P 2008-57 - Department of Economics and Business Economics

CREATES Research Paper 2008-57
Impact of time–inhomogeneous jumps and leverage type
effects on returns and realised variances
Almut E. D. Veraart
School of Economics and Management
University of Aarhus
Building 1322, DK-8000 Aarhus C
Denmark
Impact of time–inhomogeneous jumps and leverage type effects
on returns and realised variances
Almut E. D. Veraart∗
CREATES, University of Aarhus
[email protected]
November 2008
Abstract
This paper studies the effect of time–inhomogeneous jumps and leverage type effects on realised variance calculations when the logarithmic asset price is given by a Lévy–driven stochastic
volatility model. In such a model, the realised variance is an inconsistent estimator of the integrated variance. Nevertheless it can be used within a quasi–maximumlikelihood setup to draw
inference on the model parameters. In order to do that, this paper introduces a new methodology
for deriving all cumulants of the returns and realised variance in explicit form by solving a recursive system of inhomogeneous ordinary differential equations.
Keywords: Lévy processes; stochastic volatility, leverage effect; superposition; realised variance.
JEL classification: C10, C13, C14, G10, G12.
1 Introduction
Realised variance and its use for estimating and forecasting stochastic volatility has been studied
extensively in the finance literature in the last decade (see e.g. [2, 10, 11, 37, 52]). So far, such studies
have mainly focused on asset price models given by Brownian semimartingales or, more generally,
Itô semimartingales. This paper is now devoted to studying the impact of time-inhomogeneous jumps
and leverage type effects on realised variance calculations when the logarithmic asset price is given
by a stochastic integral with respect to a Lévy process.
Lévy–driven stochastic volatility models are able to cope with many stylised facts of asset returns
particularly well, e.g. they reflect the skewness and fat tails of asset return distributions more appropriately and can handle jumps and volatility smiles much better than models based on Brownian motions
alone. Recently, several types of such Lévy–based stochastic volatility models have been studied in
the financial literature. Basically, they can be divided into two groups: time–changed Lévy processes
(see e.g. [12, 21, 22]) and stochastic integrals with respect to a Lévy process (see e.g. [28, 40, 51, 53]).
Here we restrict our attention to the latter class of Lévy–based stochastic volatility models. We note
that this class of models does not generally fall into the class of affine models (as time–changed Lévy
∗
This paper is a revised part of my DPhil thesis and, therefore, I wish to thank my supervisors Neil Shephard and Matthias
Winkel for their guidance and support throughout this project. Financial support by the Rhodes Trust and by the Center
for Research in Econometric Analysis of Time Series, CREATES, funded by the Danish National Research Foundation, is
gratefully acknowledged.
1
2 MODEL DEFINITION AND TECHNICAL ASSUMPTIONS
2
processes, see e.g. [26, 39]) and, hence, explicit computations in such a modelling framework turn
out to be analytically more involved. Apart from inhomogeneous jumps, our model also allows for
very general leverage type effects or asymmetric volatility. During the last decades, many empirical
studies have revealed the fact that past stock returns tend to be negatively correlated with innovations of future volatilities. This property is often called the leverage effect — an expression which
has been derived from the hypothesis that a negative stock return might increase financial leverage
and, hence, leads to a riskier stock which results in higher volatility. [17] was probably the first to
investigate this effect, and his finding was further supported by studies by [23, 43] among others and,
more recently, by [18, 20, 35, 49, 50, 54]. While the existence of asymmetric volatility is rarely
questioned, its main determinant is still subject to vivid discussions (see e.g. [15] and the references
therein). Besides the previously mentioned leverage–hypothesis, there is also the time–varying risk
premium theory or volatility feedback theory, which essentially relies on the converse causality when
stating that increasing volatility leads to decreasing stock price returns. However, regardless of where
asymmetric volatility originates from, it is definitely an important fact, which has to be accounted
for in asset pricing, especially in the context of option pricing since the asymmetric relationship is
directly associated with implied volatility smiles. So the leverage effect is often regarded as a natural
tool for explaining smirks in option price data (see [31, 36]). Unfortunately, previous work on the
econometric properties of Lévy –driven stochastic volatility has so far only been carried out under the
no–leverage assumption ([12, 53]) which is, particularly in equity markets, not realistic. So the main
contribution of this paper is, that it overcomes this restrictive assumption and studies the impact of
both time–inhomogeneous jumps and general leverage type effects on returns and realised variance
simultaneously.
The remaining part of the paper is structured as follows. Section 2 sets up the notation and defines
the Lévy–driven stochastic volatility model, which we study in this paper. Following recent research
on stochastic volatility models, we use the so–called realised variance as a proxy for the accumulated
variance over a day. This quantity will be defined in Section 3. Section 4 and 5 contain the main
theoretical results of the paper. In Section 4, we present explicit formulae for the moments and
second order properties of the returns, the actual variance and the quadratic variation of the price
process. Section 5 addresses the first and second order properties of the realised variance where
we study in detail the influence of the jumps and the leverage effect on volatility estimation. All
these results are derived explicitly by using a novel methodology which involves solving a recursive
system of inhomogeneous ordinary differential equation. Finally, we give a brief outlook on parameter
estimation and inference in Section 6. Throughout the text, all of the mathematical proofs are relegated
to the Appendix (Section A).
2 Model definition and technical assumptions
Let (Ω, A, P) denote a probability space with filtration F = {Ft }0≤t<∞ , satisfying the usual conditions (see e.g. [47]). Let S = (St )t≥0 denote the logarithmic asset price and σ = (σt )t≥0 the
stochastic volatility (SV). We will study models of the form
Z t
σs− d(vWs + Xs ),
St = µt +
0
where X = (Xt )t≥0 denotes a pure jump Lévy processes (see e.g. [16, 45, 48]) and W = (Wt )t≥0
is a standard Brownian motion and v ∈ R is a constant, which could be 0. In that case, we would be
in a pure jump setting. Solely for ease of exposition, we will assume that EX1 = 0. Furthermore,
2 MODEL DEFINITION AND TECHNICAL ASSUMPTIONS
3
we assume that Var(X1 ) < ∞. More precisely, in order to make sure that the model is uniquely
identified, we set V arX1 + V ar(vW1 ) = 1. Otherwise, one could always multiply vW + X by a
constant and scale σ appropriately and one would still obtain the same value for the price process S.
Here we will focus on Ornstein–Uhlenbeck (OU) type stochastic volatility models. I.e., we shall
model the volatility process by a stationary Lévy–driven OU process which satisfies the following
stochastic differential equation
dσt = −λσt− dt + dYλt ,
where Y = (Yt )t≥0 denotes a pure jump subordinator (i.e. a non–decreasing Lévy process) and λ > 0
is the memory parameter. Throughout we assume that σ0 is drawn from its stationary distribution.
Similar models have been studied in detail in the Brownian motion framework (i.e. when X is a
Brownian motion) by [9] who chose σ 2 to be a OU process. However, studies by [41] have shown
that choosing σ or σ 2 to be an OU process leads to similar results. So for reasons of mathematical
tractability we have chosen the volatility process rather than the variance to be of OU–type. In such a
modelling framework, the dynamics of the squared volatility process are given by
2
dσt2 = −2λσt−
dt + 2σt− dYλt + d[Y ]λt ,
which might remind us on a jump–driven version of a square root process. Clearly, such a model
for the volatility process satisfies the essential requirement that volatility has to be non–negative. By
time–changing the subordinator Y by the memory parameter λ, we obtain a stationary distribution of
σ which does not depend on λ. Note in particular that we do not assume that σ and X are independent.
We rather choose a bivariate Lévy process (vW + X, Yλ )′ as driving process of (S, σ)′ and, hence,
can capture the leverage effect with this model. In order to be able to choose such a bivariate Lévy
process, we have to make sure that both driving processes run on the same time scale. The choice of
Yλ rather than Y in the second component is hence essential. Otherwise it would be possible that there
was already information about the price process available before there was any information about the
volatility process and vice versa, and this would possibly lead to arbitrage opportunities. Throughout
the text, we will set µ = 0, although the results can be easily generalised for a price process which
includes a drift term. Finally, we introduce the notation for the cumulants of the the driving Lévy
processes. We define Z = (Zt )t≥0 by Zt = (Xt , Yλt )′ for λ > 0, which is a bivariate pure jump
Lévy process of which the second component is a subordinator. Let ν denote the Lévy measure of
Z and νX and νYλ denote the Lévy measure of X and Yλ , respectively. Note that if X and Y are
pure jump
P Lévy processes of finite
P variation, then they can be represented as sum of their jumps:
Xt = 0≤s≤t ∆Xs ,and Yλt = 0≤s≤t ∆Yλs .
P
Remark If v = 0 and X is of finite variation, S can be written as St = 0≤s≤t ∆Ss . From [38,
Chapter IX, Proposition 3] we can deduce that its characteristic triplet is given by (0, 0, νS ), where
νS is defined by 1IF ⋆ νS = 1IF (σs− x) ⋆ νX for all F ⊆ R\{0}, where we denote by f ⋆ ν for
aR function f on a subset of R2 and random measure ν the following integral process (f ⋆ ν)t =
R×[0,t] f (x, s)ν(dx, ds).
Recall that the n–th cumulant of a stochastic process Z = (Zt )t≥0 is defined by (provided it exists)
κn (Zt ) =
1 ∂n
log (E (exp(iuZt ))) .
in ∂un
Furthermore, it is well–known (see e.g. [24, p.32, 92]) that the moments of a Lévy process can
then be expressed in terms of the corresponding cumulants by E(Zt ) = κ1 (Z1 )t, E(Zt2 ) = κ2 (Z1 )t +
3 RETURNS AND REALISED VARIANCE
4
(κ1 (Z1 )t)2 , E(Zt3 ) = κ3 (Z1 )t + 3κ1 (Z1 )κ2 (Z1 )t2 + (κ1 (Z1 )t)3 , E(Zt4 ) = κ4 (Z1 )t + 3(κ2 (Z1 )t)2 +
4κ1 (Z1 )κ3 (Z1 )t2 + 6(κ1 (Z1 )2 )κ2 (Z1 )t3 + (κ1 (Z1 )t)4 . For the cumulants of these processes, we will
use the following notation. The cumulants (denoted by κi (·), i = 1, 2, . . . ) of the random variable X1
are denoted by ξ and the cumulants of the process Y1 are denoted by η. Hence,
Z
Z
v i νY1 (dv), for i = 1, 2, . . .
ui νX (du),
ηi = κi (Y1 ) =
ξi = κi (X1 ) =
[0,∞)
R
Note that κi (Yλ ) = ληi , for i = 1, 2, . . . . Furthermore,
Z
un v m ν(du, dv),
κn,m =
R×[0,∞)
for n, m ∈ N.
Throughout the text, we will assume that at least the first four cumulants of the Lévy process Z are
finite. From the Cauchy–Schwarz inequality, we obtain the following
constraints for the cumulants of
√
Z. For n, m ∈ N, the cumulants (if they exist) satisfy κn,m ≤ ξ2n λη2m .
The cumulants of the bivariate Lévy process can be regarded as a measure of the dependence
between the two driving processes, which obviously includes the leverage effect: the measure of
dependence of first order. In the following, we will deal with the following five cumulants.
κ1,1 = E {(X1 − EX1 )(Yλ − EYλ )} = Cov(X1 , Yλ ),
κ1,2 = E (X1 − EX1 )(Yλ − EYλ )2 ,
κ1,3 = E (X1 − EX1 )(Yλ − EYλ )3 − 3E {(X1 − EX1 )(Yλ − EYλ )) E (Yλ − EYλ )2 ,
κ2,1 = E (X1 − EX1 )2 (Yλ − EYλ ) ,
κ2,2 = E (X1 Yλ ) − 2E (X1 ) E X1 Yλ2 − 2E (Yλ ) E X12 Yλ − E X12 E Yλ2
− 2 {E (X1 Yλ )}2 + 2 {E (X1 )}2 E Yλ2 + 2 {E (Yλ )}2 E X12
+ 8 E (Yλ ) E (X1 ) E (X1 Yλ ) − 6 {E (X1 )}2 {E (Yλ )}2 .
Note that if X and Yλ are independent, they have no common jumps and, hence, ν{(u, v) ∈ R ×
[0, ∞) : uv 6= 0} = 0. That means that, if X and Yλ are independent and, hence, there is no leverage
effect, then all joint cumulants κn,m = 0.
