Estimating Quadratic Variation Using a Generalized Itô
Isometry
Xisheng Yu* and Xiaoke Xie†
*
School of Economic Mathematics, Southwestern University of Finance and
Economics, Chengdu 611130, P.R.China
†
E-mail: [email protected]
Department of Economics and Management, Sichuan Vocational and
Technical College of Communications, Chengdu 611130, P.R.China
Abstract
Estimating the quadratic variation (QV) using high-frequency financial data is studied in this article and this work makes two major contributions: First the fundamental
R a
Ra
Itô isometry E ( b f (t, ω)dBt )2 = E b f 2 (t, ω)dt is generalized and the approximation error between QV and realized volatility estimator is obtained using the generalized
isometry. Second, We intuitively establish two novel estimators of QV when the volatility varies with time, using realized volatility combined with realized bipower variation
and realized quarticity respectively. To establish the estimators, the generalized Itô
isometry is employed. Further we prove that both estimators can converge to the
quadratic variation with a higher rate O(n−1 ) than existing ones and the convergence
is in mean square, not only in probability. Meanwhile the difference between the QV
and its estimator can be estimated for each estimation.
Key Words: Generalized Itô isometry; Quadratic variation; Realized volatility; Bipower
variation; Quarticity; Rate of convergence
1
1
Introduction
1.1
Itô Isometry
Let Bt := B(t, ω) (t ∈ [0, T ], ω ∈ Ω), be a standard Brownian motion on the probability
space (Ω, F, P) and Ft be the σ-algebra generated by B(s, ·) (0 ≤ s ≤ t). Suppose a function
denoted as f (t, ω): [0, T ] × Ω → R is Ft -adapted and square integrable (i.e. f ∈ L2 ([0, T ])).
Then the fundamental Itô isometry identity is given as (Kopp, 2011)
"Z
E
2 #
T
f (t, ω)dBt
Z
=E
0
T
f (t, ω)dt
2
(1)
0
As a fundamental formula in Itô integral theory, the Itô isometry is of critical importance
and intensively appears in the stochastic integrals relative to processes. It has extensive
applications in mathematical finance (e.g., Øksendal, 2003; Shreve, 2004), for example, one
of which is to enable the calculation of variance for a financial stochastic process.
As shown in (1), the fundamental isometry only concerns the expectation of squares.
In this paper, however, estimating quadratic variation (QV) and analyzing the difference
between estimator and QV involves the higher order of expectation. Due to this, the fundamental Itô isometry identity needs to be generalized.
1.2
Motivation and QV Estimation
The QV plays a crucial role in the theory of measuring the risk associated with an asset
and the research on its estimation has been largely furthered due to the availability of highfrequency data. In the financial econometrics literature (see, e.g., Anderson, et al. (2001),
Anderson, et al. (2001, 2003), Barndorff-Nielsen and Shepard (2002a, 2002b) and Comte
and Renault (1998)), a model-free volatility measurement, termed realized volatility (RV),
is extensively specified and studied. RV has been regarded as the popular estimator of the
corresponding QV since the sum of squared returns sampled at high-frequency consistently
2
estimates the QV (Andersen, et al. (2003); Barndorff-Nielsen and Shephard,(2002a)), and
the RV theoretically, in the context of semimartingale, converges in probability to the QV
in the absence of market microstructure noise. For a literature review, see Barndorff-Nielsen
and Shephard (2002a, b) and Hautscha and Podolskijb (2013)1 .
To start, some notation are established here. Suppose that τ > 0 is some fixed time
period (e.g., a trading day) and y ∗ (t) denotes the log-price of an asset for t ≥ 0, then the ith
’low-frequency’ return over such interval is defined as
yi = y ∗ (iτ ) − y ∗ ((i − 1)τ ),
i = 1, 2, · · ·
During this interval, for the ith period (equally spaced), n intra-τ ’high-frequency’ returns
are recorded as
yj,i
τ
τ ∗
= y (i − 1)τ + j − y (i − 1)τ + (j − 1) ,
n
n
∗
j = 1, 2, · · · , n
For example, if τ = 1-day and n = 288, then yj,i is the return for the jth 5-minute period
on the ith day. Obviously, the ith ’low-frequency’ return is the sum of these n intra-τ
n
X
’high-frequency’ returns, yi =
yj,i .
j=1
Then the realized volatility during this period is defined by
[yn∗ ]i
=
n
X
2
yj,i
j=1
1
Meanwhile a lot of literature on high-frequency data has paid much attention to market microstructure
noise for obtaining a more appropriate estimator of QV. Zhang, et al. (2005) are the first to propose a
so-called two scales realized volatility estimator which is consistent for QV of the latent price in the presence
of microstructure noise and the convergence rate of the estimator is O(n−1/6 ). Zhang (2006) introduces
the more complicated multiple scales realized volatility estimator with a convergence rate of O(n−1/4 ). Aı̈tSahalia, et al. (2006) modify both estimators and achieve consistency in the presence of serially correlated
microstructure noise. Another prominent consistent estimator of the QV is the realized kernel estimator
proposed by Barndorff-Nielsen, et al. (2008) which extends the unbiased but inconsistent estimator of Zhou
(1996). Kristensen (2010) and Mancini, et al. (2014) also study the kernel estimators for the spot volatility.
In addition, a general pre-averaging approach was presented in Jacod, et al. (2009) and Podolskij and Vetter
(2009) and this approach gives a consistent estimator with a convergence rate of O(n−1/4 ).
