Modeling Volatility II
(Lecture 5)
Winter Semester 2014/2015 by Lukas Vacha
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Conditional Heteroscedastic Models - II
Conditional heteroscedastic models are used for modeling ΟƒπœŽt2 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 ΟƒπœŽh2+1 is:
ΟƒπœŽh2+1 = α𝛼0 + α𝛼1 ah2 + β𝛽1 ΟƒπœŽh2
For multi-forecast, we use at2 = ΟƒπœŽt2 Ο΅πœ€t2 and since E ο€€Ο΅πœ€h2+1 ο˜ƒFh  = 1, we have,
ΟƒπœŽh2+1 = α𝛼0 + (α𝛼1 + β𝛽1 ) ΟƒπœŽh2
ΟƒπœŽh2+2 = α𝛼0 + (α𝛼1 + β𝛽1 ) ΟƒπœŽh2+1
ΟƒπœŽh2+l = α𝛼0 + (α𝛼1 + β𝛽1 ) ΟƒπœŽh2+l , l>1
ΟƒπœŽh2+l β†’ο·οŠο”’
α𝛼0
1-βˆ’Ξ±π›Ό1 -βˆ’Ξ²π›½1
, as lβ†’ο·οŠο”’βˆž.
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 β€œGhost Features”.
An m-period EWMA with smooth constant Ξ»πœ† is defined as:
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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 β€œGhost Features”.
An m-period EWMA with smooth constant Ξ»πœ† is defined as:
ΟƒπœŽt2 =
Ο„πœ-βˆ’1 2
rt -βˆ’1-βˆ’Ο„πœ
βˆ‘m
Ο„πœ=1 Ξ»πœ†
where 0<Ξ»πœ†<1
1+Ξ»πœ†+Ξ»πœ†1 +...+Ξ»πœ†m -βˆ’1
2
Ο„πœ 2
ΟƒπœŽt +1ο˜„ t = (1 -βˆ’ Ξ»πœ†) βˆ‘βˆž
Ο„πœ=0 Ξ»πœ† rt -βˆ’Ο„πœ
as nβ†’ο·οŠο”’βˆž,
Ο„πœ-βˆ’1 2
ΟƒπœŽt2+1ο˜„ t = (1 -βˆ’ Ξ»πœ†) rt2 + Ξ»πœ†(1 -βˆ’ Ξ»πœ†) ο€€βˆ‘βˆž
rt -βˆ’Ο„πœ 
Ο„πœ=1 Ξ»πœ†
using the approximation βˆ‘βˆžΟ„=1 λτ-βˆ’1 ≃
1
1-βˆ’Ξ»
we get:
ΟƒπœŽt2+1ο˜„ t = (1 -βˆ’ Ξ»πœ†) rt2 + Ξ»πœ†ΟƒπœŽt2ο˜„ t -βˆ’1 ,
where initial value could be unconditional variance in a historical sample.
In most financial applications, Ξ»πœ†=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.
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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:
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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 = ΟƒπœŽt Ο΅πœ€t ,
ΟƒπœŽt2 = α𝛼0 + β𝛽1 ΟƒπœŽt2-βˆ’1 + (1 -βˆ’ β𝛽1 ) at2-βˆ’1 ,
where 0 < β𝛽1 < 1
For α𝛼0 = 0, the IGARCH(1,1) β†’ο·οŠο”’ infinite EWMA.
Example two samples with different unconditional variance (SPX index):
Sample 1: 1/02/1950 12/31/1953:
ΟƒπœŽt2 = 0.000011 + 0.64 ΟƒπœŽt2-βˆ’1 + 0.14 at2-βˆ’1 , t-dof = 5.079.
Sample 2: 1/02/2007 12/31/2010:
ΟƒπœŽt2 = 0.0000017 + 0.9007 ΟƒπœŽt2-βˆ’1 + 0.09989 at2-βˆ’1 , t-dof = 5.651.
Sample 1: 1/02/1950 12/31/1953:
ΟƒπœŽt2 = 0.000011 + 0.64 ΟƒπœŽt2-βˆ’1 + 0.14 at2-βˆ’1 , t-dof = 5.079.
Sample 2: 1/02/2007 12/31/2010:
ΟƒπœŽt2
= 0.0000017 + 0.9007 ΟƒπœŽt2-βˆ’1
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+ 0.09989 at2-βˆ’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
-βˆ’5
-βˆ’10
0
100
200
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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 = ΞΌπœ‡ + cΟƒπœŽt2 + at
at = ΟƒπœŽt Ο΅πœ€t ,
ΟƒπœŽt2 = α𝛼0 + α𝛼1 at2-βˆ’1 + β𝛽1 ΟƒπœŽt2-βˆ’1 ,
where ΞΌπœ‡ and οš΄π’Έ are constant. οš΄π’Έ is also called risk premium
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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:
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use GARCH
rt = ΞΌπœ‡ + cΟƒπœŽt2 + at
at = ΟƒπœŽt Ο΅πœ€t ,
ΟƒπœŽt2 = α𝛼0 + α𝛼1 at2-βˆ’1 + β𝛽1 ΟƒπœŽt2-βˆ’1 ,
where ΞΌπœ‡ and οš΄π’Έ are constant. οš΄π’Έ 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)-βˆ’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
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3
2
1
0
-βˆ’1
-βˆ’2
0
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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.
