ETC4460
Financial Econometrics II
Department of Econometrics and Business Statistics
Semester 2, 2020
Week 10.
Estimation of
Volatility
Models
2
Financial Econometrics II
Outline of Weeks 6–12
Weeks 6–7: Continuous time models for asset prices
Weeks 7–8: Pricing of financial derivatives
Week 9: Nonparametric methods in financial econometrics
Week 10: Estimation of volatility models
Week 11: Long–memory models
Week 12: Review
Financial Econometrics II 2
Week 10.
Estimation of
Volatility
Models
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Financial Econometrics II
Week 10. Estimation of Volatility
Models
Financial Econometrics II 3
Week 10.
Estimation of
Volatility
Models
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Financial Econometrics II
In this part, we will be discussing the following topics:
ARCH and GARCH types of volatility
Discrete–time volatility models
Continuous–time volatility models
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ARCH & GARCH volatility
ARCH models
yt = σt εt ,
σ2t = α+β y
2
t−1,
where {εt} is a sequence of i.i.d. random errors with E[ε1] = 0 and
E[ε21 ] = 1, and α and β can be estimated by existing methods, such
as MLE.
GARCH models
yt = σt εt ,
σ2t = α+β y
2
t−1+ γ σ
2
t−1,
where (α ,β ,γ) can be estimated by existing methods, such as MLE.
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Stochastic volatility models: Discrete time
Consider a simple stochastic volatility (SV) model of the form
yt = σt εt , (10.1)
σt = ψ(σt−1;θ0)ηt , (10.2)
where {ηt} is a sequence of positive i.i.d. random errors with E[η1] = 1,
and ψ(u;θ0) is a known positive function indexed by a vector of unknown
parameters, θ0. Examples include
yt is a financial return series, and σt is the unobservable volatility
function
Once the form of ψ(u;θ0) is available, we may estimate θ0.
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Estimation of
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Models (10.1) and (10.2) can be rewritten as
ln(y2t ) = ln(σ
2
t )+ ln(ε
2
t ), (10.3)
ln(σ2t ) = ln(ψ
2(σt−1;θ0))+ ln(η2t ). (10.4)
In the case where ψ(u;θ0) = uθ0 , we have
ln(y2t ) = ln(σ
2
t )+ ln(ε
2
t ), (10.5)
ln(σ2t ) = θ0 ln(σ
2
t−1)+ ln(η
2
t ). (10.6)
Let wt = ln(y2t ), xt = ln(σ2t ), ut = ln(ε2t ) and vt = ln(η2t ). Models (10.5)
and (10.6) can be rewritten as
wt = xt + ut , (10.7)
xt = θ0 xt−1+ vt . (10.8)
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Since {xt} is not observable, and not available to the econometrician, we
rewrite models (10.7) and (10.8) as
wt = θ0 (wt−1−ut−1)+ ut + vt
= θ0wt−1+ vt + ut −θ0 ut−1 = µ+θ0wt−1+ ξt −E[ξt ], (10.9)
where µ = E[ξt ] and ξt = vt + ut −θ0 ut−1.
Model (10.9) reduces to an AR(1) model. The main purpose of the
derivations is not just about the estimation of θ0, and the ideas are useful
for volatility estimation in the continuous–time case to be discussed below.
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Stochastic volatility models: Continuous–time
Consider a continuous–time stochastic volatility (SV) model of the form
dY (t) = V (t)dB1(t) and (10.10)
dZ(t) = (µ0+ µ1Z(t))dt+(σ0+σ1Z(t))dB2(t), (10.11)
where
Y (t) is the logarithm of the financial return series (for example),
Z(t) = ln(V (t)) is the logarithm of the volatility process,
B1(t) and B2(t) are two different standard Brownian motion
processes, and B1(t) and B2(t) can be correlated,
(µ0,µ1,σ0,σ1) is a vector of unknown parameters, and
σ0 and σ1 are the volatility parameters of the log–volatility process.
Models (10.10) and (10.11) cover some important models, two of which
will be discussed in detail.
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Case 1: Geometric Brownian motion type of volatility with
µ0 = σ0 = 0. The solution of (10.11) is given by
Z(t) ∼ N
(
µ t,σ2 t
)
, (10.12)
where µ = µ1−0.5σ21 and σ = σ1.
Case 2: Ornstein–Uhlenback type of volatility with µ0 = σ1 = 0, and
µ1 ≡−c. The solution of (10.11) is given by
Z(t) =
∫ t
−∞
A(t−u)dB2(u), (10.13)
where A(u) = σ0
(
1− c∫ u0 e−c (u−v)dv).
In each of the cases, Z(t) has a closed–form expression with a Gaussian
distribution.
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Recall τn : 0= t0 < t1 < · · ·< tn−1 < tn = T . Recall also ∆i = ti− ti−1 and
∆iB= Bti −Bti−1 =
√
∆i εi, where εi ∼ N(0,1).