3
To analyze some property (e.g., the rate of convergence) of the RV estimator, due to
the fact that under the no-arbitrage assumption, price processes must follow a semimartingale (Back, 1991), many studies (see e.g., Barndorff-Nielsen and Shephard (2002a,b, 2003))
usually work within a general stochastic volatility (SV) model, which is a special case of
semimartingale so that y ∗ (t) can be decomposed as
y ∗ (t) = α∗ (t) + m∗ (t)
(2)
where the drift term α∗ (t) is a continuous process (mean process) with locally bounded
variation paths and the local martingale component m∗ (t) is a SV process having the form
∗
t
Z
m (t) =
σ(s)dBs
0
where Bs := B(s) denotes a standard Brownian motion, the spot volatility process σ(s) > 0
is càdlàg adapted and locally bounded with the integrated volatility process
2∗
Z
σ (t) =
t
σ 2 (s)ds
0
satisfying 0 < σ 2∗ (t) < ∞ on [0, ∞).
To enable the application of the generalized Itô isometry for obtaining better estimators
of QV, in this paper the volatility is assumed to be just a function of time. Then the process
y ∗ (t) with this additional assumption has all of the following properties, such as those shown
in (3)-(5).
For the above SV model (2), the quadratic variation 2 [y ∗ ](t) of y ∗ (t) is then equal to σ 2∗ (t)
2
Sometimes QV is defined by the limit (in probability) p lim
n→∞
4
n−1
X
i=0
|y ∗ (ti+1 ) − y ∗ (ti )|2 .
[y ∗ ](t) = σ 2∗ (t)
(3)
For the ith interval of time of length τ , denote the corresponding QV over this interval
by [y ∗ ]i , then we have
[y ∗ ]i = σi2 ,
i = 1, 2, · · ·
where σi2 = σ 2∗ (τ i) − σ 2∗ (τ (i − 1)) (actual volatility).
Semimartingale theory ensures that the RV measure [yn∗ ]i consistently estimates the corresponding QV [y ∗ ]i . Andersen, et al. (2001) show that the QV can be approximated to
an arbitrary accuracy using the sum of intraday squared high-frequency returns. BarndorffNielsen and Shephard (2002a) show that [yn∗ ]i converges in probability to [y ∗ ]i at a rate
O(n−1/2 ) and obtain the asymptotic distribution of the estimator
[y ∗ ]i − [y ∗ ]i L
qn P
−→ N (0, 1)
n
2
4
j=1 yj,i
3
(4)
In Barndorff-Nielsen and Shephard (2002b), the work on estimating QV using RV is reviewed
and a measure of precision of RV in an empirical examples is provided.
In addition to the RV estimator, in Barndorff-Nielsen and Shephard (2003) a higher
R τi
order power variation termed realized quarticity, τ (i−1) σ 4 (s)ds is introduced. Using this
4th-order power variation helps understand the above result and produces some improved
results. Moreover, in the context of multivariate covariation of Barndorff-Nielsen and Shephard (2004), cross terms over interval [0, t] are studied and the relation to the corresponding
realized quarticity is verified by
n−1
n X 2 2
P
yi yi+1 −→
( )
t i=1
5
Z
0
t
σ 4 (s)ds
(5)
∗
∗
where we recall that yi = y (iτ ) − y ((i − 1)τ ) and in this paper
n−1
X
2
yi2 yi+1
is called the
i=1
realized bipower variation. The convergence is also in probability and the rate is still of
O(n−1/2 ). Barndorff-Nielsen (2004) extends this to a more general realized bipower variation,
n−1
X
|yi |r |yi+1 |s , and gains some corresponding results under several assumptions.
i=1
It appears that the RV-based estimators of QV mentioned above including those in footnote 1, converge just in probability to QV and the highest rate of convergence is only of
O(n−1/2 ).
Naturally, one would ask is it possible to establish a better estimator for the QV? How fast
does it converge to the QV and what is the accuracy of the estimator? This paper concentrates
on, under the model (2) with additional assumption of σ(t) being a function of time, how to
obtain new approximations for QV from an innovative perspective by employing the proposed
generalized Itô isometry. The rate of convergence and the approximation accuracy are also
analyzed. This method is intuitive and easily understood.
This paper first uses the generalized Itô isometry to derive the rate of convergence of the
RV estimator and provides a measure of the precision of this estimator, and then establishes
two improved estimators of QV as follows. (i) Combination of realized volatility and bipower
variation. This is inspired by Corollary 2 in subsection 2.2 and the result shown in (5) (see
subsection 3.1 for details). (ii) Combination of realized volatility and quarticity. Motivated
by (4) and Theorem 3, this estimator is constructed for estimating the QV(see subsection
3.2 for details). Both novel estimators converge to the QV at a higher rate than the existing
best rate of O(n−1/2 ), in addition, the approximation errors can be estimated.
The remainder of this paper is organized as follows. The generalized Itô isometry identity
is initially proposed in section 2. Sections 3 establishes two novel estimators of QV in an
intuitive way. The conclusion is in Section 4.
6
2
The Generalized Itô Isometry
In this section, the fundamental Itô isometry is initially generalized, and the approxima-
tion error between the RV and QV is also provided. The generalized isometry plays a crucial
role in obtaining the estimators of QV in the next section.
2.1
The Generalized Itô Isometry
First a definition of function family is formally given as below.
