Leture 3: Univariate Volatility Modelling – Part II
FM321: Risk Management and Zhu
11 October 2022
LSE Finance
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Volatility modelling
Univariate volatility modelling (one financial asset)
Moving average models ARCH
Model estimation Diagnostics
Alternative approaches
Multivariate volatility modelling
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GARCH: Generalized Autoregressive Conditional Heteroskedasticity The defining equation of a GARCH(1,1) process is
σ2 =ω+αr2 +βσ2
The general specification GARCH(P,Q) allows for an arbitrary number of lags in both terms:
σ2 =ω+Xα r2
Leture 3: Univariate Volatility Modelling – Part II
GARCH(P,Q):
σ2 =ω+Xα r2
αp captures the effect of news on conditional variance.
βq correspond to memory, and they lead to persistence in volatility
estimates.
Parameter restrictions: ω > 0, αp ≥ 0, βq ≥ 0, and at least one of
the αp’s and βq’s is strictly positive.
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GARCH(1,1): unconditional variance
Assume that rt is a GARCH(1,1) process with finite unconditional variance σ2.
σ2 =ω+αr2 +βσ2
Take the unconditional expectation of the above equation,
E[σ2 ] = ω + αE[r2] + βE[σ2] t+1 t t
Recall that E[rt2] = E[σt2] = σ2.
σ2 = ω + ασ2 + βσ2
Solve for σ2,
σ2= ω 1−α−β
It is necessary to require α + β < 1 in order for the unconditional
variance to exist.
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GARCH(P,Q): unconditional variance
Similarly, for a GARCH(P,Q) process with finite unconditional variance σ2:
σ2 =ω+Xα r2
σ2 =ω+Xαpσ2 +Xβqσ2
p=1 q=1 Thus, the unconditional variance is
1 − P Pp = 1 α p − P Qq = 1 β q
In order for the unconditional variance to exist, we need
X αp + X βq < 1 p=1 q=1
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GARCH(1,1): conditional variance
What would you expect σ2 to be when there is a shock to σ2 ?
For a GARCH(1,1) process, we have
σ2 = ω + αr2
Take conditional expectations at time t,
E [σ2 ] = ω + αE [r2 ] + βE [σ2
t t+k−1 = ω + (α + β)E [σ2
Recall that σ2 = ω 1−α−β
Rearrange,
. Use it to replace ω above,
+ βσ2 t+k−1
]=σ2(1−α−β)+(α+β)E [σ2
] − σ2 = (α + β)E [σ2 ] − σ2 t t+k−1
Leture 3: Univariate Volatility Modelling - Part II
GARCH(1,1): conditional variance
Iterate backward to the current period,
] − σ2 = (α + β)E [σ2 ] − σ2 t t+k−1
Bydefinition,E[σ2 ]=σ2 ,so
t t+1 E [σ2
] − σ2 = (α + β)k−1 σ2
mean reverting to σ2.
=(α+β)2E [σ2
=(α+β)k−1E [σ2
]−σ2 ]−σ2
Given α + β < 1, we can see that conditional variance forecasts are
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GARCH(1,1): tail behavior
One can show that the unconditional kurtosis of a GARCH(1,1) process is given by
E[rt4] =3 1−(α+β)2 E[rt2]2 1−(α+β)2 −2α2
We need (α+β)2 +2α2 < 1 for this to be finite. If it is, the process exhibits unconditionally heavy tails.
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GARCH vs. ARCH
GARCH carries the same benefits of ARCH, and in addition leads to more stable volatility estimates.
This comes at a price in terms of an increase in complexity of the estimation algorithms, but usually this isn’t serious.
In fact, since one can typically use much shorter lag lengths with GARCH, the number of parameters is smaller and the estimation is faster.
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Model estimation
We use the maximum likelihood to estimate the model parameters based on the data.
Maximum likelihood asks the following question: which set of values for the parameters is more likely to have generated the observed data?
Example: suppose we have the following observed i.i.d. sample from a normally distributed random variable:
−0.2, −0.2, −0.1, 0, 0, 0.2, 0.2, 0.3
Which of the following sets of parameters is more likely to have led
to this sample being observed?
Mean Std. Dev.
A 0 0.2 B01 C 2 0.2
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Maximum likelihood: example 1
Normal density: recall that, if X is a random variable with a normal distribution N(μ,σ2), its density is given by
1 h (x−μ)2i fX(x)=σ√2πexp − 2σ2
How can we estimate (μ,σ2) in the previous example?
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Maximum likelihood: example 2
Now suppose we have four return observations, r0, r1, r2, and r3. How can we estimate ARCH(1) given zt ∼ N(0,1)?
Recallthatrt =σtzt,wherezt ∼N(0,1)andσt2 =ω+αrt2−1. The conditional density for one observation is given by
2π(ω + αr2 ) 2(ω+αrt−1)
h r2 i f(rt|rt−1)=q exp − t 2
The joint density of the observations except the first one is given by
T Yf(rt|rt−1) t=2
This expression needs to be maximized with respect to ω and α.
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ARCH(1): log likelihood
There are a number of theoretical reasons to work with log-likelihood instead of the original likelihood.