3 Returns and realised variance
Our aim is to study the econometric properties of the Lévy–type SV model defined above. So far, we
have defined a model for an asset price in continuous time. We can now use the increments of such
a process for modelling (high frequency) returns. Recall that (St )t≥0 denotes the continuous–time
log–price process of an asset. Further, let h > 0 denote the length of a fixed time interval, typically
one day. The returns of the asset price are then given by
si = Sih − S(i−1)h ,
i = 1, 2, . . . ,
where i indexes the day. Due to the availability of high frequency data, one is often interested in
modelling returns at a higher frequency than just daily data. Suppose that we are given M intra–
h observations during each time interval of length h. The time gap between these high frequency
observations is denoted by δ = h/M . Then
sj,i = S(i−1)h+jδ − S(i−1)h+(j−1)δ ,
j = 1, . . . , M
4 CUMULANTS OF RETURNS, ACTUAL VARIANCE AND INCREMENTAL QUADRATIC VARIATION5
denotes the j–th intra–h high frequency return on the i–th period of length h. Often we work with
h = 1, representing one day. Based on these high frequency returns, one can then define the realised
variance for the i–th day by
M
X
s2j,i .
[Sδ ][(i−1)h,ih] =
j=1
This quantity is often used to proxy the variability in financial markets in SV models (see e.g. [4,
10]). In this paper, we will compute the realised variance based on five–minute returns and will
therefore ignore possible market microstructure effects which come into play when analysing really
high frequency returns (i.e. one minute returns, tick by tick data). Such effects can be caused by
e.g. bid/ask spreads, irregular trading and the fact that prices are recorded in discreet time. Since
studying realised variance in the presence of market frictions is beyond the scope of this paper, we
just refer to articles by e.g. [1, 3, 5, 6, 8, 13, 33, 55–57], for recent and very detailed studies of this
research topic.R Recall that the quadratic variation of a semimartingale S = (St )t≥0 is defined by
t
[S]t = St2 − 2 0 Su− dSu . It is well–known that
ucp
[Sδ ][(i−1)h,ih] −→ [S]ih − [S](i−1)h ,
as M → ∞ (i.e. δ → 0),
where the convergence is uniform on compacts in probability (ucp) (see [45]). So the realised variance
can be used to estimate the (increments of the) quadratic variation of the price process consistently.
Hence, in our modelling framework, the realised variance can be used as a consistent estimator of
[S]ih − [S](i−1)h =
Z
ih
(i−1)h
2
σs−
d[X]s .
Rt
However, one is rather interested in estimating and forecasting the integrated variance IVt = 0 σs2 ds.
From Lévy’s theorem, we can deduce that [X]t = t if and only if X is a Brownian motion. I.e. as soon
as the Brownian motion is replaced by a more general Lévy process or semimartingale, the quadratic
variation of the SV model is not given by the integrated variance. Hence, it is important to study the
bias and the degree of inconsistency of the realised variance as proxy for the integrated variance. This
is the task we tackle in the next section.
4 Cumulants of returns, actual variance and incremental quadratic variation
Here we study the statistical properties of the following three quantities: the price process S, the
integrated variance IV and the quadratic variation of the price process [S]. From these properties,
we can then directly derive properties of the increments of the corresponding stochastic processes:
the returns of the log–price (the increments of S), the actual variance (the increments of IV ) and the
incremental quadratic variation (the increments of [S]).
4.1 Cumulants of returns
We start our theoretical study by computing the moments of the log–price process S. Since these
moments depend not only on the moments of X but also on those of σ, we calculate the moments of σ
first. In order to do that, we derive a general representation formula for the nth power of σ, for n ∈ N.
4 CUMULANTS OF RETURNS, ACTUAL VARIANCE AND INCREMENTAL QUADRATIC VARIATION6
Proposition 4.1 Let n ∈ N. As long as
σtn
−
σ0n
= −λ n
Z
Rt
0
t
σsn ds < ∞, the n–th power of σt satisfies
n
σs−
ds
0
+
n X
X
n
k
k=1
n−k
(∆Yλs )k .
σs−
0≤s≤t
Proof. Given in the Appendix. From the formula above, one can deduce the moments of σ.
Corollary 4.2 Recall that ηi = κi (Y1 ). The
four moments of the stationary distribution of
first
σ
3
1
1
2
3
4
2
3
are, hence, given by: E (σt ) = η1 , E σt = η1 + 2 η2 , E σt = η1 + 3 η3 + 2 η2 η1 , E σt =
4
3 2
1
2
4
4 η4 + η1 + 3η2 η1 + 3 η1 η3 + 4 η2 .
Now we focus on the moments of the price process S and, also, on its joint moments with the volatility
process σ. It turns out that by repeated applications of Itô’s formula and the use of the compensation
formulas for jump processes, we obtain a recursive system of inhomogeneous ordinary differential
equations, which can be solved explicitely. This methodology is described in detail in the Appendix
and will be used extensively in the remaining paper. So although our model is generally not affine
and, hence, might look complicated to tackle at first sight – at least compared to the affine models of
time changed Lévy processes (see e.g. [39]) – the new methodology proposed in this paper enables us
to derive all cumulants of interest explicitely.
Proposition 4.3 (Recursive formulae for the moments of S)
Let k, n ∈ N, with k ≤ n. If ηk < ∞ for k ≤ n, we get the following results.
1. The joint moments of S and σ are given by
Z t
−kλt
n−k k
g(u; n, k) ekλu du,
E S t σt = e
0
where
g(u; n, k) =
k X
k
j=1
+
n−k
X
j=2
+
j
ληj
Z
t
0
Z
(n − k)(n − k − 1)v 2 t n−k−2 2 n−k k−j
σs− ds +
E Ss−
E Ss
σs ds
2
0
n−k
ξj
j
k n−k
XX
j=1 l=1
Z
t
0
Z t n−1
X n j
m+n−j
k+j n−k−j
+
E σs S s
ξn−j
ds
E Ss−
σs−
j
0
j=0
Z t n−k
k
E Ssn−k−j σsj+k−l ds.
κj,l
j
l
0
2. The moments of S are given by
E (Stn )
n(n − 1) 2
v
=
2
Z
0
t
E
n−2 2
σs−
Ss−
ds +
n X
n
k=2
k
ξk
Z
0
t
n−k
k
.
E σs−
Ss−
Proof. Given in the Appendix.
Now we can recursively solve the equation above and we obtain the corresponding moments of S.
4 CUMULANTS OF RETURNS, ACTUAL VARIANCE AND INCREMENTAL QUADRATIC VARIATION7
Corollary 4.4 If ξ1 = 0, the first two moments are given by:
E (St ) = 0,
η2 t.
E St2 = v 2 + ξ2 η12 +
2
Using a Taylor expansion around 0 for moments of higher order, one obtains for t ↓ 0:
3
1
3 2
E St3 = ξ3 η13 + η3 + η1 η2 t +
v + ξ2 κ1,1 (2η12 + η2 ) + κ1,2 η1 t2 + O(t3 ),
3
2
2
1
3 2
4
4
4
2
E S t = ξ4
η4 + η1 + η2 + 3η1 η2 + η1 η3 t
4
4
3
2
3
9 2
4
2
+
3η1 + 4η1 η3 + η4 + 9η1 η2 + η2 v 2 + ξ2
4
4
3
2
3
+ 6η1 + 2η3 + 9η1 η2 κ2,1 + 3η1 + η2 κ2,2 v 2 + ξ2
2
3
+ ξ3 6η1 + 9η2 η1 + 2η3 κ1,1 + 3η2 + 6η12 κ1,2 + 2η1 κ1,3 t2 + O(t3 ).
Further, one obtains
1
3
4
V ar(St2 ) = ξ4
η4 + η22 + 3η12 η2 + η1 η3 + η14 t
4
4
3
2
3
2
2
4
+
8η1 η2 + 2η2 + 2η1 + 4η1 η3 + η4 v 2 + ξ2
4
3
2
3
+ 6η1 + 2η3 + 9η1 η2 κ2,1 + 3η1 + η2 κ2,2 v 2 + ξ2
2
3
+ξ3 6η1 + 9η2 η1 + 2η3 κ1,1 + 3η2 + 6η12 κ1,2 + 2κ1,3 η1 t2 + O(t3 ).
Since S has stationary increments, we can deduce the moments of the corresponding returns si
over a time interval of length h by setting t = h and, hence,
E(si ) = E(Sh ), E(s2i ) = E(Sh2 ), E(s3i ) = E(Sh3 ), E(s4i ) = E(Sh4 ), V ar(s2i ) = V ar(Sh2 ).
4.2 First and second order properties of the actual variance
Recent research has focused on integrated variance Ras a measure for the variability of financial mart
kets. The integrated variance is defined by IVt = 0 σs2 ds. Often, one is interested in studying the
increments of this process over a time interval of length h, say. So, we will denote these increments
by
Z ih
2
σs2 ds,
σ[(i−1)h,ih]
= IVih − IV(i−1)h =
(i−1)h
which is generally called the actual variance (AV) on the ith interval of length h. Basically, it measures the accumulated variance over a time interval (often chosen to be one day). Now we can compute
the mean, variance and covariance of the AV as given in the following proposition.
Proposition 4.5 The mean, variance and covariance of the actual variance are given by the following
formulae:
η2 2
h,
E(σ[(i−1)h,ih]
) = η12 +
2
4 CUMULANTS OF RETURNS, ACTUAL VARIANCE AND INCREMENTAL QUADRATIC VARIATION8
and
1 2
1
1
2
η + 4η1 η2 + 2η1 η3 + η4 λ h
V
= 2
λ
2 2
4
4
1 2 1
1
1
5
1 2
−λh
−2λh
2
2
+
+
− η4 − 4η1 η2 − η1 η3 − η2 ,
η1 η3 + 4η1 η2 e
η + η4 + η1 η3 e
3
4 2 8
3
8
3
4
2
ar(σ[(i−1)h,ih]
)
and
2
2
Cov(σ[(i−1)h,ih]
, σ[(i+s−1)h,(i+s)h]
)
2
1
2
2η1 η2 + η1 η3
e−λ(s+1)h + e−λ(s−1)h − 2e−λhs
= 2
λ
3
1 2
1
1
−2λ(s+1)h
−2λ(s−1)h
−2λhs
+
η + η4 + η1 η3
e
+e
− 2e
.
8 2 16
6
Proof. Given in the Appendix.
As already mentioned above, in a SV model based on a Brownian motion, the actual variance can
be consistently estimated by the realised variance. However, in a more general Lévy–based model,
the quadratic variation of the price process does not equal the integrated variance. Hence, we will
turn our attention to the quadratic variation of the log–price process and study its first and second
order properties. We will then be able to compare those with the results we have just obtained for the
integrated variance.
4.3 First and second order properties of the quadratic variation
The first and second order properties of the quadratic variation are described in the following proposition.
Proposition 4.6 Let t, s > 0 and ξ1 = 0. Then:
2
E ([S]t ) = v + ξ2
η12
1
+ η2 t,
2
V ar([S]t ) = c1 + c2 t + c3 e−λ t + c4 e−2 λ t ,
1
1
Cov([S]t , [S]t+s ) = c1 + c2 t + c3 (e−λt − e−λs + e−λ(t+s) )
2
2
1
+ c4 (e−2λt − e−2λs + e−2λ(t+s) ),
2
where ci = ci (λ, v, ξ2 , ξ4 , η1 , η2 , η3 , η4 , κ2,1 , κ2,2 ) for i = 1, . . . , 4, with
2
1 h
−40η1 η3 − 96η12 η2 − 3η4 − 6η22 v 2 + ξ2
2
24λ
+ −8η3 − 72η1 η2 − 96η13 κ2,1 + −6η2 − 12η12 κ2,2 v 2 + ξ2 ,
2
1 h
c2 =
48η1 η3 + 12η22 + 96η12 η2 + 6η4 λ v 2 + ξ2
2
24λ
+ 96η1 η2 + 96η13 + 16η3 κ2,1 + 24η12 + 12η2 κ2,2 λ v 2 + ξ2
+ 24η14 + 18η22 + 6η4 + 72η12 η2 + 32η1 η3 λ2 ξ4 ,
c1 =
4 CUMULANTS OF RETURNS, ACTUAL VARIANCE AND INCREMENTAL QUADRATIC VARIATION9
2
2
o
1 n
2
3
2
96η
η
+
32η
η
v
+
ξ
+
48η
η
+
96η
κ
v
+
ξ
1 3
2
1 2
2,1
2
1 2
1
24λ2
h
2
1
6η22 + 3η4 + 8η1 η3 v 2 + ξ2
c4 =
2
24λ
+ (8η3 + 24η1 η2 ) κ2,1 + 12η12 + 6η2 κ2,2 v 2 + ξ2 .
c3 =
From this proposition, we can easily deduce the first and second order properties for the incremental quadratic variation (IQV):
[s][(i−1)h,ih] = [S]ih − [S](i−1)h =
Z
ih
(i−1)h
2
σu−
d[vW + X]u .
They are given by the following.
