Description
We will review the econometrics of non-parametric estimation of the components of the variation of asset prices.
Variation, jumps, market frictions and
high frequency data in ?nancial econometrics
?
Ole E. Barndorff-Nielsen
The T.N. Thiele Centre for Mathematics in Natural Science,
Department of Mathematical Sciences,
University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark
[email protected]
Neil Shephard
Nu?eld College, Oxford OX1 1NF, UK
[email protected]
July 14, 2005
1 Introduction
We will review the econometrics of non-parametric estimation of the components of the variation
of asset prices. This very active literature has been stimulated by the recent advent of complete
records of transaction prices, quote data and order books. In our view the interaction of the
new data sources with new econometric methodology is leading to a paradigm shift in one of
the most important areas in econometrics: volatility measurement, modelling and forecasting.
We will describe this new paradigm which draws together econometrics with arbitrage free
?nancial economics theory. Perhaps the two most in?uential papers in this area have been
Andersen, Bollerslev, Diebold, and Labys (2001) and Barndor?-Nielsen and Shephard (2002),
but many other papers have made important contributions. This work is likely to have deep
impacts on the econometrics of asset allocation and risk management. One of our observations
will be that inferences based on these methods, computed from observed market prices and so
under the physical measure, are also valid as inferences under all equivalent measures. This puts
this subject also at the heart of the econometrics of derivative pricing.
?
Prepared for the invited symposium on Financial Econometrics, 9th World Congress of the Econometric
Society, London, 20th August 2005. We are grateful to Tim Bollerslev, Eric Ghysels, Peter Hansen, Jean Jacod,
Dmitry Kulikov, Jeremy Large, Asger Lunde, Andrew Patton, Mark Podolskij, Kevin Sheppard and Jun Yu for
comments on an earlier draft. Talks based on this paper were also given in 2005 as the Hiemstra Lecture at the
13th Annual conference of the Society of Non-linear Dynamics and Econometrics in London, the keynote address
at the 3rd Nordic Econometric Meeting in Helsinki and as a Special Invited Lecture at the 25th European Meeting
of Statisticians in Oslo. Ole Barndor?-Nielsen’s work is supported by CAF (www.caf.dk), which is funded by the
Danish Social Science Research Council. Neil Shephard’s research is supported by the UK’s ESRC through the
grant “High frequency ?nancial econometrics based upon power variation.”
1
One of the most challenging problems in this context is dealing with various forms of market
frictions, which obscure the e?cient price from the econometrician. Here we will characterise four
types of statistical models of frictions and discuss how econometricians have been attempting
to overcome them.
In section 2 we will set out the basis of the econometrics of arbitrage-free price processes,
focusing on the centrality of quadratic variation. In section 3 we will discuss central limit
theorems for estimators of the QV process, while in section 4 the role of jumps in QV will be
highlighted, with bipower and multipower variation being used to identify them and to test
the hypothesis that there are no jumps in the price process. In section 5 we write about the
econometrics of market frictions, while in section 6 we conclude.
2 Arbitrage-free, frictionless price processes
2.1 Semimartingales and quadratic variation
Given a complete record of transaction or quote prices it is natural to model prices in contin-
uous time (e.g. Engle (2000)). This matches with the vast continuous time ?nancial economic
arbitrage-free theory based on a frictionless market. In this section and the next, we will dis-
cuss how to make inferences on the degree of variation in such frictionless worlds. Section 5
will extend this by characterising the types of frictions seen in practice and discuss strategies
econometricians have been using to overcome these di?culties.
In its most general case the fundamental theory of asset prices says that a vector of log-prices
at time t,
Y
t
=
_
Y
1
t
, ..., Y
p
t
_
,
must obey a semimartingale process (written Y ? o/) on some ?ltered probability space
_
?, T, (T
t
)
t?0
, P
_
in a frictionless market. The semimartingale is de?ned as being a process
which can be written as
Y = A +M, (1)
where A is a local ?nite variation process (A ? T1
loc
) and M is a local martingale (M ? /
loc
).
Compact introductions to the economics and mathematics of semimartingales are given in Back
(1991) and Protter (2004), respectively.