In our discussion, we focus on ∆i = ∆ = Tn . In this case, each ti =
iT
n .
Some remarks follow:
Note that T is the entire time period, at which the data can be
observed;
Note also that n is the number of the partitions for the interval [0,T ];
When you have a daily, weekly, monthly, or annual dataset, you have
∆ = 1250 ,
5
250 or
20
250 , or
250
250 ; and
When T = 10, n= 2500 and ∆ = 102500 =
1
250 , for example, it means
that you have ten years daily dataset.
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We then discretize model (10.10) as
Yti −Yti−1 = Vti∆iB= Vti
√
∆εi. (10.14)
Let Wi = ln
(
(Yti−Yti−1 )2
∆
)
, Zi ≡ Zti = ln(Vti ) and Ui = ln
(
ε2i
)
.
Then model (10.14) can be written as
Wi = 2Zi+Ui, (10.15)
where we can further write Zi = Wi−Ui2 , in which
Ui = ln
(
ε2i
)
(10.16)
can be generated from a known distribution, because εi ∼ N(0,1).
In both cases outlined above where Zi has a Gaussian distribution, the
unknown parameters involved in (10.11) may then be estimated by MLE.
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We have so far discussed the following types of volatility models
ARCH models
GARCH models
Stochastic volatility models: Discrete time
Stochastic volatility models: Continuous–time
More details may be found from Tsay, R. (2005) Analysis of Financial
Time Series. 2nd Edition, John Wiley & Sons.
In the rest of this lecture, we will introduce two different types of volatility
models
Deterministic volatility models: Discrete–time
Deterministic volatility models: Continuous–time
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Deterministic time–varying model
Let us first recall a parametric ARCH model of the form: for 1≤ t ≤ n
yt = β0+β1 t+β2 t2+
√
σ0+ σ1 t+σ2 t2 εt , (10.17)
where βi for i= 0,1,2, and σ0 > 0 and σ j ≥ 0 for j = 1,2 are all unknown
parameters, and {εt} is an independent and identically distributed (i.i.d.)
random variable with E[ε1] = 0 and var[ε1] = 1.
Model (10.16) is quite popular for modelling trending and volatility
behaviours of many time series data, such as
Mean temperature;
Energy consumption;
Stock index series; and
Macroeconomic time series
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Let us now have a look at the following rewriting:
β0+β1 t+β2 t2 = α0+α1 τt +α2 τ2t ≡ m1(τt ;α), (10.18)
σ0+ σ1 t+σ2 t2 = γ0+ γ1 τt + γ2 τ2t ≡ σ21 (τt ;γ), (10.19)
for some m1(τ;α) and σ21 (τ;γ), where τt =
t
n ,
αi = ni βi for i= 0,1,2; and
γ j = n j σ j for j = 0,1,2.
In general, we may reparameterize models such that a polynomial
function of t may be rewritten as that of τt , such as
β0+β1 t+β2 t2+ · · ·+βp t p = α0+α1 τt +α2 τ2t + · · ·+αp τ pt , (10.20)
where αi = ni βi for i= 0,1,2, · · · , p.
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We define a general deterministic time–varying model of the form:
yt = m(τt)+σ(τt)εt , 1≤ t ≤ n, (10.21)
where τt = tn , m(τ) and σ(τ) > 0 are both unknown functions of τ ∈ [0,1],
and {εt} is an independent and normally distributed (i.n.d.) random
variable with E[ε1] = 0 and var[ε1] = 1.
Let m(τ) and σ2(τ) be estimated by
m̂(τ) =
n
∑
t=1
Wnt(τ)yt and σ̂2(τ) =
n
∑
t=1
Wnt(τ) (yt − m̂(τt))2 , (10.22)
with Wnt(τ) =
K( τt−τhn )
∑ns=1K(
τs−τ
hn )
, where K(u) = 1√
2pi
e−
u2
2 and hn = 1.06n−1/5 sdτ
with sdτ =
√
1
n ∑
n
t=1
(
τt − 1n ∑ns=1 τs
)2.
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Deterministic volatility models: Discrete–time
We first introduce a deterministic volatility model of the form: 1≤ t ≤ n
yt = σt εt , (10.23)
ln(σt) = µ(τt)+ ξt , (10.24)
where τt = tn , µ(τ) is an unknown function of τ ∈ [0,1], {εt} is an i.n.d.
random variable with E[ε1] = 0 and var[ε1] = 1, and {ξt} is an i.i.d.
random variable with E[ξ1] = 0 and var[ξ1] = σ21 < ∞.
Some comments are as follows:
yt can be a financial return series;
σt is an unobservable volatility, but it is driven by a type of
deterministic time–varying trend function; and
the stochastic feature is captured by the random errors: ξt .