Definition 1 Let M := M(a, b) be the family of functions f (t, ω) : [0, ∞) × Ω → R such
that,
1. (t, ω) → f (t, ω) is B × F-measurable, where B and F denote the Borel σ-algebra on
[0, ∞) and a σ-algebra on Ω, respectively.
2. f (t, ω) is Ft -adapted.
R
21
b
3. k f kM := E[ a f (t, ω)2 dt] < ∞.
The function f (t, ω) in this definition is the objective in Itô integral. To generalize the
fundamental Itô isometry (1), a well-known property of the Brownian motion Bt is required.
We write it as a lemma, see below.
Lemma 1 For the standard Brownian motion Bt with B0 = 0, its moments satisfy
E Bt2n+1 = 0,
(2n)! n
E Bt2n = n
t = (2n − 1)!!tn
2 · n!
(6)
where n ∈ Z+ and (2n − 1)!! = 1 · 3 · 5 · · · (2n − 1).
Now the fundamental isometry can be generalized as the following.
Theorem 1 Let f (t, ω) ∈ M(a, b), then the following Itô isometry holds
Z
E
2n
b
f (t, ω)dBt
Z b
n
2
= (2n − 1)!! E
f (t, ω)dt
a
a
7
(7)
for n = 1, 2, 3, · · · .
Proof. See Appendix A.1.
This theorem is of importance in deriving the estimators of QV. Setting n = 2, 3, 4 yields
the following corollary which is used in proving Corollary 2, Theorem 3 and Theorem 4.
Corollary 1 Assume that σ(s, ω) ∈ M(0, t), then
4
t
Z
E
σ(s, ω)dBs
Z t
2
2
=3 E
σ (s, ω)ds
0
6
t
Z
σ(s, ω)dBs
E
Z t
3
2
= 15 E
σ (s, ω)ds
0
Z
(9)
0
8
t
σ(s, ω)dBs
E
(8)
0
Z t
4
2
= 105 E
σ (s, ω)ds
0
(10)
0
where σ(s, ω) usually refers to the volatility function.
2.2
Convergence of RV to QV
For the log-price process y ∗ (t) in (2), the previous section discusses the RV during the ith
n
X
R τi
2
∗
τ -period interval [τ (i−1), τ i]: [yn ]i =
yj,i
and the corresponding QV: [y ∗ ]i =σi2 = τ (i−1) σs2 ds.
j=1
Naturally, how fast does the RV converge to QV? What is the approximation accuracy? The
following corollary answers these questions, thanking to the generalized Itô isometry.
Without any loss of generality3 , we assume that α∗ (t)=0 in model (2) and use time intern
X
val [0, t] instead of the ith interval [τ (i − 1), τ i], then the RV is defined by [yn∗ ]=
yi2 and
i=1
QV by [y ∗ ](t)=σ 2∗ (t), where yi is the ith recorded return over the ith subinterval [ti−1 , ti ] and
Rt
Rt
Rt
σ 2∗ (t)= 0 σ 2 (s)ds. Again volatility σ(s) is the function of time so that E 0 σ 2 (s)ds= 0 σ 2 (s)ds.
3
For further details, see the second paragraph of section 3.
8
Rt
Corollary 2 Assume y ∗ (t)= 0 σ(s)dBs , π={0=t0 < t1 < · · · < tn =t} is an equidistant
partition of interval [0, t] and σ(s) is the function of time, so that ti =i nt and σ(s) ∈ M(0, t),
then for the RV and the corresponding QV we have
E
([yn∗ ]
n
X
− [y ](t)) = 2
(Σi )2
2
∗
(11)
i=1
where Σi =σ 2∗ (ti−1 ) − σ 2∗ (ti ).
Proof. Provided in Appendix A.2.
From Corollary 2, it can be shown that if the interval is equally spaced, (ti − ti−1 )= nt for
all i=0, 1, · · · , n, and σ(s) is bounded with 0 < σ(s)2 < C, then the RV converges to the QV
in mean square, absolutely in probability. Further, the rate of
rate is O(n−1/2 )
sconvergence
n
X
and the difference between RV and QV can be estimated by 2 (Σi )2 . In fact, Equation
i=1
(11) implies that
v
v
u n Z
u n
u X
u X
∗
∗
[yn ] − [y ](t) ≈ t2
(Σi )2 = t2
i=1
since (ti+1 − ti ) =
t
n
2 r
1
2
σ 2 (s)ds ≤
tC = O(n− 2 )
n
ti−1
i=1
ti
(12)
and |σ(s)|2 < C.
Corollary 2 is about the quadratic variation, and this result can be readily extended to
higher order power variation as follows.
Rt
Theorem 2 Assume y ∗ (t)= 0 σ(s)dBs and the partition of interval [0, t] is equidistant, then
P
for the high order realized power variation ni=1 yi2m and the corresponding power variation
Pn
m
i=1 (Σi ) , the following formula holds
"
n
n
X
X
1
E
yi2m −
(Σi )m
(2m − 1)!! i=1
i=1
#2
=
for m = 1, 2, · · · , and the rate of convergence to
9
X
n
(4m − 1)!!
−1
(Σi )2m
2
((2m − 1)!!)
i=1
Pn
i=1
1
(Σi )m is O( nm−1/2
).
(13)
Proof. Similar with the proof of Corollary 2, using the Theorem 1 completes this proof.
Apparently, this result is more general. Let m = 1, Corollary 2 follows.