The Y process can exhibit jumps. It is tempting to decompose Y = Y
ct
+ Y
d
, where Y
ct
and Y
d
are the purely continuous and discontinuous sample path components of Y . However,
technically this de?nition is not clear as the jumps of the Y process can be so active that they
2
cannot be summed up. Thus we will de?ne
Y
ct
= A
c
+M
c
,
where M
c
is the continuous part of the local martingale component of Y and A
c
is A minus the
sum of its jumps
1
. Likewise, the continuous sample path subsets of classes of processes such as
o/ and /, will be denoted by o/
c
and /
c
.
Crucial to semimartingales, and to the economics of ?nancial risk, is the quadratic variation
(QV) process of (Y
, X
)
? o/. This can be de?ned as
[Y, X]
t
= p?lim
n??
t
j
?t
j=1
_
Y
t
j
?Y
t
j?1
_ _
X
t
j
?X
t
j?1
_
, (2)
(e.g. Protter (2004, p. 66–77)) for any deterministic sequence
2
of partitions 0 = t
0
< t
1
< ... <
t
n
= T with sup
j
¦t
j+1
? t
j
¦ ? 0 for n ? ?. The convergence is also locally uniform in time.
It can be shown that this probability limit exists for all semimartingales.
Throughout we employ the notation that
[Y ]
t
= [Y, Y ]
t
,
while we will sometimes refer to
_
[Y
l
]
t
as the quadratic volatility (QVol) process for Y
l
. It is
well known that
3
[Y ] = [Y
ct
] + [Y
d
], where [Y
d
]
t
=
0?u?t
?Y
u
?Y
u
(3)
with ?Y
t
= Y
t
?Y
t?
are the jumps in Y and noting that [A
ct
] = 0. In the probability literature
QV is usually de?ned in a di?erent, but equivalent, manner (see, for example, Protter (2004, p.
66))
[Y ]
t
= Y
t
Y
t
?2
_
t
0
Y
u?
dY
u
. (4)
2.2 Brownian semimartingales
In economics the most familiar semimartingale is the Brownian semimartingale (Y ? Bo/)
Y
t
=
_
t
0
a
u
du +
_
t
0
?
u
dW
u
, (5)
1
It is tempting to use the notation Y
c
for Y
ct
, but in the probability literature if Y ? SM then Y
c
= M
c
, so
Y
c
ignores A
c
.
2
The assumption that the times are deterministic can be relaxed to allow them to be any Riemann sequence
of adapted subdivisions. This is discussed in, for example, Jacod and Shiryaev (2003, p. 51). Economically this
is important for it means that we can also think of the limiting argument as the result of a joint process of Y and
a counting process N whose arrival times are the tj. So long as Y and N are adapted to at least their bivariate
natural ?ltration the limiting argument holds as the intensity of N increases o? to in?nity with n.
3
Although the sum of jumps of Y does not exist in general when Y ? SM, the sum of outer products of the
jumps always does exist. Hence [Y
d
] can be properly de?ned.
3
where a is a vector of predictable drifts, ? is a matrix volatility process whose elements are
c` adl` ag and W is a vector Brownian motion. The stochastic integral ?•W
t
, where f •g
t
is generic
notation for the process
_
t
0
f
u
dg
u
, is said to be a stochastic volatility process (? • W ? o1) —
e.g. the reviews in Ghysels, Harvey, and Renault (1996) and Shephard (2005). This vector
process has elements which are /
c
loc
. Doob (1953) showed that all continuous local martingales
with absolutely continuous quadratic variation can be written in the form of a SV process (see
Karatzas and Shreve (1991, p. 170–172))
4
. The drift
_
t
0
a
u
du has elements which are absolutely
continuous — an assumption which looks ad hoc, however arbitrage freeness plus the SV model
implies this property must hold (Karatzas and Shreve (1998, p. 3) and Andersen, Bollerslev,
Diebold, and Labys (2003, p. 583)). Hence Y ? Bo/ is a rather canonical model in the ?nance
theory of continuous sample path processes. Its use is bolstered by the facts that Ito calculus
for continuous sample path processes is relatively simple.