Theorem 4.7 Let t, s > 0 and ξ1 = 0 and v 2 + ξ2 = 1. Then:
2
E([s][(i−1)h,ih] ) = E(σ[(i−1)h,ih]
),
3 2
4
1
4
V ar([s][(i−1)h,ih] ) = V
+ ξ4
+ η2 + η1 + η1 η3 + η4 h
4
3
4
h
1
+ 2 κ2,1 4η13 + 2η1 η2 e−λh − 1 + λh
λ
1
1
1
−2λh
2
e
− 1 + 2λh ,
+ κ2,1 η1 η2 + η3 + κ2,2 η1 + η2
3
2
2
2
ar(σ[(i−1)h,ih]
)
3η12 η2
and
2
2
Cov([s][(i−1)h,ih] , [s][(i+s−1)h,(i+s)h] ) = Cov(σ[(i−1)h,ih]
, σ[(i+s−1)h,(i+s)h]
)
1
+ 2 [κ2,1 2η13 + η1 η2 (e−λ((s+1)h) − 2e−λhs + e−λ((s−1)h) )
λ
1
1
1
1
2
η3 + η1 η2 + κ2,2
+ κ2,1
η1 + η2
·
6
2
4
2
(e−2λ(s+1)h − 2e−2λhs + e−2λ(s−1)h )].
When we compare the first and second order properties of the AV with those of the IQV, we
observe the following. Firstly, in the variance of IQV there is an extra summand given by ξ4 Eσ 4 h.
Note here that weRhave assumed that X is a pure jump Lévy process without drift. Hence, νX1 6= 0
and, hence, ξ4 = R x4 νX1 (dx) > 0, so this factor will never disappear. Secondly, both the variance
and the covariance of the IQV have an extra term which is due to a possible leverage–type effect in the
model. Clearly, in the absence of this effect (i.e. when X and Y are independent), then κ2,1 = κ2,2 =
0, and hence this extra term would not exist. So, altogether, we can say that, by choosing a pure jump
Lévy process as a driving process for the asset price, we observe an extra term in the variance of the
IQV compared to the variance of the AV. If one additionally allows for a leverage type effects, both
the variance and the covariance of the IQV have to be generalised by an additional leverage term.
4.4 Covariation of returns
We conclude this section by studying the covariance between returns, squared returns and IQV. We
will consider returns over a time interval of length h, which are denoted by si = Sih − S(i−1)h .
5 FIRST AND SECOND ORDER PROPERTIES OF THE REALISED VARIANCE
10
Proposition 4.8 Let i, s ∈ N. For h → 0, one obtains with ξ1 = 0:
Cov(si , si+s ) = 0,
Cov(si , s2i−s ) = 0,
1 n
Cov(si , s2i+s ) = 2 8η12 κ1,1 e−λ(s+1)h − 2e−λhs + e−λ(s−1)h
4λ
+ (κ1,1 η2 + κ1,2 η1 ) e−2λ(s+1)h − 2e−2λhs + e−2λ(s−1)h
o
v 2 + ξ2
Cov(s2i , s2i+s ) = Cov([s][(i−1)h,ih] , [s][(i+s−1)h,(i+s)h] )
+ κ1,1 2(η2 + 2η12 )κ1,1 + 3κ1,2 η1 v 2 + ξ2 h3 + O(h4 ).
Besides,
Cov(s2[(i−1)h,ih] − [s][(i−1)h,ih] , s2[(i+s−1)h,(i+s)h] − [s][(i+s−1)h,(i+s)h] ) = 0.
Proof. Given in the Appendix.
So we see that the asset returns are uncorrelated (under the assumption that ξ1 = 0). Further,
we observe that the covariance between returns and squared returns basically depends on the two
leverage parameters κ1,1 , κ2,1 , which denote the covariation between X1 and Yλ and the joint centred
moment of X12 and Yλ , respectively. Recall that the squared returns can also be used for estimating the
variance (although such an estimate is noisier than one based on realised variance). This covariation
will damp down exponentially with the lag length s and so will the influence of a possible leverage
effect. Finally, we observe that the covariance between squared returns can be approximated by the
covariance between the IQV and by terms of lower order which depend on parameters κ of possible
leverage .
5 First and second order properties of the realised variance
Finally, we can apply the results we have deduced so far for computing the first and second order
properties of the realised variance (RV) and for studying the degree of inconsistence of the RV as
estimator for the integrated variance.
5.1 First and second order properties of Realised Variance Error
Let
Ht =
St2
− IVt =
Proposition 5.1 Let t, s > 0 and ξ1 = 0 and
v2
St2
−
Z
0
t
σs2 ds.
+ ξ2 = 1. Then (for t → 0):
E(Ht ) = 0,
3
4
1
η4 + η22 + 3η12 η2 + η1 η3 + η14 t
E(Ht2 ) = (ξ4 − 2)
4
4
3
3
+
6η1 + 2η3 + 9η1 η2 κ1,1 + 6η12 + 3η2 κ1,2 + 2κ1,3 η1 ξ3
9 2 3
4
3
2
4
2
η3 + 4η1 + 6η1 η2 κ2,1 + 2η1 + η2 κ2,2 + 3η1 + 4η1 η3 + 9η1 η2 + η2 + η4 t2
+
3
4
4
5 FIRST AND SECOND ORDER PROPERTIES OF THE REALISED VARIANCE
11
Proof. Given in the Appendix.
Since both S 2 and IV are stationary, we can easily deduce the results for the corresponding
increments of the returns and the IV by setting t = h and, hence,
2 2
2
2
2
E si − σ[(i−1)h,ih] = E (Hh ) ,
si − σ[(i−1)h,ih]
E
= E Hh2 .
So we observe that even in the presence of leverage the expectation of H is zero and, hence, there
is no bias. However, leverage–type effects (of higher order) do affect the mean square error. We start
by studying the properties of the difference between the squared log–price process and the quadratic
variation.
Proposition 5.2 Let t, s > 0 and ξ1 = 0. Define Gt = St2 − [S]t . Then (for t → 0):
E(Gt ) = 0,
2
1
3 2
8
4
2
η1 η3 + 2η1 + η4 + η2 + 6η1 η2 v 2 + ξ2
V ar(Gt ) =
3
2
2
2
4
3
2
2
+
η3 + 4η1 + 6η1 η2 κ2,1 + 2η1 + η2 κ2,2 v + ξ2 t + O(t3 ),
3
3
9
2
3
Cov(Gt , [S]t ) = ξ3 κ1,1 3η1 + η3 + η1 η2 + κ1,2 3η1 + η2 + κ1,3 η1 t2 + O(t3 ).
2
2
Proof. Given in the Appendix.
Recall that the realised variance error (when estimating the IQV by the RV) is given by
[Sδ ][(i−1)h,ih] −
2
σ[(i−1)h,ih]
=
M
X
j=1
2
(s2j,i − σj,i
).
Using
L
sj,i = Sδ
and
L
[s]j,i = [S]δ ,
we obtain the following result for the squared returns and the IQV.
Corollary 5.3 Let ξ1 = 0 and v 2 + ξ2 = 1. Then:
E([Sδ ][(i−1)h,ih] − [s][(i−1)h,ih] ) = 0,
8
1
3 2
4
2
V ar([Sδ ][(i−1)h,ih] − [s][(i−1)h,ih] ) =
η1 η3 + 2η1 + η4 + η2 + 6η1 η2
3
2
2
4
3
2
+ κ2,1
η3 + 4η1 + 6η1 η2 + κ2,2 2η1 + η2
h2 M −1 + O(M −2 ),
3
and
Cov([Sδ ][(i−1)h,ih] − [s][(i−1)h,ih] , [Sδ ][(i+s−1)h,i+sh] − [s][(i+s−1)h,(i+s)h] ) = 0.
So we observe the following. RV is an unbiased estimate for IQV; the variance of the RV error is
of O(M −1 ) and the RV errors are uncorrelated. These findings correspond to similar results in the
Brownian motion case (see [10]).
5 FIRST AND SECOND ORDER PROPERTIES OF THE REALISED VARIANCE
12
5.2 Cumulants of realised variance
Using the results above, we can now derive the mean, variance and covariance of the realised variance.
Proposition 5.4 Let ξ1 = 0 and v 2 + ξ2 = 1. The first and second order properties of the realised
variance are then given by
1
E[Sδ ][(i−1)h,ih] = η12 + η2 h,
2
and
and
8
1
3
η1 η3 + 2η14 + η4 + η22 + 6η12 η2
V ar([Sδ ][(i−1)h,ih] ) = V ar([s][(i−1)h,ih] ) +
3
2
2
)
(
κ21,1
κ
κ
4
1,2
1,1
η1 + κ2,1
η3 + 4η13 + 6η1 η2 + κ2,2 2η12 + η2
η2 + 3η12 + 3
+ 2
λ
λ
3
+ξ3 κ1,1 6η13 + 2η3 + 9η1 η2 + κ1,2 6η12 + 3η2 + 2κ1,3 η1 h2 M −1
"
o
κ21,1 n 2 −2λh
−2λh
−λh
e
−
1
e
+
4e
−
5
+
η
η
+
2
1
λ2
3
−2λh
+ 2 κ1,2 κ1,1 η1 −1 + e
hM −1 + O M −2 ,
2λ
Cov([Sδ ][(i−1)h,ih] , [Sδ ][(i+s−1)h,(i+s)h] ) = Cov([s][(i−1)h,ih] , [s][(i+s−1)h,(i+s)h] )
κ1,1 h
+ 2 8κ1,1 η12 −2e−λhs + e−λ(s−1)h + e−λ(s+1)h
4λ
i
2
+ 2κ1,1 (η1 + η2 ) + 3κ1,2 η1 −2e−2λhs + e−2λ(s+1)h + e−2λ(s−1)h hM −1 + O M −2 .
Proof. Given in the Appendix.
5.3 Comparing the autocorrelation functions of realised variance, quadratic variation
and integrated variance
Now we briefly study some implications of our results for autocorrelations of RV, QV, and IV. Hereby
we follow [12], who have studied the same question in the framework of a time–changed Lévy process.
From our results above, we can deduce that:
lim Cor [Sδ ][(i−1)h,ih] , [Sδ ][(i+s−1)h,(i+s)h] = Cor [s][(i−1)h,ih] , [s][(i+s−1)h,(i+s)h]
M →∞
=
where the leverage part in the covariance is denoted by
LC =
2
2
Cov(σ[(i−1)h,ih]
, σ[(i+s−1)h,(i+s)h]
) + LC
,
2
+ ξ4 Eσ 4 + LV
V ar σ[(i−1)h,ih]
−λ((s+1)h)
1 h
3
−λhs
−λ((s−1)h)
κ
2η
+
η
η
e
−
2e
+
e
2,1
1 2
1
λ2
1
1
1
1
2
η3 + η1 η2 + κ2,2
η1 + η2
+ κ2,1
6
2
4
2
5 FIRST AND SECOND ORDER PROPERTIES OF THE REALISED VARIANCE
i
e−2λ(s+1)h − 2e−2λhs + e−2λ(s−1)h ,
and the leverage part in the variance is denoted by
LV =
13
−λh
1 h
3
κ
4η
+
2η
η
e
−
1
+
λh
2,1
1 2
1
λ2
1
1
1
−2λh
2
e
− 1 + 2λh .
+ κ2,1 η1 η2 + η3 + κ2,2 η1 + η2
3
2
2
In the absence of leverage type effects (hence LC = LV = 0), we obtain exactly the same results as
derived by [12] for time–changed Lévy processes:
• The acf of the RV is monotonically decreasing in ξ4 .
• For M → ∞, the acf of the RV is given by
2
2
Cov(σ[(i−1)h,ih]
, σ[(i+s−1)h,(i+s)h]
)
lim Cor [Sδ ][(i−1)h,ih] , [Sδ ][(i+s−1)h,(i+s)h] =
M →∞
2
+ ξ4 Eσ 4
V ar σ[(i−1)h,ih]
2
2
< Cor σ[(i−1)h,ih]
, σ[(i+s−1)h,(i+s)h]
, since ξ4 > 0.
which implies that the ACF of RV systematically underestimates the ACF of the actual variance.
• And for ξ4 → ∞, we obtain
lim Cor [Sδ ][(i−1)h,ih] , [Sδ ][(i+s−1)h,(i+s)h] = 0.
ξ4 →∞
However, if we allow for leverage–type effects, we observe the following for the acf of the RV:
• Dependencies between X and (higher) moments of Y (i.e. κ1,1 , κ1,2 , κ1,3 ) are asymptotically
negligible. In particular, the quantity κ1,1 , which describes the classical leverage effect, has
asymptotically no influence on the acf of the RV.
• Dependencies between X 2 and (higher) moments of Y (i.e. κ2,1 , κ2,2 ) do influence the acf of
the RV.