In summary, this section proposes the generalized Itô isometry. Furthermore, how the
RV of a log-price process y ∗ (t) converges to the QV is described. Corollary 2 suggests that
RV can converge to QV in mean square, not only in probability with a rate of O(n−1/2 ) if
σ(s) is bounded and the measure of the precision is also provided. Now one would ask if this
approximation can be improved and how? The next section intents to answer this question
by presenting two improved estimators for the QV.
3
Estimators of QV
Based on the previous section, this section constructs two novel estimators of QV in an
intuitive and succinct way.
Recall that, notation yi represents the ith low-frequency return over an interval of [(i −
1)τ, iτ ] and yj,i the jth intra-τ high-frequency return in such interval. To estimate the QV, an
P
2
then is usually computed for each τ -period
estimator, for example, the RV: [yn∗ ]i = nj=1 yj,i
separately. Hence from a theoretical point of view, we only need to think about a single
period, starting from time 0 until time t, i.e. [0, t], but working with n equally spaced highP
frequency returns to calculate RV as [yn∗ ]= ni=1 yi2 where yi is the ith return over the ith
interval of time of length t/n. Additionally, in the model (2), we assume the continuous
mean process α∗ (t)=0 since the QV of y ∗ (t) equals that of m∗ (t), not influenced by α∗ (t).
Such treatments allow us to use a rather simpler natation and this section just follows these
simple notation.
3.1
Combination of Realized Volatility and Bipower Variation
An estimator, termed here combination of realized volatility and bipower variation (hereafter RVB) is constructed in this subsection.
10
Realized bipower variation is a cross term estimator and defined as
n−1
X
2
yi2 yi+1
i=1
and Barndorff-Nielsen (2004) shows that (see also formula (5)),
n−1
n X 2 2
P
( )
yi yi+1 −→
t i=1
Z
t
σ 4 (s)ds
(14)
0
From Corollary 2
E
([yn∗ ]
n
n Z
X
X
2
− [y ](t)) = 2
(Σi ) = 2
∗
2
i=1
i=1
2
ti
σ(s)ds
ti−1
Following this, we try to replace the RHS of (14) by
t
≈ 2( )E
n
n
([yn∗ ]
2t
Z
t
σ 4 (s, ω)ds
0
− [y ∗ ](t))2 . Inspired by this,
to derive a new estimator of QV, we consider the formula
"
E 2
n−1
X
#2
2
yi2 yi+1
− ([yn∗ ] − [y ∗ ](t))2
(15)
i=1
and fortunately, we actually obtain an improved estimator described in the Theorem 3.
For simplicity of the proof of Theorem 3, however, we first give a lemma below, assuming
σ(s) ≡ 1, that’s, y ∗ (t) = Bt .
Lemma 2 The following estimator
v
v
u n−1
u n−1
n
u X
u X
X
2
yi2 yi+1
:=
(Bti − Bti−1 )2 ± t2
(Bti − Bti−1 )2 (Bti+1 − Bti )2
[yn∗ ] ± t2
i=1
i=1
(16)
i=1
converges in mean square to the QV, i.e. t, for Brownian motion Bt , at a rate of O(n−1 ).
The interval [0, t] here is equally spaced.
Proof. The proof is given in Appendix A.3.
11
With the similar way, the Theorem 3 is obtained.
Theorem 3 For the process y ∗ (t) =
Rt
0
σ(s)dBs with the interval [0, t] spaced equally and
σ(s) ∈ M(0, t), the formula (15)
"
E 2
n−1
X
#2
2
− ([yn∗ ] − [y ∗ ](t))2
yi2 yi+1
i=1
converge to 0 at a rate of O(n−1 ). Further, when σ(s) is constant, the rate of convergence
reaches O(n−2 ). As a consequence, the RVB estimator
v
u n−1
u X
∗
[y ] ± t2
y2y2
n
i
i+1
(17)
i=1
converges in mean square to the QV, [y ∗ ](t), at a rate of O(n−1/2 ) and a rate of O(n−1 ) if
σ(s) is constant.
Similarly, the the difference between QV and RVB estimator can also be estimated.
3.2
Combination of Realized Volatility and Quarticity
2
As shown in Section 3.1, the RVB estimator involves the cross term yi2 yi+1
. Can we try
to consider the quarticity term yi4 instead of the cross term? Computing quarticity term
seems easier since no cross terms appears. Following this, this subsection proposes another
estimator of QV, termed here Combination Estimator of Realized Volatility and Quarticity
(hereafter RVQ).
The realized quarticity is defined as
n
X
yi4
i=1
P
where yi = ni=1 (y ∗ (ti ) − y ∗ (ti−1 )). Barndorff-Nielsen and Shephard (2002a) derive that (see
12
also formula (4) )
[yn∗ ] − [y ∗ ](t) L
q P
−→ N (0, 1)
n
2
4
y
i=1 i
3
(18)
The rate of convergence to the QV [y ∗ ](t), however, is still of O(n−1/2 ). With the encouragement of the above, to derive another estimator of QV, we consider the formula
n
E
2X 4
y − ([yn∗ ] − [y ∗ ](t))2
3 i=1 i
!2
(19)
Fortunately, the proposed RVQ estimator has a better convergence rate, as stated in the
following theorem.