If Y ? Bo/ then
[Y ]
t
=
_
t
0
?
u
du
the integrated covariance process, while
dY
t
[T
t
? N (a
t
dt, ?
t
dt) , where ?
t
= ?
t
?
t
, (6)
where T
t
is the natural ?ltration – that is the information from the entire sample path of Y up
to time t. Thus a
t
dt and ?
t
dt have clear interpretations as the in?nitesimal predictive mean
and covariance of asset returns. This implies that A
t
=
_
t
0
E(dY
u
[T
u
) du while, centrally to our
interests,
d[Y ]
t
= Cov (dY
t
[T
t
) and [Y ]
t
=
_
t
0
Cov (dY
u
[T
u
) du.
Thus A and [Y ] are the integrated in?nitesimal predictive mean and covariance of the asset
prices, respectively.
2.3 Jump processes
There is no plausible economic theory which says that prices must follow continuous sample path
processes. Indeed we will see later that statistically it is rather easy to reject this hypothesis even
for price processes drawn from very thickly traded markets. In this paper we will add a ?nite
activity jump process (this means there are a ?nite number of jumps in a ?xed time interval)
J
t
=
Nt
j=1
C
j
, adapted to the ?ltration generated by Y , to the Brownian semimartingale model.
4
An example of a continuous local martingale which has no SV representation is a time-change Brownian
motion where the time-change takes the form of the so-called “devil’s staircase,” which is continuous and non-
decreasing but not absolutely continuous (see, for example, Munroe (1953, Section 27)). This relates to the work
of, for example, Calvet and Fisher (2002) on multifractals.
4
This yields
Y
t
=
_
t
0
a
u
du +
_
t
0
?
u
dW
u
+
Nt
j=1
C
j
. (7)
Here N is a simple counting process and the C are the associated non-zero jumps (which we
assume have a covariance) which happen at times 0 = ?
0
< ?
1
< ?
2
< ... . It is helpful to
decompose J into J = J
A
+ J
M
, where, assuming J has an absolutely continuous intensity,
J
A
t
=
_
t
0
c
u
du, and c
t
= E(dJ
t
[T
t
). Then J
M
is the compensated jump process, so J
M
? /,
while J
A
? T1
ct
loc
. Thus Y has the decomposition as in (1), with A
t
=
_
t
0
(a
u
+c
u
) du and
M
t
=
_
t
0
?
u
dW
u
+
Nt
j=1
C
j
?
_
t
0
c
u
du.
It is easy to see that [Y
d
]
t
=
Nt
j=1
C
j
C
j
and so
[Y ]
t
=
_
t
0
?
u
du +
Nt
j=1
C
j
C
j
.
Again we note that E(dY
t
[T
t
) = (a
t
+c
t
) dt, but now,
Cov (?
t
dW
t
, dJ
t
[T
t
) = 0, (8)
so
Cov (dY
t
[T
t
) = ?
t
dt + Cov (dJ
t
[T
t
) ,= d[Y ]
t
.
This means that the QV process aggregates the components of the variation of prices and so is
not su?cient to learn the integrated covariance process.
To identify the components of the QV process we can use the bipower variation (BPV)
process introduced by Barndor?-Nielsen and Shephard (2006). So long as it exists, the p p
matrix BPV process ¦Y ¦ has l, k-th element
_
Y
l
, Y
k
_
=
1
4
__
Y
l
+Y
k
_
?
_
Y
l
?Y
k
__
, l, k, = 1, 2, ..., p, (9)
where, so long as the limit exists and the convergence is locally uniform in t,
5
_
Y
l
_
t
= p?lim
??0
t/?
j=1
¸
¸
¸Y
l
?(j?1)
?Y
l
?(j?2)
¸
¸
¸
¸
¸
¸Y
l
?j
?Y
l
?(j?1)
¸
¸
¸ . (10)
5
In order to simplify some of the later results we consistently ignore end e?ects in variation statistics. This can
be justi?ed in two ways, either by (a) setting Yt = 0 for t < 0, (b) letting Y start being a semimartingale at zero
at time C < 0 not at time 0. The latter seems realistic when dealing with markets open 24 hours a day, borrowing
returns from small periods of the previous day. It means that there is a modest degree of wash over from one
days variation statistics into the next day. There seems little econometric reasons why this should be a worry.
Assumption (b) can also be used in equity markets when combined with some form of stochastic imputation,
adding in arti?cal simulated returns for the missing period — see the related comments in Barndor?-Nielsen and
Shephard (2002).