5.4 Superposition model
Let us briefly mention a method for generalising our model slightly. Many empirical studies have indicated that one–factor stochastic volatility models cannot fit empirical data very satisfactorily. Hence,
a standard approach for tackling this problem is to study at least a two–factor (or a multi–factor)
stochastic volatility model (see e.g. [14], [18]). Often, one uses the class of so–called superposition
models where the volatility is not just given by a single OU process (as in our modelling framework),
but rather by a convex combination of independent OU processes (see e.g. [7], [10] and the references therein). We assume that the volatility process
Pis given by a weighted sum of independent OU
processes. For J ∈ N and i = 1..J, let wi ≥ 0 and Ji=1 wi = 1. Then we define
σt =
J
X
i=1
(i)
wi τt ,
(i)
(i)
dτ (i) = −λ(i) τt dt + dYλ(i) t ,
6 MODEL ESTIMATION AND INFERENCE
14
where we assume that the Yi are independent (but not necessarily identically distributed). However,
(i)
as in the one–factor model, we allow for dependence between Xt and Yλ(i) t . In particular, since X is
a Lévy process, there is a sequence of independent identically distributed random variables XJ,k for
law
k = 1, . . . , J such that X = XJ,1 + · · · + XJ,J . So, we write
Z
(i)
un v m νXJ,i ,Y (i) (du, dv),
κn,m =
R×R+
for the corresponding cumulants of the bivariate Lévy process. The mean, variance and covariance of
the realised variance when the volatility process is given by a superposition model can be derived in
a similar way as in the one–factor model. However, due to the fact that we allow for dependencies
between the driving process of the asset price and the driving processes of the different components
of σ, these formulae become rather lengthy. So, we will just present them in the Appendix.
6 Model estimation and inference
6.1 Quasi–likelihood estimation based on realised variance
Finally, we turn our attention to estimating the parameters of our Lévy –driven stochastic volatility
model. It is well–known that parameter estimation in such a model framework is difficult since one
cannot easily compute the exact likelihood function. Here we follow [12] and use a quasi–maximum
likelihood approach (see e.g. [30, Chapter 5]) based on the Gaussian density function. This methodology leads to consistent and asymptotically normally distributed set of estimators. Alternative estimation techniques include method of moment (e.g. [19]) and simulation based methods. For instance,
independent work by [46], [29] and [32] have focused on Markov chain Monte Carlo methodology for
Bayesian inference in OU stochastic volatility models. Recall that we have shown that we can write
the mean, the variance and the covariance of the vector of realised variances [Sδ ] = ([Sδ ]1 , . . . , [Sδ ]n )′
as function of the model parameters, which we write in terms of a vector θ, say. We choose the following quasi–maximum likelihood (QML) approach for estimating the parameters. Let
n
1
l(θ) = log L(θ) = − log(2π) − log det(Cov([Sδ ]))
2
2
1
− ([Sδ ] − E([Sδ ]))′ (Cov([Sδ ]))−1 ([Sδ ] − E([Sδ ])) (1)
2
denote the Gaussian realised quasi–likelihood function and let θ̂ = arg maxθ log l(θ) denote the QML
estimate. In order to find this estimate, one has to compute the inverse and the determinant of the RV
vector, which would be in general an operation of order O(n3 ). However, since σ is stationary, [Sδ ] is
itself stationary. Hence, Cov([Sδ ]) is a Toeplitz matrix, which can be inverted by using the Levinson–
Durbin algorithm (see [42] and [27]) in O(n2 ). Basically, one uses a Choleski decomposition of the
covariance matrix (see e.g. [25]) with Cov([Sδ ]) = LDL′ = P P ′ , where L is lower diagonal, with
ones on the diagonal and D is a diagonal matrix with the variances of the residuals (which are denoted
by E) as entries. So the likelihood function (1) can be written as
n
1
1
l(θ) = − log(2π) − log(det(D)) − E ′ E,
2
2
2
where the residuals E are given by
E = D −1/2 L−1 [Sδ ] − E [Sδ ] = P −1 [Sδ ] − E [Sδ ] .
6 MODEL ESTIMATION AND INFERENCE
15
Remark We can express the likelihood function in terms of the mean, variance and covariance of
the linear predictions of the RV. Assume that f denotes the joint density of the time series of the RV.
Straightforwardly, we get
f ([Sδ ]1 , . . . , [Sδ ]n ) = f ([Sδ ]1 )
n
Y
i=2
f ([Sδ ]i | [Sδ ]i−1 , . . . , [Sδ ]1 ) .
Now let EL (yi |Fi−1 ) and V arL (yi |Fi−1 ) denote the mean and the variance of the linear prediction.
In order to construct a quasi–likelihood, we assume that f is given by a Gaussian density and hence
n
n
1X
l(θ) = log L(θ, y) = − log(2π) −
log V arL ([Sδ ]i |Fi−1 )
2
2
i=1
−
n
1 X ([Sδ ]i − EL ([Sδ ]i |Fi−1 ))2
.
2
V arL ([Sδ ]i |Fi−1 )
i=1
So we observe that the entries in the diagonal matrix D in the Choleski decomposition are exactly the
variances of the best linear, unbiased one–step ahead forecast of the RV.
So far, we have only discussed how the model parameters can be estimated. However, in the remaining
part of this section we will briefly describe how one can make inference on the model parameters. Let
2
1
∂l(θ)
∂ l(θ)
1
J = lim
Cov
.
,
and
I = lim − E
n→∞ n
n→∞ n
∂θ
∂θ∂θ ′
It is well–known (see e.g. [30]) that not only the maximum likelihood estimator but also the QML
estimator is asymptotically normally distributed with an adjusted covariance matrix (compared to the
MLE setting) and hence
√
d
n(θ̂ − θ) → N (0, I −1 J I −1 ).
Based on this asymptotic result, we can construct 95 % confidence intervals for θ, which are of the
form
1/2
1/2
1.96 −1
1.96 −1
θ̂ − √
I J I −1
≤ θ ≤ θ̂ + √
I J I −1
,
n
n
where the square root of a positive (semi-)definite matrix Σ, say, is defined by the matrix Σ1/2 such
′
that Σ1/2 Σ1/2 = Σ. Estimating the so–called sandwich matrices I, which only appear in a QML
setting and accounting for the fact, that the estimation was not based on the true density function, does
not cause any problems, whereas estimating the covariance matrix J is more complicated. Here we
∂
l(xt , θ̂) and
have used spectral methods based on the approach popularised by [44]. I.e., let ĥt = ∂θ
let m denote the number of nonzero autocorrelations of ht (h). Then,
m n
X
j
1 X ′
′
ˆ
1−
ĥt ĥt−j .
J = Ω̂0 +
Ω̂j + Ω̂j , Ω̂j =
m+1
n
j=1
t=j+1
The sandwich matrices can be estimated straightforwardly by
n
1 ∂2
1 X ∂2
Î = −
l(x, θ̂) = −
l(xi , θ̂).
n ∂θ∂θ ′
n
∂θ∂θ ′
i=1
6 MODEL ESTIMATION AND INFERENCE
16
6.2 Empirical study
3.0
Log−prices
3.5
4.0
(a)
(c)
2002
2003
2004
Years
2005
2006
2002
2003
2004
Years
2005
2006
Realised volatilities
0.02
0.04
0.06
2001
2001
Correlogram for rv
Correlogram for returns
0.0 0.1 0.2 0.3 0.4 0.5 −0.01 0.00
0.01
0.02
In order to illustrate our results further, we have also carried out an empirical study. We have used
General Motors (GM) intra–day TAQ database, available at WRDS, from 2 January 2001 to 28 April
2006. Before analysing the data, we have cleaned the data. Following methods used by [34], we
concentrate on quote data from one stock exchange only. Here we have chosen the NYSE. We only
consider quotes, where both the bid–size and the ask–size are greater than 0, and which are quoted in
a normal trading environment (quote condition = 12 in the TAQ database). Since data at the beginning
and at the end of a trading day differ quite a lot from the quotes during the day, we concentrate
on data from 9.35 am until 15.55pm only. Further, if there were not any quote data for more than
five minute, we have interpolated the missing data via linear interpolation. Besides, we have deleted
obvious outliers, which we have chosen to be data points which differ by more than 0.15 from the
prevailing (log-) price level. Finally, we have deleted data from those days where there were too
many data missing at the beginning of a day (no data before 9.45 am) or which were just half trading
days. Furthermore, we have focused on the bid–prices only. In order to construct a time series of
five minute returns of the log–bid–prices, we use the previous tick sampling method. After we have
cleaned the data, we have a data set consisting of 1,308 business days with 76 five minute returns
per day, hence 99,408 returns. We provide a plot of the cleaned GM log–price data (based on five
(b)
0
0
(d)
1
2
3
Lags in days
50
100
Lags in days
4
150
5
200
Figure 1: GM data: (a) cleaned log–bid–prices (plotted every 5 minutes for 1308 business days
from 2 January 2001 – 28 April 2006; (b) autocorrelation between five minute returns; (c) time series of realised volatilities (i.e. square root of realised variances) based on five minute returns; (d)
autocorrelation between realised variances.
minute data), the corresponding autocorrelation function and the time series of the realised variances
with their autocorrelation in Figure 1. We have estimated the model parameters of the one–factor
model and the superposition model (with two factors). We have carried out the estimation three
times: once, when ξ2 = ξ3 = ξ4 = 0 and all the leverage parameters are set to 0 (this corresponds
to a Brownian semimartingale framework), once when allowing for jumps but setting all leverage
parameters to zero (and, hence, assuming that there is no leverage) and once with jumps and arbitrary
leverage parameters (which just satisfy the moment conditions described previously). The model fit
of the Brownian motion based model was very poor, hence we do not report the exact details. By
allowing for jumps, the model fit has improved significantly and the corresponding estimation results
6 MODEL ESTIMATION AND INFERENCE
Table 1.
Estimation results for the one–factor
model, without κ
Parameter
Estimate
Robust
17
Table 2.
Estimation results for the one–factor
model, with κ
Parameter
Estimate
standard error
Robust
standard error
λ
0.037
0.026
λ
0.037
0.026
ξ4
0.353
0.141
ξ3
-0.009
1.153
η1
1.770
0.249
ξ4
0.356
0.132
η2
0.153
0.198
η1
1.771
0.223
η3
2.197
0.891
η2
0.145
0.269
η4
3.195
4.441
η3
2.153
0.261
Quasi-L
-3264.13
η4
3.188
0.950
BP
28.789
κ11
-0.009
0.181
m
35
κ12
-0.009
2.459
κ13
-0.009
1.078
κ21
0.008
0.074
κ22
0.011
0.668
Quasi-L
-3264.13
BP
28.789
m
35
are given in the following tables. Note here that in the superposition model the QML estimates for
the third and fourth cumulant of Y , i.e. η3 and η4 are on the boundaries rather than in the interior of
the parameter space. So in these cases, we do not report the robust standard errors. In order to assess
the model fit, we provide the following plots and statistics: Figure 2 and Figure 3 show the empirical
versus the fitted autocorrelation function of the realised variance (a), the estimated (based on the one
step ahead forecast) and the empirical realised variance (b), the time series of the scaled residuals (c)
and their autocorrelation function (d). Furthermore we have computed the Box–Pierce statistic based
on 20 lags which measured the degree of dependence in the scaled residuals. It is really striking that
the model fit seems to be already quite good for the one–factor model, when we allow for jumps in
the asset price. Incorporating a second factor leads to another (small) improvement. One could also
study multifactor models with more than two factors. But from our results for one and two factors,
we would not expect that this would lead to a big improvement anymore. Note here that, in order
to estimate the parameters of the superposition model, we have used O(M −1 )–approximations (as
given in the Appendix) for the variance and covariance of the realised variance (rather than O(M −2 )–
approximations), since this reduces the number of parameters which have to be estimated from 20
to 12. Studying the higher order approximation and carrying out a numerical optimisation over 20
unknown parameters, which are partly very weakly identified, will be subject to future research. Let
us now turn our attention to the estimated leverage parameters. We observe that accounting for the
leverage parameters in this particular example does not lead to an improvement (but also not to a
6 MODEL ESTIMATION AND INFERENCE
0.5
18
(a)
(b)
50
Estimated acf
Empirical acf
0.4
40
0.3
30
0.2
Estimate
RV
20
0.1
10
0.0
0
(c)
50
100
150
200
10
2001
(d)
2002
2003
2004
2005
2006
0.1
5
0.0
0
−0.1
2001
2002
2003
2004
2005
2006
0
50
100
150
200
Figure 2: (a)Empirical autocorrelation of realised variance and estimated (for J=1) acf for
01/2001–04/2006; (b) estimated variance and realised variance; (c) scaled residuals; (d) acf of
scaled residuals.