Rt
Theorem 4 For the process y ∗ (t) =
0
σ(s)dBs with the interval [0, t] spaced equally and
σ(s) ∈ M(0, t), the formula (19)
n
E
2X 4
y − ([yn∗ ] − [y ∗ ](t))2
3 i=1 i
!2
converge to 0 at a rate of O(n−2 ). As a consequence, the RVQ estimator
v
u n
u2 X
∗
y4
[yn ] ± t
3 i=1 i
(20)
converges in mean square to the QV, [y ∗ ](t), at a rate of O(n−1 ).
Proof. It is provided in Appendix A.4.
This theorem shows that the RVQ estimator has a rate of convergence of O(n−1 ), which
1
is better than O(n− 2 ) and the difference between QV and RVQ can be estimated, since the
formula (19) is expressed as
n
E
2X 4
yi − ([yn∗ ] − [y ∗ ](t))2
3 i=1
!2
13
=8
n
X
i=1
!2
Σ2i
n
16 X 4
−
Σ
3 i=1 i
(21)
as given in the proof in Appendix A.4.
In particular, when the process y ∗ (t) is the Brownian motion Bt (i.e. σ(s) ≡ 1), then we
obtain one more estimator of the QV of Bt , as below.
Corollary 3 The following estimator
v
u n
u2 X
2
(Bti − Bti−1 ) ± t
(Bti − Bti−1 )4
3
i=1
i=1
n
X
(22)
converges in mean square to the QV, i.e. t, for Brownian motion Bt , at a rate of O(n−1 ).
The interval [0, t] here is equally spaced.
Indeed, by direct calculation following formula (21),

E
2
3
n
X
i=1
(Bti − Bti−1 )4 −
!2 2
n
X
(Bti − Bti−1 )2 − t
i=1
4
4
 = 8t − 16t ,
n2
3n3
the LHS converges to 0 at a rate of O(n−2 ). Setting the LHS equal to 0, the RVQ estimator
(22) is obtained. The symbol ± can be explained by the reflection principle for Brownian
motion.
Obviously from Lemma 2 and Corollary 3, for the Brownian motion Bt , both novel estimators of RVQ and RVB have a higher rate of convergence than the RV estimator.
In brief, two new estimators of QV, termed RVB and RVQ, are established in this section
using realized volatility, realized bipower variation and realized quarticity. Both estimators
improve the convergence rate and the approximation accuracy can be estimated for each
estimator.
4
Conclusion and Further Research
This paper provides an insight into volatility estimation in the field of high-frequency
data. We study how to establish two novel estimators for the quadratic variation of a special
14
semimartingale process, including analyzing the rate of convergence and the approximation
accuracy.
First of all, to achieve the objectives above, the fundamental Itô isometry identity is
initially generalized in this paper.
Second, we show that the realized volatility for high-frequency data not only converges,
as well-known in probability to quadratic variation, but also converges in mean square under
some condition as stated in Corollary 2. Furthermore the expression for the approximation
error is obtained.
Third and more importantly, two novel estimators of the quadratic variation are constructed in an intuitive and easily-understood way, employing realized volatility, realized bipower
variation and realized quarticity. We prove that under some condition, both estimators can
converge in mean square to the quadratic variation at a rate of O(n−1 ), which is better than
1
those in previous studies such as O(n− 2 ). Meanwhile the difference between each estimator
and the quadratic variation is also estimated so that the precision of each estimator can be
determined. Finally, for a special case, the Brownian motion, some corresponding results for
estimating its quadratic variation are provided.
Further research is required to consider a more general case of semimartingale process for
estimating the quadratic variation. Another direction is to investigate the usefulness of this
work using the high-frequency data.
15
Appendix
A
Proof of Lemma ,Theorems and Corollary
A.1
Proof of Theorem 1
First define a simple function (or simple process) ø(t, ω) ∈ M(a, b),
ø(t, ω) :=
m−1
X
øi (ω) · 1[ti ,ti+1 ) (t)
(A.1-1)
i=0
where øi (ω) is Fti -measurable with a partition π = {a = t0 < t1 < · · · < tm = b} denoting
the value on [ti , ti+1 ), and 1[ti ,ti+1 ) (t) is the indicator function defined on [ti , ti+1 ). Then the
Itô integral of simple process ø(t, ω) is defined as
b
Z
ø(t, ω)dBs (ω) :=
a
=
m−1
X
i=0
m−1
X
øi (ω)[Bi+1 (ω) − Bi (ω)]
øi (ω)∆Bi (ω)
(A.1-2)
i=0
For an arbitrary function f (t, ω) ∈ M(a, b), there exists a sequence of simple function
(øn (t, ω))n having the form of (A.1-1) which approximates f (t, ω), that is, lim k øn − f kM = 0
n→∞
so that the Itô integral of f (t, ω) can be defined by
Z
b
Z
f (t, ω)dBs (ω) := lim
a
n→∞
b
øn (t, ω)dBs (ω) (in L2 − lim)
(A.1-3)
a
Now it remains to prove that equation (7) holds for the simple process ø(t, ω) in (A.1-1)
since the integral of f (t, ω) can be approximated by the sequence of integrals of (øn (t, ω))n
as specified above.
16
Note that from formula (A.1-2),
2n
b
Z
ø(t, ω)dBt
a
"
=
m−1
X
#2n
øi (ω)∆Bi (ω)
#2n
"i=0
m−1
X
:=
øi ∆Bi
i=0
X
=
i0 +i1 +···+im−1 =2n
(2n)!