5
Here ¸x| is the ?oor function, which is the largest integer less than or equal to x. Combining
the results in Barndor?-Nielsen and Shephard (2006) and Barndor?-Nielsen, Graversen, Jacod,
Podolskij, and Shephard (2005) if Y is the form of (7) then, without any additional assumptions,
µ
?2
1
¦Y ¦
t
=
_
t
0
?
u
du,
where µ
r
= E[U[
r
, U ? N(0, 1) and r > 0, which means that
[Y ]
t
?µ
?2
1
¦Y ¦
t
=
Nt
j=1
C
j
C
j
.
At ?rst sight the robustness of BPV looks rather magical, but it is a consequence of the fact
that only a ?nite number of terms in the sum (10) are a?ected by jumps, while each return
which does not have a jump goes to zero in probability. Therefore, since the probability of
jumps in contiguous time intervals goes to zero as ? ? 0, those terms which do include jumps do
not impact the probability limit. The extension of this result to the case where J is an in?nite
activity jump process is discussed in Section 4.4.
2.4 Forecasting
Suppose Y obeys (7) and introduce the generic notation
y
t+s,t
= Y
t+s
?Y
t
= a
t+s,t
+m
t+s,t
, t, s > 0.
So long as the covariance exists,
Cov (y
t+s,t
[T
t
) = Cov (a
t+s,t
[T
t
) + Cov (m
t+s,t
[T
t
)
+Cov (a
t+s,t
, m
t+s,t
[T
t
) + Cov (m
t+s,t
, a
t+s,t
, [T
t
) .
Notice how complicated this expression is compared to the covariance in (6), which is due to the
fact that s is not necessarily dt and so a
t+s,t
is no longer known given T
t
— while
_
t+dt
t
a
u
du was.
However, in all likelihood for small s, a makes a rather modest contribution to the predictive
covariance of Y .
This suggests using the approximation that
Cov (y
t+s,t
[T
t
) · Cov (m
t+s,t
[T
t
) .
Now using (8) so
Cov (m
t+s,t
[T
t
) = E([Y ]
t+s
?[Y ]
t
[T
t
) ?E
_
__
t+s
t
c
u
du
___
t+s
t
c
u
du
_
[T
t
_
.
6
Hence if c or s is small then we might approximate
Cov (Y
t+s
?Y
t
[T
t
) · E([Y ]
t+s
?[Y ]
t
[T
t
)
= E([? • W]
t+s
?[? • W]
t
[T
t
) + E([J]
t+s
?[J]
t
[T
t
) .
Thus an interesting forecasting strategy for covariances is to forecast the increments of the QV
process or its components. As the QV process and its components are themselves estimable,
though with substantial possible error, this is feasible. This approach to forecasting has been
advocated in a series of in?uential papers by Andersen, Bollerslev, Diebold, and Labys (2001),
Andersen, Bollerslev, Diebold, and Ebens (2001) and Andersen, Bollerslev, Diebold, and Labys
(2003), while the important earlier paper by Andersen and Bollerslev (1998a) was stimulating
in the context of measuring the forecast performance of GARCH models. The use of forecast-
ing using estimates of the increments of the components of QV was introduced by Andersen,
Bollerslev, and Diebold (2003). We will return to it in section 3.9 when we have developed an
asymptotic theory for estimating the QV process and its components.
2.5 Realised QV & BPV
The QV process can be estimated in many di?erent ways. The most immediate is the realised
QV estimator
[Y
?
]
t
=
t/?
j=1
_
Y
j?
?Y
(j?1)?
_ _
Y
j?
?Y
(j?1)?
_
,
where ? > 0. This is the outer product of returns computed over a ?xed interval of time of
length ?. By construction, as ? ? 0, [Y
?
]
t
p
? [Y ]
t
. Likewise
_
Y
l
?
_
t
=
t/?
j=1
¸
¸
¸Y
l
?(j?1)
?Y
l
?(j?2)
¸
¸
¸
¸
¸
¸Y
l
?j
?Y
l
?(j?1)
¸
¸
¸ , l = 1, 2, ..., p, (11)
_
Y
l
?
, Y
k
?
_
=
1
4
__
Y
l
?
+Y
k
?
_
?
_
Y
l
?