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Let us discuss how to estimate σt by directly estimating µ(τ) in the
following way:
ln(y2t ) = 2ln(σt)+ ln(ε
2
t ), (10.25)
ln(σt) = µ(τt)+ ξt , (10.26)
which implies
zt ≡ 12
(
ln(y2t )− ln(ε2t )
)
= µ(τt)+ ξt . (10.27)
Using (10.22), we can estimate µ(τ) by
µ̂(τ) =
n
∑
t=1
Wnt(τ) zt , (10.28)
after {yt : 1≤ t ≤ n} becomes available to the econometrician and
εt ∼ N(0,1) can be generated.
We can finally estimate σt by
σ̂t = eµ̂(τt ), 1≤ t ≤ n. (10.29)
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We may also use an alternative to models (10.23) and (10.24) as follows:
yt = σt εt , (10.30)
σt = ρ(τt) ·ηt , (10.31)
where εt ∼ N(0,1), {ηt} is an independent and distributed distributed
(i.i.d.) random variable with E[η21 ] = 1, and is independent of {εt}.
Models (10.30) and (10.31) can be rewritten as
yt = ρ(τt) ·ηt · εt ≡ ρ(τt)ξt , (10.32)
where ξt = ηt · εt defines an i.i.d. random error term with E[ξ1] = 0 and
E[ξ 21 ] = 1.
Model (10.32) can further be written as
y2t = ρ
2(τt) ·ξ 2t , (10.33)
which suggests estimating ρ2(τ) by
ρ̂2(τ) =
n
∑
t=1
Wnt(τ)y2t . (10.34)
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Deterministic volatility models: Continuous–time
Consider a continuous–time stochastic volatility (SV) model of the form
dY (t) = V (t)dB1(t) and (10.35)
dZ(t) = m(t)dt+σ(t)dB2(t), (10.36)
where
Y (t) is the logarithm of the financial return series (for example),
Z(t) = ln(V (t)) is the logarithm of the volatility process,
B1(t) and B2(t) are two different standard Brownian motion
processes, and B1(t) and B2(t) are independent of each other,
m(t) and σ(t) are unknown functions of t, and
m(·) and σ(·) are the drift and volatility functions of the log–volatility
process.
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By the generalized Ito lemma, the solution of dZ(t) = m(t)dt+σ(t)dB2(t)
is given by
Z(t) = Z(0)+
∫ t
0
m(s)ds+
∫ t
0
σ(s)dB2(s). (10.37)
which then gives an expression for the volatility process:
V (t) = eZ(t) = exp
(
Z(0)+
∫ t
0
m(s)ds+
∫ t
0
σ(s)dB2(s)
)
. (10.38)
There are a couple of issues about (10.38)
Both m(·) and σ(·) are unknown; and
The data process Y (t) has not been used.
We will therefore discuss how to use the observations of Y (t) to estimate
µ(·) and σ(·) as well as then V (t).
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Recall what we have done previously in (10.14). We approximate the
continuous–time models:
dY (t) = V (t)dB1(t) and (10.39)
dZ(t) = m(t)dt+σ(t)dB2(t), (10.40)
by
Yti −Yti−1 = Vti∆iB= Vti
√
∆εi, (10.41)
Zti −Zti−1 = m(ti)∆+σ(ti)
√
∆ηi, (10.42)
where ti = iTn , ti− ti−1 = ∆ = Tn , εi ∼ N(0,1) and ηi ∼ N(0,1).
Define the following random variables:
Wi = ln
(
(Yti−Yti−1 )2
∆
)
; Zi ≡ Zti = ln(Vti );
µi = ln
(
ε2i
)
and νi = ηi∆−1/2.
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Models (10.41) and (10.42):
(Yti −Yti−1 )2
∆
= V 2ti ε
2
i , (10.43)
Zti −Zti−1 = m(ti)∆+σ(ti)
√
∆ηi, (10.44)
can then be written as
Wi = 2Zi+ µi, (10.45)
Zi−Zi−1 = (m(ti)+σ(ti)νi) ∆, (10.46)
which can be written as
yi = m(ti)+σ(ti)νi, (10.47)
where yi = 12∆ ((Wi−Wi−1)− (µi−µi−1)).
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Since µi = ln
(
ε2i
)
can be generated from a known distribution, due to
εi ∼ N(0,1), the actual values of yi become available to you.
By the proposed estimation method, we may estimate m(·) and σ(·)
by m̂(·) and σ̂(·), respectively
We may then estimate V (t) by
V̂ (t) = exp
(
Z(0)+
∫ t
0
m̂(s)ds+
∫ t
0
σ̂(s)dB2(s)
)
. (10.48)
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Examples
Consider the S&P 500 data set from January 1928 to January 1997.
Let yt be the index value at time t;
Let xt = ln(yt/yt−1) be the log–return series.
We will apply one of the models to estimate
1 the trend of yt ; and
2 the volatility of xt .
We will go through this together during the lecture.
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