(ø0 ∆B0 )i0 (ø1 ∆B1 )i1 · · · (øm−1 ∆Bm−1 )im−1
(i0 )!(i1 )! · · · (im−1 )!
Then we have the following via direct calculations,
Z
2n
b
ø(t, ω)dBt
X
E
a
(2n)!
i0
i1
im−1
=
E
(ø0 ∆B0 ) (ø1 ∆B1 ) · · · (øm−1 ∆Bm−1 )
(i
)!(i
)!
·
·
·
(i
)!
0
1
m−1
i0 +i1 +···+im−1 =2n
X
(2n)!
=
E[(ø0 ∆B0 )i0 ]E[(ø1 ∆B1 )i1 ] · · · E[(øm−1 ∆Bm−1 )im−1 ]
(i
)!(i
)!
·
·
·
(i
)!
0
1
m−1
i +i +···+i
=2n
0
1
m−1
Following the Lemma 1, we can let ik = 2jk (k = 0, 1, · · · , m − 1), hence the formula above
follows
(2n)!
E[(ø0 ∆B0 )2j0 ]E[(ø1 ∆B1 )2j1 ] · · · E[(øm−1 ∆Bm−1 )2jm−1 ]
(2j
)!(2j
)!
·
·
·
(2j
)!
0
1
m−1
j0 +j1 +···+jm−1 =n
X
(2n)!
j0
j1
0
1
(2j0 − 1)!!E(ø2j
(2j1 − 1)!!E(ø2j
=
0 )(∆t0 )
1 )(∆t1 )
(2j0 )!(2j1 )! · · · (2jm−1 )!
j0 +j1 +···+jm−1 =n
h
i
2jm−1
jm−1
· · · (2jm − 1)!!E(øm−1 )(∆tm−1 )
X
(2n)!
2jm−1
2j1
j0
j1
jm−1
0
E(ø2j
=
0 )(∆t0 ) E(ø1 )(∆t1 ) · · · E(øm−1 )(∆tm−1 )
n
2
(j
)!(j
)!
·
·
·
(j
)!
0
1
m−1
j0 +j1 +···+jm−1 =n
X
(2n)!
n!
= n
[E(ø20 ∆t0 )]j0 [E(ø21 ∆t1 )]j1 · · · [E(ø2m−1 ∆tm−1 )]jm−1
2 n! j +j +···+j =n (j0 )!(j1 )! · · · (jm−1 )!
0
1
"m−1 m−1
#n
X
= (2n − 1)!!
E(øi (ω)2 ∆ti )
i=0Z b
n
2
= (2n − 1)!! E
ø (t, ω)dt
=
X
a
17
A.2
Proof of Corollary 2
Calculating the LHS of Equation (11) yields,
E ([yn∗ ] − [y ∗ ]t )2
n
X
=E
!2
(y ∗ (ti ) − y ∗ (ti−1 ))2 − σ 2∗ (t)
i=1
"
n Z
X
2
ti
Z
#2
t
−E
σ 2 (s)ds
ti−1
0
" i=1
#2 Z
2
Z
n
2
ti
t
X
2
=E
σ (s)ds
σ(s)dBs
+ E
0
i=1 " ti−1
2 # Z t
n Z ti
X
2
− 2E
σ(s)dBs
σ (s)ds
E
ti−1
0
i=1

2 Z
Z ti
4 X
Z ti
n
n
X

σ(s)dBs
E
σ(s)dBs +
=
E
=E
σ(s)dBs
ti−1
i=1
ti−1
i6=j
" n Z
2
Z t
X
σ 2 (s)ds − 2E
+ E
0
ti
σ(s)dBs

tj−1
2 # Z t
2
σ(s)dBs
σ (s)ds
E
ti−1
i=1
!2 
tj
0
By the fundamental Itô isometry Equation (1) and Corollary 1, it comes to
n Z
X
=3
E
2
ti
σ(s)ds
ti−1
i=1
n Z
X
+
E
0
+
i=1
n X
2
ti
σ(s)ds
ti−1
Z
E
i=1
ti−1
n Z
X
+
E
σ(s)ds
ti−1
Z
Z
!
tj
σ(s)ds
σ(s)ds
E
tj−1
! Z
t
2
2
σ (s)ds
E
σ (s)ds
ti
0
tj
E
2
ti
ti−1
i=1
ti
ti−1
i6=j
Z t
2
n Z
X
2
+ E
σ (s)ds − 2 E
n Z
X
=2
E
ti
σ(s)ds
!
σ(s)ds
tj−1
i6=j
Z t
2
Z t
Z t
2
2
2
+ E
σ (s)ds − 2 E
σ (s)ds
E
σ (s)ds
0
0
0
!2 Z
2
2
Z ti
n
n Z ti
t
X
X
2
=2
E
σ(s)ds +
E
σ(s)ds − E
σ (s)ds
i=1
ti−1
ti
i=1
n
X
ti−1
n Z
X
=2
E
=2
2
ti−1
i=1
σ(s)ds
(Σi )2
i=1
18
0
A.3
Proof of Lemma 2
Denote ∆Bi = Bti − Bti−1 , and ti − ti−1 = t/n for i = 1, 2, · · · , n, first six formulas below
are derived employing Lemma 1.