(a)
(b)
50
Estimate
RV
Estimated acf
Empirical acf
40
0.4
30
0.2
20
10
0.0
0
(c)
50
100
150
200
10
2001
(d)
2002
2003
2004
2005
2006
0.1
5
0.0
0
−0.1
2001
2002
2003
2004
2005
2006
0
50
100
150
200
Figure 3: (a)Empirical autocorrelation of realised variance and estimated (for J=2) acf for
01/2001–04/2006; (b) estimated variance and realised variance; (c) scaled residuals; (d) acf of
scaled residuals.
deterioration) of the model fit. We hypothesise here that estimating the leverage parameters by our
QML method is difficult since they only appear in terms of low order. So they are not easy to identify
and since their corresponding confidence intervals all cover the 0, it is not possible to deduce clearly
whether leverage (in the sense of non–zero joint cumulants) is present in the data. These findings are
in line with other empirical studies, which focused on estimating the leverage effect in single stocks.
However, when studying index data, one often observes a leverage effect, which is present for several
days (see e.g. [20]). Finally, we provide a plot (Figure 4) of the empirical cross–correlation between
returns and realised variances which we denote by
P
((si − s)([Sδ ]i+s − [Sδ ]))
q
L(s) = P i
P
2
2
i ([Sδ ]i+s − [Sδ ]))
i ((si − s)
for s ∈ {−20, . . . , 20}, where s and [Sδ ] denote the sample mean of the returns and the realised
variance, respectively. Similarly to [20], this function can be interpreted as a kind of leverage correlation function. In addition to this function, we plot√the Bartlett confidence bounds of the hypothesis
that there is no leverage, which are given by 1.96/ T = 0.054. Although we can clearly spot that
7 CONCLUSION
19
Table 3.
Estimation results for the two–factor model,
without κ
Parameter
Estimate
Robust
Table 4.
Estimation results for the two–factor model,
with κ
Parameter
Estimate
standard error
λ(1)
0.130
Robust
standard error
0.175
λ(1)
0.130
0.198
0.011
0.039
λ(2)
0.011
0.017
λ(2)
w1
0.478
0.156
w1
0.478
0.893
ξ4
0.344
0.210
ξ4
0.344
0.356
η1
1.667
0.482
η1
1.667
0.511
η2
2.360
0.863
η2
2.360
3.536
η3
1e-007
η3
1.0002e-007
η4
1e-007
η4
1.0374e-007
Quasi-L
-3261.956
BP
27.949
m
35
(1)
κ21
(1)
κ22
(2)
κ21
(2)
κ22
2.3994e-008
1.263
2.4014e-008
5.570
2.3972e-008
3.630
4.1144e-007
22.086
Quasi-L
-3261.956
BP
27.948
m
35
returns and future realised variance seem to be slightly negatively correlated for a couple of days, the
correlation is really small and not statistically significant.
7 Conclusion
In this paper, we have studied the impact of time–inhomogeneous jumps and leverage type effects
on returns and realised variance in Lévy–driven stochastic volatility models. In particular, we have
derived explicit expressions for the cumulants of the returns and the realised variance by solving a
recursive system of inhomogeneous ordinary differential equations. This seems to be a very powerful
technique and might be applicable to a much wider class of asset price models. This aspect will be
investigated further in future research. Although the realised variance is an inconsistent estimator
of the integrated variance in the Lévy–driven stochastic volatility model studied in this paper, we
have shown how it can be used in a quasi–maximumlikelihood framework for estimating the model
parameters of our model.
A TECHNICAL APPENDIX
−0.04
−0.02
L(s)
0.00
0.02
0.04
20
−20
−15
−10
−5
0
5
s
10
15
20
Figure 4: This plot shows the leverage correlation function L(s) = Cor(si , [Sδ ]i+s ) for s ∈
{−20, . . . , 20} .
A
Technical appendix
A.1 Proofs
Proof of Proposition 4.1 From the binomial formula one obtains:
∆σtn
=
σtn
=
n X
−
k=1
n
σt−
n
= (σt− + ∆σt ) −
n n−k
σ
(∆Yλt )k .
k t−
n
σt−
=
n X
n
k=0
k
n
n−k
(∆σt )k − σt−
σt−
Applying Itô’s formula to f (x) = xn with f ′ (x) = nxn−1 , f ′′ (x) = n(n − 1)xn−2 , one gets
σtn − σ0n = f (σt ) − f (σ0 )
Z t
Z
X
1 t ′′
f ′ (σs− )dσs +
=
f (σs− )d[σ, σ]cs +
(f (σs ) − f (σs− ) − f ′ (σs− )∆σs )
2
0
0
0<s≤t
Z t
Z t
X
n−1
n−1
n
∆Yλs )
dYλs +
(∆σsn − nσs−
σs−
σs−
ds + n
= −λ n
0
0
= −λ n
Z
0
t
n
σs−
ds +
n X
k=1
0<s≤t
n X
k
n−k
(∆Yλs )k .
σs−
0≤s≤t
Proof of Proposition 4.3 Note that, for n ∈ N,
∆Stn
=
n X
n
k=1
k
n−k
(σt− ∆Xt )k St−
,
A TECHNICAL APPENDIX
21
and, by Itô’s formula:
Stn
=n
Z
t
0
n(n − 1) 2
v
+ Xs ) +
2
n−1
σs− d(vWs
Ss−
Z
t
0
n−2 2
σs− ds
Ss−
n X X
n
n−k
+
.
(σs− ∆Xs )k Ss−
k
0≤s≤t k=2
Hence,
E (Stn )
Z
n(n − 1) 2
=
v
2
t
E
0
n−2 2
σs−
Ss−
ds +
n X
n
k
k=2
ξk
Z
t
0
n−k
k
.
E σs−
Ss−
From the integration by parts formula, it follows that, for k, n ∈ N and k ≤ n,
Z t
Z t
k
n−k
k
n−k
n−k k
+ [S n−k , σ k ]t = I + II + III.
σs−
dSs−
Ss− dσs +
S t σt =
0
0
From Proposition 4.1, one can deduce that
I = −kλ
Z
t
0
n−k k
σs− ds
Ss−
+
k X
X
k
j=1
j
n−k k−j
σs− (∆Yλs )j ,
Ss−
0≤s≤t
and
II = (n − k)
+
Z
n−k
X
j=2
t
0
k+1 n−k−1
d(vWs
Ss−
σs−
n−k
j
X
(n − k)(n − k − 1)v 2
+ Xs ) +
2
Z
t
0
n−k−2 2
σs− ds
Ss−
k+j n−k−j
σs−
Ss−
(∆Xs )j ,
0≤s≤t
and
III =
X
∆Ssn−k ∆σsk
=
k n−k
XX
j=1 l=1
0≤s≤t
n−k
j
X
k
n−k−j j+k−l
σs− (∆Xs )j (∆Yλs )l .
Ss−
l
0≤s≤t
When taking the expectation and applying the Master formula and Fubini’s theorem, one obtains:
Z t Z t k X
k
n−k k
n−k k−j
n−k k
E Ss− σs− ds +
ληj
σs− ds
E Ss−
E St σt = −kλ
j
0
0
j=1
+
(n − k)(n − k
2
− 1)v 2
Z
t
0
n−k
X n − k Z t n−k−2 2
E Ss
σs ds +
ξj
E σsk+j Ssn−k−j
j
0
j=2
Z t Z t k n−k
n−1 XX
X
n−k
k
n
j
m+n−j
ds +
E Ssn−k−j σsj+k−l ds.
κj,l
E Ss− σs−
ξn−j
+
j
l
j
0
0
j=0
j=1 l=1
The equation above can be written as
d n−k k n−k k
σt− = g(t; n, k),
E St σt + kλE St−
dt
(2)
A TECHNICAL APPENDIX
22
where g(·; n, k) is defined as in Proposition 4.3. Since the processes above are continuous in probability, we can write
n−k k
σt− = E Stn−k σtk .
E St−
Clearly, equation (2) is a inhomogeneous ordinary differential equation of first order. From solving
(2) with initial value 0 at 0, one obtains
Z t
g(u; n, k)ekλu du.
E Stn σtk = e−kλt
0
Proof of Proposition 4.5 First, we have to compute the mean, variance and covariance of the integrated variance IV. We use Fubini’s Theorem and the results we have already derived for computing
the moments of the volatility process σ. Hence,
Z t
Z t
η2 2
t.
E σs2 ds = η12 +
σs− ds =
E (IVt ) = E
2
0
0
The second moment is given by
E
where for u ≥ s,
IVt2
=
Z tZ
0
t
0
E σs2 σu2 ds du,
E σs2 σu2 = E σs4 + E σs2 σu2 − σs2 .
Using the SDE representation of σ 2 and Itô’s formula, one obtains the SDE
d
E σs2 σu2 = −2λ E σs2 σu2 + 2η1 E σs2 σu + η2 E σs2 ,
du
which can be solved using the initial value s, E σs4 . Similarly,
d
E σs2 σu = −λ E σs2 σu + η1 E σs2 ,
du
with initial value s, E σs3 . By writing
and
E IVt2 =
Z tZ
0
u
0
E σs2 σu2 ds du +
E (IVt IVt+s ) = E
IVt2
+
Z tZ
0
Z tZ
0
t+s
t
u
t
E σs2 σu2 ds du,
E σx2 σu2 dx du,
we can immediately deduce the variance and the covariance of IV. The mean, variance and covariance
of σ[(i−1)h,ih] can be directly derived from the corresponding results for IV.
A TECHNICAL APPENDIX
23
Proof of Proposition 4.6 First, we compute the mean of [S]. Using the formula for the quadratic
variation of a stochastic integral (see e.g. [45] Theorem II.29), the Master Formula and Fubini’s Theorem, one can deduce that
Z t
Z
t
η12
1 η2
2
2
2
2
σs− d[vW + X]s = v + ξ2
E ([S]t ) = E
E σs ds = v + ξ2
+
t.
λ2 2 λ
0
0
Second, we compute second moment of [S]. Using Itô’s formula, we deduce that
Z t
Z t
2
2
d[vW + X]u + [[S]]u
[S]t = 2 [S]u− d[S]u + [[S]]u = 2 [S]u− σu−
0
0
Z t
X
X
2
2
4
= 2v 2
[S]u− σu−
ds + 2
[S]u− σu−
(∆Xu )2 +
σu−
(∆Xu )4 .
0
0≤u≤t
0≤u≤t
Hence (using again the Master Formula and Fubini’s Theorem),
Z t
Z
t
E [S]2t = 2 v 2 + ξ2
E σu4 du.
E [S]u σu2 du + ξ4
0
0
So we have to compute E [S]u σu2 . We apply the integration by parts formula and obtain
Z u
Z u
2
2
2
σx−
d[S]x + [[S], σ 2 ]u
[S]x− dσx +
[S]u σu =
0
0
Z u
Z u
Z u
2
[S]x− d[Y ]λx
[S]x− σx dYλx +
[S]x− σx dx + 2
= −2λ
0
Z u
Z u 0
Z0 u
2
4
4
σx−
d[[X], σ 2 ]x .
+ v2
σx−
d[X]x +
σx−
dx +
0
0
0
Applying the Master formula and Fubini’s theorem, we get
Z u
Z u
Z u
E ([S]x ) dx
E ([S]x σx ) dx + η2
E [S]u σu2 = −2λ
E [S]x σx2 dx + 2η1
0
0
0
Z u
Z u
Z
u
3
4
2
E σx2 dx.
E σx dx + κ2,2
E σx dx + 2κ2,1
+ v + ξ2
0
0
0
So we have to compute E ([S]u σu ) first and, then, we can solve the differential equation
(E[S]u σu2 )′ = −2λE [S]u σu2 + 2η1 E ([S]u σu ) + η2 E ([S]u )
+ v 2 + ξ2 E σu4 + 2κ2,1 E σu3 + κ2,2 E σu2 ,
with initial value (0, 0). So similarly as above, we can compute E ([S]u σu ). From the integration by
parts formula, we get
Z u
Z u
[S]u σu =
[S]x− dσx +
σx− d[S]x + [[S], σ]x
0
0
Z u
Z u
Z u
Z u
2
3
σx−
d[[X], σ]x .
σx− d[vW + X]x +
[S]x− dYλx +
[S]x− σx− dx +
= −λ
0
0
0
Hence, we deduce from the Master formula and Fubini’s theorem that
0
A TECHNICAL APPENDIX
E ([S]u σu ) = −λ
Z
24
u
E ([S]x σx ) dx + η1
0
Z
u
E ([S]x ) dx + v 2 + ξ2
0
Z
u
0
E σx3 dx
Z u
E σx2 dx.