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Examples of Two Regime AR (1) model
Two-Regime AR(1) model is represented by:
xt =ο€ž
α𝛼1 + β𝛽1 xt -βˆ’1 + Ο΅πœ€t xt -βˆ’1 < k
α𝛼2 + β𝛽2 xt -βˆ’1 + Ο΅πœ€t xt -βˆ’1 β‰₯ k
Parameters are initially set to: α𝛼1 = α𝛼2 = 0, β𝛽1 = -βˆ’1.5 and β𝛽2 = 0.5 to obtain following Two
Regime AR(1) process:
xt =ο€ž
-βˆ’1.5 xt -βˆ’1 + Ο΅πœ€t xt -βˆ’1 < 0
0.5 xt -βˆ’1 + Ο΅πœ€t xt -βˆ’1 β‰₯ 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 -βˆ’ k
0.32
α𝛼1
-βˆ’0.49
β𝛽1
-βˆ’0.99
α𝛼2
0
β𝛽2
0.66
New Random Case
Export Simulated Series
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2
0
-βˆ’2
-βˆ’4
0
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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 -βˆ’1 + Ο΅πœ€t xt -βˆ’1 < k
0.9 xt -βˆ’1 + Ο΅πœ€t xt -βˆ’1 β‰₯ k
and see the mixture of short-memory and long-memory processes
xt =ο€ž
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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 -βˆ’1 + Ο΅πœ€t xt -βˆ’1 < k
0.9 xt -βˆ’1 + Ο΅πœ€t xt -βˆ’1 β‰₯ k
and see the mixture of short-memory and long-memory processes
xt =ο€ž
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 = Ο•πœ‘0 + Ο•πœ‘1 rt -βˆ’1 + at ,
at = ΟƒπœŽt Ο΅πœ€t
α𝛼0 + α𝛼1 at2-βˆ’1 + β𝛽1 ΟƒπœŽt2-βˆ’1 at -βˆ’1 ≀ 0
ΟƒπœŽt2 = ο€ž
α𝛼2 + α𝛼3 at2-βˆ’1 + β𝛽2 ΟƒπœŽt2-βˆ’1 at -βˆ’1 > 0
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AR(1)-βˆ’TAR-βˆ’GARCH(1,1)
Simulated series
Simulated Volatility
treshold -βˆ’ k
1.7
Ο•πœ‘0
0.6
Ο•πœ‘1
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
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10
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0
-βˆ’5
0
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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):
ΟƒπœŽt2 = 0.00005 + 0.665 ΟƒπœŽt2-βˆ’1 + 0.244 at2-βˆ’1 I (at -βˆ’1 < 0)
Estimated volatility eq. (t innovations): ΟƒπœŽt2 = 0.00004 + 0.71 ΟƒπœŽt2-βˆ’1 + 0.240 at2-βˆ’1 I (at -βˆ’1 < 0)
where I (at -βˆ’1 < 0) = 1 if at -βˆ’1 < 0.
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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 = ΟƒπœŽt Ο΅πœ€t
lnο€€ΟƒπœŽt2  = α𝛼0 +
1+α𝛼1 L +...+α𝛼s L s
g (Ο΅πœ€t -βˆ’1 )
1-βˆ’Ξ²π›½1 L ...-βˆ’Ξ²π›½m L m
g (Ο΅πœ€t ) = ΞΈπœƒ Ο΅πœ€t + γ𝛾[ο˜ƒΟ΅πœ€t ο˜„ -βˆ’ E ο˜ƒΟ΅πœ€t ο˜„]
where α𝛼0 > 0, ΞΈπœƒ, γ𝛾 ∈ β„οž΅ are real constants. For the standard Gaussian random variable Ο΅πœ€t ,
E ο˜ƒΟ΅πœ€t ο˜„ =
2/βˆ•Ο€πœ‹ .
Function g (.) captures the asymmetric relations between stock returns and volatility. It is
both function of magnitude and sign of Ο΅πœ€t . Each component of g (Ο΅πœ€t ) has mean zero. Thus,
g (Ο΅πœ€t ) is a zero mean i.i.d. random sequence.
When Ο΅πœ€t is positive (negative) g (Ο΅πœ€t ) is linear in Ο΅πœ€t with slope ΞΈπœƒ+γ𝛾 (ΞΈπœƒ-γ𝛾). Thus, g (Ο΅πœ€t ) allows
for asymmetrical response of conditional variance (ΟƒπœŽt2 ) to positive or negative returns.
Example EGEARCH(1,0)
(1 -βˆ’ β𝛽1 L ) lnο€€ΟƒπœŽt2  = α𝛼0 + g (Ο΅πœ€t -βˆ’1 )
REMARK: Check how your software estimates EGARCH. For Example in Eviews and the
model is defined as:
lnο€€ΟƒπœŽt2  = Ο‰πœ” + β𝛽 lnο€€ΟƒπœŽt2-βˆ’1  + α𝛼
at -βˆ’1
ΟƒπœŽt -βˆ’1
 + γ𝛾
ο˜ƒat -βˆ’1 ο˜„
ΟƒπœŽt -βˆ’1

where Ο‰πœ” > 0, α𝛼, γ𝛾 ∈ β„οž΅ 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. β€œIs 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 [email protected]
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).
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