n−1
X
2
2
E(∆Bi+1
∆Bi2 ∆Bj+1
∆Bj2 ) = (n2 − 5n + 6)
i,j=1;|j−i|≥2
n−1 X
n
X
t4
n4
2
∆Bi2 ∆Bj2 ∆Bk2 ) = (n3 + 9n2 + 14n − 24)
E(∆Bi+1
i=1 j,k=1
n−1
n
XX
2
∆Bi2 ∆Bj2 ) = (n2 + 3n − 4)
E(∆Bi+1
i=1 j=1
n
X
E(∆Bi2 ∆Bj2 ∆Bk2 ∆Bl2 )
i,j,k,l=1
n
X
E(∆Bi2 ∆Bj2 ) = n(n + 2)
i,j=1
t3
n3
t4
= n(n + 12n + 44n + 48) 4
n
3
E(∆Bi2 ∆Bj2 ∆Bk2 ) = n(n2 + 6n + 8)
i,j,k=1
n
X
t4
n4
2
t3
n3
t2
n2
With these formulas and Equation (6), now we can prove the Lemma 2. Since

n−1
X

E 2
(Bti+1 − Bti )2 (Bti − Bti−1 )2 −
!2 2
n
X
(Bti − Bti−1 )2 − t 
i=1
i=1
=E
2
n−1
X
i=1
i=1
2
∆Bi+1
∆Bi2 )2 − 4E
i,j=1
n
X
n−1
X
n
X
!
n
X
2
∆Bi+1
∆Bi2 (
∆Bj2 − t)2 + E(
∆Bi2 − t)4
i=1
i=1
i=1
" n−1 j=1
!#
n
n
n−1
X
X
X
X
2
2
2
= 4E(
∆Bi+1
∆Bi2 ∆Bj+1
∆Bj2 ) − 4E
∆Bi+1
∆Bi2 (
∆Bj2 )2 − 2t
∆Bj2 + t2
= 4E(
n−1
X
n
X
2
∆Bi+1
∆Bi2 − (
∆Bi2 − t)2
!2
i=1
+ E(
i=1
i=1
 i=1
n−1
n−2
X
X
4
4
2
∆Bi+1
∆Bi2 ) + 4
= 4
E(∆Bi+1
∆Bi4 ) + 8
E(∆Bi+2
i=1
j=1
j=1
n
n
n
X
X
X
2 4
2 3
2 2 2
∆Bi ) − 4E(
∆Bi ) t + 6E(
∆Bi ) t − 4E(
∆Bi2 )t3 + t4
i=1
i=1
n−1
X
i,j=1;|j−i|≥2
19

2
2
∆Bj2 )
E(∆Bi+1
∆Bi2 ∆Bj+1
−
4
+
n−1 X
n
X
n−1
X
n−1 X
n
X
2
2
2
E(∆Bi+1
∆Bi2 )
E(∆Bi+1
∆Bi2 ∆Bj2 ∆Bk2 ) − 8t
E(∆Bi+1
∆Bi2 ∆Bj2 ) + 4t2
i=1 j,k=1
i=1
i=1 j=1
!
!
!
n
n
n
X
X
X
E(∆Bi2 ∆Bj2 )
E(∆Bi2 ∆Bj2 ∆Bk2 ) + 6t2
E(∆Bi2 ∆Bj2 ∆Bk2 ∆Bl2 ) − 4t
i,j=1
i,j,k=1
i,j,k,l=1
n
X
3
E∆Bi2 + t4
− 4t
i=1
n−1
n−2
X
X
t
t
t
t 4
= 36
( ) + 24
( )4 + 4(n2 − 5n + 6)( )4 − 4(n3 + 9n2 + 14n − 24)( )4
n
n
n
n
i=1
i=1
n−1
X
t
t
t4
+ 8t(n2 + 3n − 4)( )3 − 4t2
( )2 + (n3 + 12n2 + 44n + 48) 3
n
n
n
i=1
n
Xt
t
t
+ t4
−4tn(n2 + 6n + 8)( )3 + 6t2 n(n + 2)( )2 − 4t3
n
n
n
i=1
= ( n82 +
36 4
)t
n4
= O(n−2 )
Hence,
n−1
X
2
(Bti+1 − Bti )2 (Bti − Bti−1 )2 −
i=1
!2
n
X
(Bti − Bti−1 )2 − t
i=1
converges in mean square to 0, at a rate O(n−2 ). Set
2
n−1
X
(Bti+1 − Bti )2 (Bti − Bti−1 )2 −
n
X
!2
(Bti − Bti−1 )2 − t
= 0,
i=1
i=1
we have that the estimator
v
u n−1
u X
2
(Bti − Bti−1 ) ± t2
(Bti − Bti−1 )2 (Bti+1 − Bti )2
n
X
i=1
i=1
converges to t at a rate of O(n−1 ).
A.4
!
Proof of Theorem 4
R ti 2
∗
∗
σ(s)dB
,
y
=
y
(t
)
−
y
(t
)
and
Σ
=
σ (s)ds, then the
s
i
i
i−1
i
0
ti−1
P
quadratic variation [y ∗ ](t) = σ 2∗ (t) = ni=1 Σi . Besides, the realized volatility and realized
P
P
quarticity are defined by [yn∗ ] = ni=1 yi2 and ni=1 yi4 , respectively.
Recall that y ∗ (t) =
Rt
By Corollary 1 and the fundamental Itô isometry, we first have the following,
20
Eyi2 = Σi , Eyi4 = 3(Σi )2 , Eyi6 = 15(Σi )3 , Eyi8 = 105(Σi )4
Now we can calculate the following expression,
#2
n
2X 4
E
y − ([yn∗ ] − [y ∗ ](t))2
3 i=1 i

!2 2
n
n
X
X
2
yi4 −
yi2 − σ 2∗ (t) 
=E
3 i=1
i=1

!