+ κ2,1
0
So one obtains E ([S]u σu ) by solving the differential equation
d
E ([S]u σu ) = −λE ([S]u σu ) + η1 E ([S]u ) + v 2 + ξ2 E σu3 + κ2,1 E σu2 ,
du
with initial value (0, 0). Finally, we derive the covariance of [S].
Z t+s
Z
2
E ([S]t ([S]t+s − [S]t )) = E [S]t
σu d[vW + X]u = v 2 + ξ2
t
[S]t σu2
t+s
t
E [S]t σu2 du.
In order to compute E
, we use once again integration by parts, the Master Formula and
Fubini’s Theorem so that in the end we just have to solve the following two differential equations for
u ≥ t:
d
E ([S]t σu ) = −λE ([S]t σu ) + λη1 E ([S]t ) ,
du
with initial value (t, E[S]t σt ) and
d
E[S]t σu2 = −2λE[S]t σu2 + 2λη1 E[S]t σu + λη2 E[S]t ,
du
with initial value t, E[S]t σt2 . Combining these results, we obtain the covariance from
Cov ([S]t , [S]t+s ) = E{[S]t [S]t+s } − E([S]t ) E([S]t+s )
= E([S]t )2 + E{[S]t ([S]t+s − [S]t ) − E([S]t ) E([S]t+s ).
Proof of Proposition 4.8 In order to proof Proposition 4.8, i.e. the covariations between, returns and
squared returns, we have to compute the corresponding covariations of the price process S.
In the following, we will only sketch the proof. Basically, we will always apply Itô’s formula, the
Master Formula and Fubini’s Theorem in order to derive a first order ordinary differential equations
(ODE) for the expectation we want to compute. This ODE can then be easily solved.
Covariation of returns
Cov(St , St+s ) = ESt2 + E(St (St+s − St ))
Z t+s
2
St σs− d(vWs + Xs ) = E St2 = V ar (St ) .
= E St + E
(3)
t
From equation (3), we can immediately deduce that
Cov(si , si+s ) = E (si si+s ) = E(Sih − S(i−1)h )(S(i+s)h − S(i+s−1)h )
= Cov(Sih , S(i+s)h ) − Cov(Sih , S(i+s−1)h ) − Cov(S(i−1)h , S(i+s)h )
+ Cov(S(i−1)h , S(i+s−1)h ) = 0.
A TECHNICAL APPENDIX
25
Covariation of returns and squared returns
2
2
Cov(St , St+s
) = E St3 + E St (St+s
− St2 )
Z t+s
Z t+s
3
2
= E St + E St 2
σu− d[vW + X]u
Su− dSu +
t
t
Z t+s
E St σu2 du.
= E St3 +
(4)
t
In the following, we will always assume that u ≥ t.
Computing E (St σu )
Z
St σu = St σt + St (σt − σu ) = St σt +
u
St dσu = St σt +
t
Hence,
E (St σu ) = E (St σt ) − λ
So, E (St σu ) is the solution to
Z
Z
u
St (−λσt dt + dYλt ).
t
u
E (St σu ) du.
t
d
E (St σu ) = −λE (St σu ) ,
du
with initial value (t, E (St σt )).
Computing E St σu2
St σu2
=
St σt2
+
Z
Similarly,
u
t
St dσx2
=
St σt2
− 2λ
Hence,
E St σu2 = E St σt2 − 2λ
So E St σu2 is the solution to
Z
Z
u
t
u
t
2
St σx−
dx
+2
Z
u
St σx− dYλx +
u
St d[σ]x .
t
t
E St σx2 dx + 2η1
Z
Z
u
E (St σx ) dx.
t
d
E St σu2 = −2λE St σu2 + 2η1 E (St σu ) ,
du
with initial value (t, E St σt2 ).
Computing E (Sa Sb Sc ) for a ≤ b ≤ c
Sa Sb Sc = Sa Sb (Sb + (Sc − Sb )) =
Hence, ESa Sb Sc = ESa Sb2 .
Sa Sb2
+
Z
c
Sa Sb σx− d(vWx + Xx ).
b
A TECHNICAL APPENDIX
26
So we get
2
2
2
2
− E S(i−1)h S(i+s)h
+ E Sih S(i+s−1)h
− 2E Sih S(i+s−1)h
Cov(si , s2i+s ) =E Sih S(i+s)h
2
2
+ 2E S(i−1)h S(i+s−1)h
− E S(i−1)h S(i+s−1)h
1 = 4 η2 −2 λ e−2 λ sh + λ e−2 λ (sh−h) + λ e−2 λ (h+sh)
4λ
+8η12 κ1,1 −2 e−λ sh + e−λ (h+sh) + e−λ (sh−h)
+ κ1,2 λ e−2 λ (h+sh) η1 + λ e−2 λ (sh−h) η1 − 2 λ e−2 λ sh η1 = O(h2 ).
Covariation between squared returns
2
2
2
Cov(St2 , St+s
) = E St4 + E St2 (St+s
− St2 ) − E St2 ESt+s
Z
t+s
2
4
2
E St2 σu2 du − E St2 E St+s
= E S t + v + ξ2
,
t
and
Cov(s2i , s2i+s ) = E(s2i s2i+s ) − E s2i E s2i+s ,
and
2 2
2 2
2
2
S(i+s)h − 2 Sih
E(s2i s2i+s ) = E(Sih
S(i+s)h S(i+s−1)h + Sih
S(i+s−1)h − 2 Sih S(i−1)h S(i+s)h
2
2
2
+ 4 Sih S(i−1)h S(i+s)h S(i+s−1)h − 2 Sih S(i−1)h S(i+s−1)h
+ S(i−1)h
S(i+s)h
2
2
2
− 2 S(i−1)h
S(i+s)h S(i+s−1)h + S(i−1)h
S(i+s−1)h
).
Computing E Sa2 Sb Sc for a ≤ b ≤ c
Sa2 Sb Sc = Sa2 Sb (Sb + (Sc − Sb )) = Sa2 Sb2 +
Hence,
Z
c
b
Sa2 Sb σx− d(vWx + Xx ).
E Sa2 Sb Sc = E Sa2 Sb2 .
Computing E Sa Sb Sc2 for a ≤ b ≤ c Similarly,
Z c
Z c
2
Sa Sb σx−
d[X]x .
Sa Sb Sx− σx− d(vWx + Xx ) +
Sa Sb Sc2 = Sa Sb (Sb2 + (Sc2 − Sb2 )) = Sa Sb3 + 2
b
b
Hence,
E
Sa Sb Sc2
=E
Sa Sb3
2
+ v + ξ2
Z
b
c
E Sa Sb σx2 dx
A TECHNICAL APPENDIX
27
Computing ESa Sb3 for a ≤ b Sa Sb3 = Sa4 + Sa (Sb3 − Sa3 ). From Itô’s formula, we deduce that
Z b
Z t
X
2
2
3
3
2
2
St− dSt + 3v
Sb − Sa = 3
St− σt−
ds +
(∆St3 − 3St−
∆St ),
a
0
a≤t≤b
where
∆St3
=
2 X
n
j
j=0
j
2−j
3
2
2
St−
σt−
(∆Xt )2−j = σt−
(∆Xt )3 + 3St− σt−
(∆Xt )2 + 3St−
σt− ∆Xt .
Hence,
ESa Sb3
=
ESa4
+ ξ3
Computing E St σu3 for u ≥ t
St σu3
St σt3
=
+
St (σu3
−
σt3 )
=
+3
Z
Z
b
a
St σt3
u
t
Sa σt3 dt
2
+ 3 v + ξ2
+ St −3λ
Z
u
t
X
2
σx−
dYλx +
E
=E
St σt3
− 3λ
Z
Z
b
a
ESa St σt2 dt.
3
σx−
dx

2
(∆Yλx )3 + 3σx− (∆Yλx )2 + 3σx−
∆Yλx  .
t≤x≤u
Hence,
St σu3
u
E
t
St σx3
Z
u
E St σx2 dx
t
Z
Z u
E (St σx ) dx + 3η1
+ 3η2
dx + 3η1
t
t
3
u
So E St σu is the solution to the the ODE
E St σx2 dx.
d
E St σu3 = −3λE St σu3 + 3η1 E St σu2 + 3η2 E (St σu ) + 3η1 E St σu2 ,
du
with initial value t, E St σt3 .
Computing E (St Su σu ) for u ≥ t
St Su σu = St2 σt + St (St σt − Su σu )
Z
Z u
2
Sx− σx− dx +
= St σt + St (−λ
t
u
Sx− dYλx +
t
Z
u
σx− dSx +
t
Z
u
σx− d[X, σ]x ).
t
Hence,
E (St Su σu ) = E
St2 σt
−λ
Z
u
E (St Sx σx ) dx + η1
t
Z
t
u
E (St Sx ) dx + κ1,1
| {z }
=E(St2 )
Z
u
E (St σx ) dx.
t
A TECHNICAL APPENDIX
28
Computing E St Su σu2 for u ≥ t
St Su σu2 = St2 σt2 + St (Su σu2 − St σt2 ),
where
Su σu2
−
St σt2
Z
u
2
Sx− σx−
dx
Z
Z
u
u
Sx− d[Y ]λx
Sx− σx− dYλx +
+2
= −2λ
t
t
t
Z u
X
X
2
3
∆Xx ∆Yλx +
σx− ∆Xx (∆Yλx )2 .
d(vWx + Xx ) + 2
σx−
+
σx−
t
t≤x≤u
t≤x≤u
Hence,
Z u
Z u
Z u
E St Su σu2 = E St2 σt2 − 2λ
E St Sx σx2 dx + 2η1
E (St Sx σx ) dx + η2
E (St Sx ) dx
t
t
t
Z u
Z u
E (St σx ) dx.
E St σx2 dx + κ1,2
+ 2κ1,1
t
t
Computing Sa Sb σu for u ≥ b ≥ a
Sa Sb σu = Sa Sb σb + Sa Sb (σu − σb )
Z u
−λσx− dx + dYλx ).
= S a S b σb + S a S b (
b
Hence,
Z
E (Sa Sb σu ) = E (Sa Sb σb ) − λ
u
E (Sa Sb σx ) dx + η1
b
Z
u
b
E Sa2 dx.
Computing Sa Sb σu2 for u ≥ b ≥ a
Sa Sb σu2 = Sa Sb σb2 + Sa Sb (σu2 − σb2 )
Z
Z u
2
σx−
dx + 2
= Sa Sb σb2 + Sa Sb (−2λ
b
u
b
σx− dYλx + [σ]u − [σ]b ).
Hence,
E
Sa Sb σu2
=E
Sa Sb σb2
− 2λ
Z
u
E
b
Sa Sb σx2
dx + 2η1
Z
u
E (Sa Sb σx ) dx + η2
b
Z
u
E (Sa Sb ) dx.
b
Computing E (Sa Sb Sc Sd ) for a ≤ b ≤ c ≤ d
Sa Sb Sc Sd = Sa Sb Sc2 + Sa Sb Sc (Sd − Sc ),
so,
E (Sa Sb Sc Sd ) = E Sa Sb Sc2 .
A TECHNICAL APPENDIX
29
From the results above, Cov(s2i , s2i+s ) follows straightforwardly .
A.1.1 Proofs for Section 5
Proof of Proposition 5.1 Recall that Ht = St2 − IVt . Since we have already computed the moments
of S and IV , we only have to compute the joint moments of these two processes. The computations
are analogous to the ones carried out in the proof of Proposition 5.2, so we will refer to that proof for
more details.
Proof of Proposition 5.2 Recall that Gt = St2 − [S]t . Since we have already computed the moments
of S 2 and [S], we basically just have to compute the joint moments of these two processes in order to
derive the mean, variance and covariance of G.
Mean The mean is given by
E (Gt ) = E St2 − E ([S]t ) = 0.
Variance Next, we compute the second moment of G.
G2t = St4 − 2St2 [S]t + [S]2t ,
where
St2 [S]t
=
Z
t
0
2
2
Ss−
σs−
d[vW
+ X]s +
Z
t
0
2
[S]s− (2Ss− σs− d(vWs + Xs )+ σs−
d[vW + X]s )+ [S 2 , [S]]t ,
and
2
[S , [S]]t =
Z
t
0
2
σs−
d[S 2 , [vW + X]]s =
2
2
σs−
(2Ss− σs− ∆Xs + σs−
(∆Xs )2 )(∆Xs )2
0≤s≤t
X
=2
X
3
Ss− σs−
(∆Xs )3 +
0≤s≤t
X
4
σs−
(∆Xs )4 .
0≤s≤t
Hence,
E
St2 [S]t
=
Z
0
t
E
Ss2 σs2
2
ds + v + ξ2
Z
t
0
E [S]s σs2 ds
Z t
Z t
E Ss σs3 ds + ξ4
E σs4 ds.