For ARCH(1), this means
T−1 1XT logL=− log(2π)−
log(ω+αr2 )− t−1
t 2 ω+αrt−1
Leture 3: Univariate Volatility Modelling - Part II
Parameter estimation: numerical optimization
Suppose we’re trying to estimate a single parameter θ, which belongs to a parameter space Θ.
Typically, no closed-form solution exists, which implies we need to use a numerical algorithm to find the optimal value of θ.
Often, black boxes exist to do so (say, in R or Matlab).
Numerical problems often arise in this process, rendering the solution uncertain.
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Optimization in one dimension
Suppose we’re trying find θ that maximizes f (θ). The general process works as follows:
choose an initial value θ0 for the parameters
compute f ′(θ0)
if f ′(θ0) > 0, should increase θ; if f ′(θ0) < 0, should decrease θ increment θ0 accordingly (details depends on algorithm) to find θ1 iterate until convergence
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Optimization in one dimension: idea case
With a smooth function that has a clear unique global maximum, the process usually works well:
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Optimization in one dimension: issues
If local maxima are not unique, problems can arise:
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Optimization in one dimension: issues
If the objective function is flat near the maximum, we can also have problems.
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Numerial optimization considerations
There are no general solutions to these types of issues.
The simpler the model, the less likely you’ll run into problems (GARCH of low orders is usually OK)
For multivariate models (next topic), problems are common.
This problem makes it particularly important to make sure that estimates are sensible.
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Diagnostics
There are a number of tools one can use to evaluate models:
Parameter tests
Likelihood ratios
Residual analysis
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Parameter tests
Consider two nested models (that is, one is a restricted version of the other)
Example: ARCH(1) is a restricted version of GARCH(1,1) in which β = 0.
Which specification is better?
Estimation output usually produces a test statistic (valid asymptotically) that can be used to test the significance of individual parameters.
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Likelihood ratio tests
Denote the likelihood ratio values for the restricted and unrestricted models LR and LU , respectively.
We always have LR ≤ LU ; if the constraints actually hold in the data, we would have LR = LU .
Even if the constraint holds for the parameters, in the sample we can observe a discrepancy due to statistical variation.
The difference between LR and LU in the sample can be used as the basis for a statistical test of the validity of the hypothesis.
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Likelihood ratio tests
Under the null hypothesis that the constraints hold, one can show that asymptotically
LR = −2(logLR − logLU) ∼ χ2(k)
where k is the number of constraints involved in the null hypothesis.
For example, if we’re comparing ARCH(1) with GARCH(1,1), then k = 1 (as the only constraint is β = 0).
Result holds asymptotically, so the more data we have the more precise the test is.
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Residual analysis
We usually make a distributional assumption to be able to estimate our models.
For instance, standardized residuals are normal.
Once we estimate parameters, we can compute the model’s estimates for conditional variance.
For instance, with GARCH(1,1) we have:
1 − αˆ − βˆ
σˆ2 =ωˆ+αˆr2 +βˆσˆ2 fort=2,3,...,T t t−1 t−1
Leture 3: Univariate Volatility Modelling - Part II
Residual analysis
With a series for σˆt2, we can compute the model’s standardized residuals:
Question: does {zˆ } satisfy our distributional assumptions?
Can apply the methods from Lecture 1 to evaluate this question:
Ljung-Box test for serial correlation
Jarque-Bera test for normality
QQ-Plots can help determine the distribution of {zˆt}
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Extensions of GARCH
GARCH models tend to produce reasonable estimates for conditional volatility, and for this reason it is often used as a workhorse model.
GARCH has been extended in many different ways, to account for a number of economic, financial and statistical effects.
We’ll go over some of those extensions below.
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Extensions of GARCH
Non-Normal GARCH: Instead of assuming that zt is conditionally normal, we can adopt other specifications.
For example, we could have zt ∼ tν .
The parameter ν would need to be estimated, which typically
requires lots of observations.
Note that we assume that Et−1[zt2] = 1; since the Student-t distribution does not have unit variance, the zt is a normalized version of that distribution.
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Extensions of GARCH: GARCH-in-Mean
GARCH-in-Mean: accounts for the possibility that in times of higher risk expected returns may be higher.
In some cases, the zero mean assumption isn’t appropriate. One possible specification is
rt =μt +σtzt
σ2 =ω+αr2 +βσ2
t t−1 t−1 μ t = δ σ t2
where δ is an additional parameter to estimate.
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Extensions of GARCH: GJR-GARCH
Leverage Effects: in many settings, it is observed that positive and negative shocks have different impact on conditional volatility.
Effectively, this means that the ARCH term should read
if r > 0 t−1
if r < 0 t−1
whereα′ >α>0.
Using the indicator function I[rt−1<0] (which equals 1 if rt−1 < 0 and
0 otherwise), we can write the model as
σ2 =ω+ α+γI r2 +βσ2
t [rt−1<0] t−1 t−1
where γ = α′ − α.
This specification is known as the GJR-GARCH model.
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Model choice
There is no cookbook recipe for how to choose models. Researchers make choices of models based on:
Statistical considerations
Economic performance characteristics
Knowledge of factors outside the model that may be relevant
When choosing a model, bear in mind that:
Models with more parameters or more complex specification are harder to estimate, and require more data.
More parameters usually imply lower precision in parameter estimates, and more model uncertainty.
Data from a long time ago isn’t as representative of how markets operate today.
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