?Y
k
?
__
and ¦Y
?
¦
p
? ¦Y ¦.
In practice, the presence of market frictions can potentially mean that this limiting argument
is not really available as an accurate guide to the behaviour of these statistics for small ?. Such
di?culties with limiting arguments, which are present in almost all areas of econometrics and
statistics, do not invalidate the use of asymptotics, for it is used to provide predictions about
?nite sample behaviour. Probability limits are, of course, coarse and we will respond to this by
re?ning our understanding by developing central limit theorems and hope they will make good
predictions when ? is moderately small. For very small ? these asymptotic predictions become
poor guides as frictions bite hard and this will be discussed in section 5.
7
In ?nancial econometrics the focus is often on the increments of the QV and realised QV
over set time intervals, like one day. Let us de?ne the daily QV
V
i
= [Y ]
hi
?[Y ]
h(i?1)
, i = 1, 2, ...
while it is estimated by the realised daily QV
´
V
i
= [Y
?
]
hi
?[Y
?
]
h(i?1)
, i = 1, 2, ....
Clearly
´
V
i
p
? V
i
as ? ? 0. The l-th diagonal element of
´
V
i
, written
´
V
l,l
i
is called the realised
variance
6
of asset l, while its square root is its realised volatility. The latter estimates the
_
V
l,l
i
,
the daily QVol process of asset l. The l, k-th element of
´
V
i
,
´
V
l,k
i
, is called the realised covariance
between assets l and k. O? these objects we can de?ne standard dependence measures, like
realised regression
´
?
l,k
i
=
´
V
l,k
i
´
V
k,k
i
p
? ?
l,k
i
=
V
l,k
i
V
k,k
i
,
which estimates the QV regression and the realised correlation
´?
l,k
i
=
´
V
l,k
i
_
´
V
l,l
i
´
V
k,k
i
p
? ?
l,k
i
=
V
l,k
i
_
V
l,l
i
V
k,k
i
,
which estimates the QV correlation. Similar daily objects can be calculated o? the realised BPV
process
´
B
i
= µ
?2
1
_
¦Y
?
¦
hi
?¦Y
?
¦
h(i?1)
_
, i = 1, 2, ...
which estimates
B
i
=
_
Y
ct
¸
hi
?
_
Y
ct
¸
h(i?1)
=
_
hi
h(i?1)
?
2
u
du, i = 1, 2, ...
Realised volatility has a very long history in ?nancial economics. It appears in, for exam-
ple, Rosenberg (1972), O?cer (1973), Merton (1980), French, Schwert, and Stambaugh (1987),
Schwert (1989) and Schwert (1998), with Merton (1980) making the implicit connection with the
case where ? ? 0 in the pure scaled Brownian motion plus drift case. Of course, in probability
theory QV was discussed as early as Wiener (1924) and L´evy (1937) and appears as a crucial
object in the development of the stochastic analysis of semimartingales which occurred in the
second half of the last century. For more general ?nancial processes a closer connection between
realised QV and QV, and its use for econometric purposes, was made in a series of independent
and concurrent papers by Comte and Renault (1998), Barndor?-Nielsen and Shephard (2001)
and Andersen, Bollerslev, Diebold, and Labys (2001). The realised regressions and correlations
6
Some authors call
V
l,l
i
the realised volatility, but throughout this paper we follow the tradition in ?nance of
using volatility to mean standard deviation type objects.
8
were de?ned and studied in detail by Andersen, Bollerslev, Diebold, and Labys (2003) and
Barndor?-Nielsen and Shephard (2004).
A major motivation for Barndor?-Nielsen and Shephard (2002) and Andersen, Bollerslev,
Diebold, and Labys (2001) was the fact that volatility in ?nancial markets is highly and unstably
diurnal within a day, responding to regularly timed macroeconomic news announcements, social
norms such as lunch times and sleeping or the opening of other markets. This makes estimating
lim
??0
_
[Y ]
t+?
?[Y ]
t
_
/?
extremely di?cult. The very stimulating work of Genon-Catalot, Lar´edo, and Picard (1992),
Foster and Nelson (1996), Mykland and Zhang (2002) and Mykland and Zhang (2005) tries to
tackle this problem using a double asymptotics, as ? ? 0 and ? ? 0. However, in the last ?ve years
many econometrics researchers have mostly focused on naturally diurnally robust quantities like
the daily or weekly QV.