+ 2ξ3
0
0
All quantities above are already known, so the result for the variance follows directly from above. A TECHNICAL APPENDIX
30
Covariance
2
Cov(Gt , Gt+s ) = Cov(St2 − [S]t , St+s
− [S]t+s )
2
2
= E((St2 − [S]t )(St+s
− [S]t+s )) − E(St2 − [S]t )E(St+s
− [S]t+s ),
where
2
2
2
E((St2 − [S]t )(St+s
− [S]t+s )) = ESt2 St+s
− E[S]t St+s
− E[S]t+s St2 + E[S]t [S]t+s .
2
Computing ESt2 St+s
2
E St2 St+s
= E St2 (St + (St+s − St ))2
= E St4 + 2 E St3 (St+s − St ) +E(St2 (St+s − St )2 ).
|
{z
}
=0
2
(St+s − St ) = 2
Hence,
E
Z
t+s
t
St2 (St+s
Computing E St2 σu2 for u ≥ t
St2 σu2
Hence,
=
Su− dSu − 2
St2 σt2
=
St2 σt2
=
St2 σt2
+
− St )
St2 (σu2
+2
Z
− 2λ
u
t
Z
−
t
St dSu +
t
= v + ξ2
σt2 )
St2 σt2
=
2
St2 σs−
ds
+
Z
t+s
t
2
St2 σs− dσs
u
t+s
Z
+
u
t
+2
Z
Z
t+s
Z
u
t
t
2
σu−
d[vW + X]u
E St2 σu2 du.
St2 σu
=
St2 σt
=
St2 σt
+
St2 (σu
−λ
Z
u
t
St2 dσs2
St2 d[σ]s
u
t
St2 σs− dYλs +
Z
u
t
St2 d[σ]s .
d
E St2 σu2 = −2λE St2 σu2 + 2η1 E St2 σu + η2 E St2 .
du
Computing E St2 σu for u ≥ t
Hence,
2
Z
− σt ) =
St2 σt
St2 σs− ds
Z
+
+
u
t
Z
u
t
St2 dσs
St2 dYλs .
d
E St2 σu = −λE St2 σu + η1 E St2 .
du
A TECHNICAL APPENDIX
2
Computing E [S]t St+s
31
2
2
[S]t St+s
= [S]t St2 + [S]t (St+s
− St2 )
Z t+s
2
Su− dSu + [S]t+s − [S]t ).
= [S]t St + [S]t (2
t
Hence,
2
E[S]t St+s
Computing E [S]t+s St2
=
E[S]t St2
2
+ v + ξ2
Z
t+s
t
2
E[S]t σu−
du.
[S]t+s St2 = [S]t St2 + St2 ([S]t+s − [S]t )
Z t
2
2
St2 σu−
d[vW + X]u .
= [S]t St +
0
Hence,
E
[S]t+s St2
Computing E [S]t [S]t+s
=E
[S]t St2
2
+ v + ξ2
Z
t+s
t
E St2 σu2 du.
We have already shown that
E ([S]t [S]t+s ) = E
So altogether, we obtain:
[S]2t
2
+ v + ξ2
Z
t+s
t
E [S]t σu2 du.
2
2
2
E((St2 − [S]t )(St+s
− [S]t+s )) = E St2 St+s
− E [S]t St+s
− E [S]t+s St2 + E ([S]t [S]t+s )
= E (St2 − [S]t )2 = E G2t .
So
Cov(Gt , Gt+s ) = E G2t − E (Gt ) E (Gt+s ) = V ar(Gt ) = O(t2 ),
for t → 0, which does not depend on s.
Computing Cov(Gt , [S]t )
Cov(Gt , [S]t ) = E (Gt [S]t ) − E (Gt ) E ([S]t ) = E (Gt [S]t )
= E St2 [S]t − E [S]2t .
A TECHNICAL APPENDIX
32
Note here that the corresponding properties for the increments of G follow directly from the results above.
Proof of Proposition 5.4 The first and second order properties of the realised variance can be derived
from the corresponding results of the squared returns:
E[Sδ ][(i−1)h,ih] =
M
X
j=1
V ar [Sδ ][(i−1)h,ih] =
M
X
E s2j,i ,
X
Cov(s2j,i , s2k,i ).
X
Cov(s2j,i , s2k,i+s ).
V ar(s2j,i ) + 2
i=1
1≤j<k≤M
And for s > 0:
Cov [Sδ ][(i−1)h,ih] , Sδ ][(i+s−1)h,(i+s)h] =
1≤j, k≤M
Finally, we express the (quite lengthy) formulae of the variance and covariance of the realised variance
in form of a Taylor series expansion and focus on the first terms only. These leads to the results given
in Proposition 5.4.
A.1.2 Superposition Model
Now we assume that σ is given by a superposition model as defined in Section 5.4. As already mentioned, we can derive the mean, variance and covariance even in that more general model, but due to
the fact that the driving processes of the asset price and the volatility components are dependent, the
formulae become quite lengthy. However, here we focus the case J = 2 and we compute the quadratic
variation of the price process and use that as an O(M −1 )–approximation of the corresponding formulae of the realised variance. Since we only consider the case J = 2 here, we can write the weights as
w1 = w, w2 = 1 − w.
Proposition A.1 Let v 2 + ξ2 = 1. Then:
1
2
2
E ([S]t ) =
w −w+
η2 + η1 h,
2
and
(1)
(2)
(1)
(2)
V ar([S]) = a1 + a2 h + a3 e−λ h + a4 e−λ h + a5 e−2λ h + a6 e−2λ h
(1)
(2)
(1)
(1)
(2)
+a7 e−(λ +λ h) + a8 + a9 h + a10 e−λ h + a11 e−λ h + a12 e−2λ h
(2)
(1)
(2)
+a13 e−2λ h + a14 e−(λ +λ h) + ξ4 h a15 ,
where
a1 =
−
4
λ(2) 2
+8
w
λ(2) 2
w2
w2
− 4 (2) 2 − 4 (1) 2
λ
λ
η2 η12
+ −
5
3λ(2) 2
−5
w2
w
+ 5 (2) 2
(2)
2
λ
λ
A TECHNICAL APPENDIX
5 w3
5 w3
+
−
3 λ(2) 2 3 λ(1) 2
η3 η1 −
33
1 −4 λ(1) 4 w3 + 14 λ(1) 2 λ(2) 2 w2 + λ(1) 4 w4
4
−4 λ(1) 4 w + λ(1) 4 + 10 λ(1) 2 λ(2) 2 w4 − 4 λ(1) 2 λ(2) 2 w + 6 λ(1) 4 w2 + λ(2) 4 w4
+2 λ(1) λ(2) 3 w4 + λ(1) 2 λ(2) 2 + 2 λ(1) 3 λ(2) − 20 λ(1) 2 λ(2) 2 w3 + 2 λ(1) 3 λ(2) w4
2 −1
(1) 3 (2) 3
(1) 3 (2) 2
(1) 3 (2)
2
(1)
(2)
(1) 2 (2) 2
−8 λ
λ w + 12 λ
λ w − 8λ
λ w η2 λ
λ +λ
λ
1
1 w3
3 w2
1 w
1
1
4
w +
−
+
−
η4 ,
+
− (2) 2 −
2 λ(2) 2 4 λ(2) 2 2 λ(2) 2 8 λ(2) 2
8λ
8 λ(1) 2
n
1
a2 = (1) (2) (1)
16 λ(2) 2 w2 + 16 λ(1) 2 − 32 λ(1) 2 w − 32 λ(1) λ(2) w
λ λ
λ + λ(2)
+16 λ(1) λ(2) + 32 λ(1) λ(2) w2 + 16 λ(1) 2 w2 η2 η12 + 8 λ(1) 2 + 8 λ(1) λ(2)
+24 λ(1) 2 w2 + 8 w3 λ(2) 2 + 24 λ(1) λ(2) w2 − 24 λ(1) λ(2) w − 8 w3 λ(1) 2
−24 λ(1) 2 w η3 η1 + 12 λ(1) w4 λ(2) − 8 w3 λ(1) 2 + 2 λ(1) 2 + 12 λ(1) 2 w2
−8 λ(1) λ(2) w + 2 w4 λ(1) 2 + 2 λ(1) λ(2) − 8 λ(1) 2 w + 20 λ(1 ) λ(2 ) w 2
2
2
2
−24 λ(1 ) λ(2 ) w 3 + 2 w 4 λ(2 ) η2 2 + λ(1 ) λ(2 ) − 4 λ(1 ) w − 4 w 3 λ(1 )
4 (3 η1 η2 + η3 w) w2 η1
,
3 λ(1) 2
4 −3 w2 + 3 w + w3 − 1 η3 η1
4 6 w − 3 w2 − 3 η2 η12
−
,
=−
3λ(2) 2
3λ(2) 2
w3 η3 η1
w4 η22
w4 η4
=
+
+
,
3λ(1) 2
4λ(1) 2 8λ(1)2
−24 w3 + 36 w2 + 6 w4 − 24 w + 6 η22
8 − 24 w + 24 w2 − 8 w3 η3 η1
+
=
24λ(2) 2
24λ(2) 2
4
3
2
−12 η4 w + 3 w η4 − 12 η4 w + 18 w η4 + 3 η4
+
(2) 2
24λ
2
2
2
η w 1 − 2w + w
=2 2
,
2
λ(1) + λ(2)
"
w w λ(1) 2 + 2 λ(1) λ(2) + λ(2) 2 η13
− w 4 λ(1) 2 w3 − 3 w2 λ(1) 2 + 5 λ(2) 2 w2
= −4
2
λ(1) 2 λ(1) + λ(2)
+10 w2 λ(1) λ(2) + 2 w λ(1) 2 − 8 w λ(1) λ(2) − 4 λ(2) 2 w + 4 λ(1) λ(2) + 2 λ(2) 2 η2 η1
#
2 −1 1 w4 η
(1)
3
(1)
(2)
(1) 2
−
λ +λ
λ
κ
3 λ(1) 2 2,1
)
(
1 w2 η1 2 1 2 w2 − 2 w + 1 w2 η2
(1)
κ2,2
−
+ −
2
(1
)
2 λ(1 ) 2
4
λ
a3 =
a4
a5
a6
a7
a8
+6 λ(1) 2 w2 + w4 λ(1) 2 + 6 λ(1) λ(2) w2 + 2 λ(1) w4 λ(2) − 4 λ(1) λ(2) w3
o
−4 λ(1) λ(2) w + λ(1) 2 + w4 λ(2) 2 η4 ,
A TECHNICAL APPENDIX
34
n + 4 −λ(2) 2 w2 + w λ(1) 2 + 2 w λ(1) λ(2) + 2 λ(2) 2 w − λ(1) 2 − 2 λ(1) λ(2) − λ(2) 2
2 −1 3
(1)
(2)
(2) 2
η1 λ
λ +λ
+ −4 λ(2) 2 w4 + 10 w3 λ(1) λ(2) + 5 λ(1) 2 w3 + 13 λ(2) 2 w3
−17 λ(2) 2 w2 − 22 w2 λ(1) λ(2) − 11 w2 λ(1) 2 + 9 w λ(1) 2 + 18 w λ(1) λ(2) + 11 λ(2) 2 w
−3 λ(1) 2 − 6 λ(1) λ(2) − 3 λ(2) 2
)
2 −1 1 w4 − 4 w3 + 6 w2 − 4 w + 1 η
3
(2)
η2 η1 λ(2) 2 λ(1) + λ(2)
κ2,1
−
3
λ(2) 2
(
)
1 w2 − 2 w + 1 η12 1 2 w4 − 6 w3 + 7 w2 − 4 w + 1 η2
(2)
+ −
−
κ2,2 ,
2
4
λ(2) 2
λ(2) 2
o−1 2
n
2 (1) (1)
+ κ2,1 6 λ(1) w3
a9 = κ2,1 6 λ(2) + 6 λ(1) w w η13 λ(1) λ(1) + λ(2)
3
3
o−1
n
+3 λ(1) w − 3 w2 λ(1) − 6 λ(2) w + 9 λ(2) w2 + 3 λ(2) w η2 η1 λ(1) λ(1) + λ(2)
2
(1) 2 2
(1)
(1)
(1) 3
(2) 3
2
1 κ2,2 2 w − 2 w + 1 w η2
2 κ2,1 λ w + λ w w η3 κ2,2 w η1
+
+
+
3
2
λ(1)
λ(1)
λ(1) λ(1) + λ(2)
(2)
(2) 2
(2)
(2)
(1)
(1)
3
2 κ2,1 6 λ w + 6 λ − 12 λ w + 6 λ − 6 λ w η1
+
3
λ(2) λ(1) + λ(2)
2 (2)
+ κ2,1 6 w4 λ(2) + 21 w2 λ(1) − 18 λ(1) w + 6 λ(2) − 9 λ(1) w3 − 21 λ(2) w3
3
o−1
n
−21 λ(2) w + 6 λ(1) + 30 λ(2) w2 η2 η1 λ(2) λ(1) + λ(2)
2 (2) + κ2,1 −4 λ(1) w + 6 w2 λ(1) + λ(1) w4 + λ(1) − 4 λ(1) w3 − 4 λ(2) w3 + λ(2)
3
o−1
n
+6 λ(2) w2 − 4 λ(2) w + w4 λ(2) η3 λ(2 ) λ(1) + λ(2)
2
(2)
(2)
2
4
3
2
1 κ2,2 2 w − 4 w + 2 η1
1 κ2,2 2 w − 6 w + 7 w − 4 w + 1 η2
+
+
,
2
λ(2)
λ(2)
)
(2
2 w2 − 2 w + 1 w η2 η1
w η3
(1)
κ2,1 ,
a10 = 4 (1)12 + 2
λ
λ(1) 2
(
)
−1 + 2 w3 − 4 w2 + 3 w η2 η1
(−2 + 2 w) η13
(2)
a11 = −2
κ2,1 ,
−2
(2)
2
(2)
2
λ
λ
)
!