!2 
!4
2
n
n
n
n
X
X
X
X
4
4
= E
yi4 − E 
yi4
yi2 − σ 2∗ (t)  + E
yi2 − σ 2∗ (t)
9
3
i=1
i=1
 i=1  i=1 !

2
n
n
n
n
X
X
X
X
4
4
yi4 
= E
yi4 yj4 − E 
yi2 − 2σ 2∗ (t)
yi2 + (σ 2∗ (t))2 
9 i,j=1
3
i=1
i=1
i=1
!4
!3
!2
n
n
n
n
X
X
X
X
2
2∗
2
2∗
2
2
2∗
3
+E
yi
− 4σ (t)E
yi
+ 6(σ (t)) E
yi
− 4(σ (t)) E
yi2 + (σ 2∗ (t))4
i=1
i=1
i=1
!
! i=1
n
n
n
n
n
n
X
X
X
X
X
X
4
4
yi6 yj2
yi4 yj2 yk2 + 2
yi4 yj4 − E
yi4 yj4 +
yi8 +
yi8 +
= E
9
3
i=1
i,j=1;i6
i,j,k=1;i6=j6=k;i6=k
i,j=1;i6=j
i,j=1;i6=j
! i=1
! =j
n
n
n
n
n
X
X
X
X
X
8
4
+ σ 2∗ (t)E
yi4 yj2 − (σ 2∗ (t))2 E
yi6 +
yi4 + E
yi6 yj2
yi8 + 4
3
3
i=1
i=1
i=1
i,j=1;i6=j
i,j=1;i6
! =j
n
n
n
X
X
X
+E 3
yi4 yj4 + 6
yi4 yj2 yk2 +
yi2 yj2 yk2 yl2
i,j=1;i6=j
i,j,k=1;i6=j6=k;i6=k
i,j,k,l=1;i6=j6=k6!
=l;i,j6=k,l
n
n
n
X
X
X
− 4σ 2∗ (t)E
yi6 + 3
yi4 yj2 +
yi2 yj2 yk2
i=1
i,j=1;i6=j
!i,j,k=1;i6=j6=k;i6=k n
n
n
X
X
X
yi2 + (σ 2∗ (t))4
+ 6(σ 2∗ (t))2 E
yi4 +
yi2 yj2 − 4(σ 2∗ (t))3 E
"
i=1
i=1
i,j=1;i6=j
21
4
=
9
105
n
X
Σ4i
+9
i=1
4
−
3
n
X
n
X
i,j=1;i6=j
n
X
!
Σ2i Σ2j
n
X
!
Σ2i ΣjΣk + 30
Σ3i Σj
i=1
i,j,k=1;i6
=j6=k;i6=k
i,j=1;i6=j
! i,j=1;i6=j n
!
!
n
n
n
n
X
X
X
X
X
8 2∗
4
+ σ (t) 15
Σ3i + 3
Σ2i + 105
Σ4i + 60
Σ2i Σj − (σ 2∗ (t))2 3
Σ3i Σj
3
3
i=1
i=1
i=1 !
i,j=1;i6=j
i,j=1;i6=j
n
n
n
X
X
X
+ 27
Σ2i Σ2j + 18
Σ2i Σj Σk +
Σi Σj Σk Σl
i,j=1;i6=j
i,j,k=1;i6=j6=k;i6=k
i,j,k,l=1;i6=j6=k6=!
l;i,j6=k,l
n
n
n
X
X
X
− 4σ 2∗ (t) 15
Σ3i + 9
Σ2i Σj +
Σi Σj Σk
i=1
i,j=1;i6=j!
i,j,k=1;i6=j6=k;i6=k
n
n
n
X
X
X
2
2
2∗
3
+ 6Σ 3
Σi +
Σi Σj − 4(σ (t))
Σi + (σ 2∗ (t))4
105
Σ4i + 9
i=1
=
n
8X
3
=8
Σ4i + 8
i=1
n
X
i=1
Σ2i Σ2j + 3
n
X
i=1
i,j=1;i6=j
n
X
Σ2i Σ2j
!2 i,j=1;i6=j n
16 X 4
Σ
Σ2i
−
3 i=1 i
R ti 2
σ (s)ds, ti − ti−1 = nt (1 ≤ i ≤ n) and σ(s) is bounded, then
Note that Σi = E ti−1
P
Pn
2
4
the former term 8 ( ni=1 (Σi )2 ) can be expressed as O(n−2 ) and the latter − 16
i=1 (Σi )
3
expressed as O(n−3 ).
As a result,
n
2X 4
y − ([yn∗ ] − [y ∗ ](t))2
3 i=1 i
!2
−→0
in mean square at a rate of O(n−2 ).
Let
n
2X 4
y − ([yn∗ ] − [y ∗ ](t))2 = 0
3 i=1 i
then we get,
v
u n
u2 X
∗
y 4 = [y ∗ ](t)
[yn ] ± t
3 i=1 i
22
Consequently, we can say that the RVQ estimator
v
u n
u2 X
∗
[yn ] ± t
y4
3 i=1 i
converges in mean square to [y ∗ ](t) at a rate of O(n−1 ).
23
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26