2.6 Changes in probability law, QV and BPV
The observed price process Y ? o/, governed by its data generating process or measure P,
is not uniquely interesting. In ?nancial economics the stochastic behaviour of Y under risk
neutral versions P
?
(i.e. so called equivalent martingale measures) are also important for they
determine the price of contingent assets based on Y . An interesting question is whether [Y ]
computed under P tells us anything about the behaviour of [Y ] under P
?
. To discuss this recall
that the notation P
?
0.
Building on the earlier CLT of Barndor?-Nielsen and Shephard (2006), Barndor?-Nielsen,
Graversen, Jacod, Podolskij, and Shephard (2005) have established a CLT which covers this
situation when Y ? Bo/. We will only present the univariate result, which has that as ? ? 0
so
?
?1/2
(¦Y
?
¦
t
?¦Y ¦
t
) ? µ
2
1
_
(2 +?)
_
t
0
?
2
u
dB
u
, (31)
where B ?? Y , the convergence is in law stable as a process and
? =
_
?
2
/4
_
+? ?5 · 0.6090.
This result, unlike Theorem 1, has some quite technical conditions associated with it in order to
control the degree to which the volatility process can jump; however we will not discuss those
issues here. Extending the result to cover the joint distribution of the estimators of the QV and
the BPV processes, they showed that
?
?1/2
_
µ
?2
1
¦Y
?
¦
t
?µ
?2
1
¦Y ¦
t
[Y
?
]
t
?[Y ]
t
_
L
? MN
__
0
0
_
,
_
(2 +?) 2
2 2
__
t
0
?
4
u
du
_
,
a Hausman (1978) type result as the estimator of the QV process is, of course, fully asymptoti-
cally e?cient when Y ? Bo/. Consequently
?
?1/2
_
[Y
?
]
t
?µ
?2
1
¦Y
?
¦
t
_
¸
?
_
t
0
?
4
u
du
L
? N (0, 1) , (32)
which can be used as the basis of a test of the null of no jumps.
4.2 Multipower variation
The “standard” estimator of integrated quarticity, given in (18), is not robust to jumps. One
way of overcoming this problem is to use a multipower variation (MPV) measure — introduced
by Barndor?-Nielsen and Shephard (2006). This is de?ned as
¦Y ¦
[r]
t
= p?lim
??0
?
(1?r
+
/2)
t/?
j=1
_
I
i=1
¸
¸
Y
?(j?i)
?Y
?(j?1?i)
¸
¸
r
i
_
,
where r
i
> 0, r = (r
1
, r
2
, ..., r
I
)
for all i and r
+
=
I
i=1
r
i
. The usual BPV process is the special
case ¦Y ¦
t
= ¦Y ¦
[1,1]
t
.
If Y obeys (7) and r
i
< 2 then
¦Y ¦
[r]
t
=
_
I
i=1
µ
r
i
_
_
t
0
?
r
+
u
du,
32
This process is approximated by the estimated MPV process
¦Y
?
¦
[r]
t
= ?
(1?r
+
/2)
t/?
j=1
_
I
i=1
¸
¸
Y
?(j?i)
?Y
?(j?1?i)
¸
¸
r
i
_
.
In particular the scaled realised tri and quadpower variation,
µ
?4
1
¦Y
?
¦
[1,1,1,1]
t
and µ
?3
4/3
¦Y
?
¦
[4/3,4/3,4/3]
t
,
respectively, both estimate
_
t
0
?
4
u
du consistently in the presence of jumps. Hence either of these
objects can be used to replace the integrated quarticity in (32), so producing a non-parametric
test for the presence of jumps in the interval [0, t]. The test is conditionally consistent, meaning
if there is a jump, it will detected and has asymptotically the correct size. Extensive small
sample studies are reported in Huang and Tauchen (2005), who favour ratio versions of the
statistic like
?
?1/2
_
µ
?2
1
¦Y
?
¦
t
[Y
?
]
t
?1
_
¸
¸
¸
_
?
¦Y
?
¦
[1,1,1,1]
t
(¦Y
?