(
2
w3 η1 η2 1 w4 η3
1 3 − 6 w + 6 w2 w2 η2
1 w2 η12
(1)
(1)
a12 =
+
+
κ2,2 ,
κ2,1 +
(1)
2
(1)
(1)
2
(1)
2
3 λ
2λ
12
λ
λ
(
)
1 12 − 12 w3 − 36 w + 36 w2 η2 η1
1 4 + 24 w2 − 16 w3 − 16 w + 4 w4 η3
a13 =
+
12
12
λ(2) 2
λ(2) 2
(
)
2 − 12 w η 2
3 + 21 w2 + 6 w4 η
6
+
6
w
3
−
12
w
−
18
w
1
1
2
(2)
(2)
1
κ2,1 +
+
κ2,2 ,
(2)
2
(2)
2
12
12
λ
λ
A TECHNICAL APPENDIX
a14
a15
(
)
w −1 + 2 w3 − 4 w2 + 3 w η2 η1
w (−2 + 2 w) η1 3
(1)
= 2
κ2,1
2 + 2
2
(1
)
(2
)
(1
)
(2
)
λ +λ
λ +λ
)
(
w −1 + 2 w3 − 4 w2 + 3 w η2 η1
w (−2 + 2 w) η1 3
(2)
κ2,1 ,
+ 2
2 + 2
2
(1)
(2)
(1)
(2)
λ +λ
λ +λ
4
4
2
2
2
= η1 + 3 − 6 w + 6 w η2 η1 + 4 w + − 4 w η3 η1
3
3 2 1 4
1
3
3
4
2
3
2
+ 6 w − 3 w + − 6 w + 3 w η2 + −w + − w + w + w η4 ,
4
4
2
2
(1)
(1)
(1)
Cov([S]t , [S]t+s ) = b1 e−λ (s+1)h − 2 e−λ sh + e−λ (s−1)h
(2)
(2)
(2)
+ b2 e−λ (s+1)h − 2 e−λ sh + e−λ (s−1)h
(1)
(1)
(1)
+ b3 e−2λ (s+1)h − 2 e−2λ sh + e−2λ (s−1)h
(2)
(2)
(2)
+ b4 e−2λ (s+1)h − 2 e−2λ sh + e−2λ (s−1)h
(1)
(2)
(1)
(2)
(1)
(2)
+ b5 e−(s+1)h(λ +λ ) − 2 e−sh(λ +λ ) + e−(s−1)h(λ +λ ) ,
where
3
(1)
2 3
2
2
3
2
2
w
η
η
+
w
η
η
+
2
w
η
+
2
w
−
2
w
+
w
η
η
κ
2 1
3 1
2 1
1
2,1 ,
3
λ(1) 2
2 2 3
1
2
2
2
2 w − 4 w + 2 η2 η1 + −2 w + − w + 2 w η3 η1
= (2) 2
3 3
λ
(2) i
+ (−2 w + 2) η13 + 4 w2 − 2 w3 − 3 w + 1 η2 η1 κ2,1 ,
1
1 4
1
1 3
1
1 3
(1)
w η3 η1 + w4 η22 +
w η4 +
w η2 η1 + w4 η3 κ2,1
= (1) 2
6
8
16
2
6
λ
1 2 2
1 4 1 3 1 2
(1)
+
w η1 +
w − w + w η2 κ2,2 ,
4
4
4
8
1 3 1 1 2 1
1
(1) 4 (2)
3
κ2,2 η2 w
− w + + w − w η3 η1
= (2) 2 −36 λ
6
6 2
2
λ
1
1
1 3
1
1
1
+ − w3 − w + + w2 + w4 η22 − η4 w +
η4
2
2
8 4
8
4
16
1
3 2
1 1 3
3 2 3
1 4
3
w − w + − w η2 η1
+ w η4 − η4 w + w η4 +
16
4
8
2
2
2 2
2
1
1
2 3
(2)
+ − w + w 2 − w + w4 +
η3 κ2,1
3
3
6
6
1
1 1 2 2
1 1 4
7 2 3 3 1
(2)
+
− w + + w η1 +
w − w − w + + w η2 κ2,2 ,
2
4 4
8
4
2
8 4
4
1
3
2
2
2
3
4
3
2
=
2 w − 2 w + w η2 + 2 w − 2 w η1 + 2 w − 4 w + 3 w
λ(1) + λ(2)
(2) i
(1)
−w) η2 η1 } κ2,1 + 2 w2 − 2 w η13 + 2 w4 − 4 w3 + 3 w2 − w η2 η1 κ2,1 .
b1 =
b2
b3
b4
b5
1
35
A TECHNICAL APPENDIX
36
Proof of Proposition A.1 In the following we will assume that 1 ≤ i, j ≤ J and v 2 + ξ2 = 1.
Moments of σ The moments of σ can be straightforwardly derived from the corresponding moments
of τ (i) by
J
X
(j)
wj τt .
σt =
j=1
(i)
In the following we use the notation ηk = λ(i) ηk , for i = 1, . . . , J and k ∈ N.
Mean
E ([S]t ) = E
Variance
[S]2t
E
Z
=2
Computing E ([S]u σu )
Z
t
E
0
t
0
2
σu−
d[X]u
[S]u σu2
= E σ02 t.
du + ξ4
Z
t
0
E σu4 du.
J
X
[S]u wi τu(i) ,
[S]u σu =
i=1
where
Z
(i)
(i)
E [S]u τu = −λ
u
0
Z
(i)
(i)
E [S]x τx dx + η1
u
E ([S]x ) dx +
0
Z
u
E (τx(i) )3 dx
0
Z u (i)
+ κ2,1
E (τx(i) )2 dx.
0
Computing E [S]u σu2
[S]u σu2
= [S]u
J
X
i=1
wi τu(i)
!2
=
J
X
i=1
[S]u wi2
2
τu(i) + 2
X
[S]u wj τu(j) wk τu(k) ,
1≤j<k≤J
where
d (i)
E [S]u τu(i) 2 = −2λ(i) E [S]u τu(i) 2 + 2η1 E [S]u τu(i) + E σu2 τu(i) 2
du
(1)
(i)
+ 2κ2,1 E σu2 τu(i) + κ2,2 E σu2 .
A TECHNICAL APPENDIX
37
(i) (k)
for i 6= k
Computing E [S]u τu τu
τu(i) τu(k)
Z
u
Z
t
τs(k) dτs(i) + [τ (i) , τ (k) ]u
0
Z0 u
Z u
(i)
(i)
(k)
(k)
(k)
(i)
τs(k) −λ(i) τs− ds + dYλ(i) s
τs −λ τs− ds + dYλ(k) s +
=
0
0
X
(i)
(k)
+
∆Yλ( i) ∆Yλ( k) .
=
τs(i) dτs(k)
+
0≤s≤u
Z
(i)
(k)
=− λ +λ
u
0
(i) (k)
τs− τs− ds
+
Z
0
u
(k)
(i)
τs− dYλ(k) s
+
Z
u
0
(k)
(k)
τs− dYλ(i) s + [τ (i) , τ (k) ]u .
Note: From the independence of the Y (i) we get:
{(u, v) ∈ R2+ : ν(Y (i)
,Y
λ(i)
(k)
λ(k)
(u, v)
> 0} ⊂ {(u, v) ∈ R2+ : uv = 0}.
So we obtain
[S]u τu(i) τu(k)
=
Z
u
[S]s− d
Z0 u
τs(i) τs(k)
+
Z
u
0
(i) (k)
τs− τs− d[S]s + [[S], τ (i) τ (k) ]u
(k)
(i)
(k)
(i) (k)
(k)
[S]s− − λ(i) + λ(k) τs− τs− ds + τs− dYλ(k) s + τs− dYλ(i) s + d[τ (i) , τ (k) ]s
=
0
Z u
Z u
(i) (k)
(i) (k) 2
2
σs−
d[[X], τs− τs− ]s .
τs− τs− σs− d[X]s +
+
0
0
Hence,
Z u (k)
(i) (k)
E [S]s τs(i) ds
ds + η1
E [S]s τs τs
− λ +λ
0
0
Z u Z u (i)
E τs(i) τs(k) σs2 ds
+ η1
E [S]s τs(k) ds +
0
0
Z u
Z u (i)
(k)
2 (i)
Eσs2 τs(k) ds,
+ κ2,1
E σs τs ds + κ2,1
Z
(i) (k)
=
E [S]u τu τu
u
(i)
(k)
0
0
since
Z
u
0
2
σs−
d[[X]], τ (i) τ (k) ] =
X
0≤s≤u
(k)
(i)
2
σs−
(∆Xs )2 (τs− ∆τs(k) + τs− ∆τs(i) + ∆ τs(i) τs(k) ).
(i)
Computing E σs2 τs
[S]u τu(i)
Z
u
Z
u
(i)
τs− d[S]s + [[S], τ (i) ]u
0
0
Z u
Z
Z u
(i)
(i)
(i)
[S]s− dYλ(i) s +
= −λ
[S]s− τs− ds +
=
[S]s− dτs(i)
0
+
0
0
u
(i)
2
σs−
τs− d[X]s
+
Z
0
u
2
σs−
d[[X], τ (i) ]s .
REFERENCES
38
Hence,
Z
E [S]u τu(i) = −λ(i)
u
0
Z
(i)
E [S]s τs(i) ds + η1
0
u
E ([S]s ) ds +
Z
0
u
E σs2 τs(i) ds
Z
(i)
+ κ2,1
u
0
Eσs2 ds.
The computations for the covariance are completely analogous to the ones above and the corresponding calculations in the one–factor setting.
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Research Papers
2008
2008-41:
Lars Stentoft: American Option Pricing using GARCH models and the
Normal Inverse Gaussian distribution
2008-42:
Ole E. Barndorff-Nielsen, Silja Kinnebrock and Neil Shephard:
Measuring downside risk — realised semivariance
2008-43:
Martin Møller Andreasen: Explaining Macroeconomic and Term
Structure Dynamics Jointly in a Non-linear DSGE Model
2008-44:
Christian M. Dahl, Henrik Hansen and John Smidt: The cyclical
component factor model
2008-45:
Christian M. Dahl and Yu Qin: The limiting behavior of the estimated
parameters in a misspecified random field regression model
2008-46:
Bent Jesper Christensen, Christian M. Dahl and Emma M. Iglesias:
Semiparametric Inference in a GARCH-in-Mean Model
2008-47:
Charlotte Christiansen: Mean Reversion in US and International Short
Rates
2008-48:
Tim Bollerslev, George Tauchen and Hao Zhou: Expected Stock
Returns and Variance Risk Premia
2008-49:
Tim Bollerslev: Glossary to ARCH (GARCH)
2008-50:
Giuseppe Cavaliere, Anders Rahbek and A.M.Robert Taylor: Testing
for Co-integration in Vector Autoregressions with Non-Stationary
Volatility
2008-51:
Bent Jesper Christensen and Michael Sørensen: Optimal inference in
dynamic models with conditional moment restrictions
2008-52:
Katarzyna Lasak: Likelihood based testing for no fractional
cointegration
2008-53:
Katarzyna Lasak: Maximum likelihood estimation of fractionally
cointegrated systems
2008-54:
Andrew J. Patton and Allan Timmermann: The Resolution of
Macroeconomic Uncertainty: Evidence from Survey Forecast
2008-55:
Carlos Capistrán and Allan Timmermann: Forecast Combination With
Entry and Exit of Experts
2008-56:
Carlos Capistrán and Allan Timmermann: Disagreement and Biases in
Inflation Expectations
2008-57:
Almut E. D. Veraart: Impact of time–inhomogeneous jumps and
leverage type effects on returns and realised variances

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