¦
t
)
2
L
? N (0, 1) ,
which has pretty reasonable ?nite sample properties. They also show that this test tends to
under reject the null of no jumps in the presence of some forms of market frictions.
It is clearly possible to carry out jump testing on separate days or weeks. Such tests are
asymptotically independent over these non-overlapping periods under the null hypothesis.
To illustrate this methodology we will apply the jump test to the DM/Dollar rate, asking
if the hypothesis of a continuous sample path is consistent with the data we have. Our focus
will mostly be on Friday January 15th 1988, although we will also give results for neighbouring
days to provide some context. In Figure 5 we plot 100 times the change during the week of the
discretised Y
?
, so a one unit uptick represents a 1% change, for a variety of values of n = 1/?,
as well as giving the ratio jump statistics
´
B
i
/
´
V
i
with their corresponding 99% critical values.
In Figure 5 there is a large uptick in the D-mark against the Dollar, with a movement of
nearly two percent in a ?ve minute period. This occurred on the Friday and was a response to
the news of a large fall in the U.S. balance of payment de?cit, which led to a large strengthening
of the Dollar. The data for January 15th had a large
´
V
i
but a much smaller
´
B
i
. Hence the
statistics are attributing a large component of
´
V
i
to the jump, with the adjusted ratio statistic
being larger than the corresponding 99% critical value. When ? is large the statistic is on the
borderline of being signi?cant, while the situation becomes much clearer as ? becomes small.
This illustration is typical of results presented in Barndor?-Nielsen and Shephard (2006) which
showed that many of the large jumps in this exchange rate correspond to macroeconomic news
33
0
1
2
3
4
Change in Y
?
during week
using ?=20 minutes
Mon Fri Thurs Wed Tues
0.50
0.75
1.00
Ratio jump statistic and 99% critical values
Mon Tues Wed Thurs
Fri
(µ
1
)
?2 ^
B
i
/
^
V
i
99% critical values
0
1
2
3
4
Change in Y
?
during week
using ?=5 minutes
Mon Tues Wed
Thurs Fri
0.25
0.50
0.75
1.00
Ratio jump statistic and 99% critical values
Mon Tues
Wed
Thurs
Fri
(µ
1
)
?2 ^
B
i
/
^
V
i
99% critical values
Figure 5: Left hand side: change in Y
?
during a week, centred at 0 on Monday 11th January
and running until Friday of that week. Drawn every 20 and 5 minutes. An up tick of 1 indicates
strengthening of the Dollar by 1%. Right hand side shows an index plot of
´
B
i
/
´
V
i
, which should
be around 1 if there are no jumps. Test is one sided, with criticial values also drawn as a line.
announcements. This is consistent with the recent economics literature documenting signi?cant
intraday announcement e?ects, e.g. Andersen, Bollerslev, Diebold, and Vega (2003).
4.3 Grids
It is clear that the martingale based CLT for irregularly spaced data for the estimator of the
QV process can be extended to cover the BPV case. We de?ne
¦Y
Gn
¦
t
=
t
j
?t
j=1
¸
¸
Y
t
j?1
?Y
t
j?2
¸
¸
¸
¸
Y
t
j
?Y
t
j?1
¸
¸
p
? ¦Y ¦
t
.
Using the same notation as before, we would expect the following result to hold, due to the fact
that H
G
is assumed to be continuous,
?
?1/2
_
µ
?2
1
¦Y
Gn
¦
t
?µ
?2
1
¦Y ¦
t
[Y
?
]
t
?[Y ]
t
_
L
? MN
__
0
0
__
(2 +?) 2
2 2
__
t
0
_
?H
G
u
?u
_
?
4
u
du
_
.
34
The integrated moderated quarticity can be estimated using µ
?4
1
¦Y
?
¦
[1,1,1,1]
t
, or a grid version,
which again implies that the usual feasible CLT continues to hold for irregularly spaced data.
This is the expected result from the analysis of power variation provided by Barndor?-Nielsen
and Shephard (2005c).
Potentially there are modest e?ciency gains to be had by computing the estimators of BPV
on multiple grids and then averaging them. The extension along these lines is straightforward
and will not be detailed here.
4.4 In?nite activity jumps
The probability limit of realised BPV is robust to ?nite activity jumps. A natural question to
ask is: (i) is the CLT also robust to jumps, (ii) is the probability limit also una?ected by in?nite
activity jumps, that is jump processes with an in?nite number of jumps in any ?nite period of
time. Both issues are studied by Barndor?-Nielsen, Shephard, and Winkel (2004) in the case
where the jumps are of L´evy type, while Woerner (2004) looks at the probability limit for more
general jump processes.
Barndor?-Nielsen, Shephard, and Winkel (2004) ?nd that the CLT for BPV is a?ected by
?nite activity jumps, but this is not true of tripower and high order measures of variation. The
reason for the robustness of tripower results is quite technical and we will not discuss it here.
However, it potentially means that inference under the assumption of jumps can be carried out
using tripower variation, which seems an exciting possibility. Both Barndor?-Nielsen, Shephard,
and Winkel (2004) and Woerner (2004) give results which prove that the probability limit of
realised BPV is una?ected by some types of in?nite activity jump processes. More work is
needed on this topic to make these result de?nitive. It is somewhat related to the parametric
study of A¨?t-Sahalia (2004). He shows that maximum likelihood estimation can disentangle a
homoskedastic di?usive component from a purely discontinuous in?nite activity L´evy component
of prices. Outside the likelihood framework, the paper also studies the optimal combinations
of moment functions for the generalized method of moment estimation of homoskedastic jump-
di?usions. Further insights can be found by looking at likelihood inference for L´evy processes,
which is studied by A¨?t-Sahalia and Jacod (2005a) and A¨?t-Sahalia and Jacod (2005b).
4.5 Testing the null of no continuous component
In some stimulating recent papers, Carr, Geman, Madan, and Yor (2003) and Carr and Wu
(2004), have argued that it is attractive to build SV models out of pure jump processes, with
no Brownian aspect. This is somewhat related to the material we discuss in section 5.6. It is
clearly important to be able to test this hypothesis, seeing if pure discreteness is consistent with
35
observed prices.
Barndor?-Nielsen, Shephard, and Winkel (2004) showed that
?
?1/2
_
¦Y
?
¦
[2/3,2/3,2/3]
t
?[Y
ct
]
t
_
has a mixed Gaussian limit and is robust to jumps. But this result is only valid if ? > 0, which
rules out its use for testing for pure discreteness. However, we can arti?cially add a scaled
Brownian motion, U = ?B, to the observed price process and then test if
?
?1/2
_
¦Y
?
+U
?
¦
[2/3,2/3,2/3]
t
??
2
t
_
is statistically signi?cantly greater than zero. In principle this would be a consistent non-
parametric test of the maintained hypothesis of Peter Carr and his coauthors.
4.6 Alternative methods for identifying jumps
Mancini (2001), Mancini (2004) and Mancini (2003) has developed robust estimators of
_
Y
ct
¸
in the presence of ?nite activity jumps. Her approach is to use truncation
t/?
j=1
_
Y
j?
?Y
(j?1)?
_
2
I(
¸
¸
Y
j?
?Y
(j?1)?
¸
¸
< r
?
), (33)
where I (.) is an indicator function. The crucial function r
?
has to have the property that
_
? log ?
?1
r
?1
?
? 0. It is motivated by the modulus of continuity of Brownian motion paths that
almost surely
lim
??0
sup
0?s,t?T
|t?s|
We will review the econometrics of non-parametric estimation of the components of the variation of asset prices.
Variation, jumps, market frictions and
high frequency data in ?nancial econometrics
?
Ole E. Barndorff-Nielsen
The T.N. Thiele Centre for Mathematics in Natural Science,
Department of Mathematical Sciences,
University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark
[email protected]
Neil Shephard
Nu?eld College, Oxford OX1 1NF, UK
[email protected]
July 14, 2005
1 Introduction
We will review the econometrics of non-parametric estimation of the components of the variation
of asset prices. This very active literature has been stimulated by the recent advent of complete
records of transaction prices, quote data and order books. In our view the interaction of the
new data sources with new econometric methodology is leading to a paradigm shift in one of
the most important areas in econometrics: volatility measurement, modelling and forecasting.
We will describe this new paradigm which draws together econometrics with arbitrage free
?nancial economics theory. Perhaps the two most in?uential papers in this area have been
Andersen, Bollerslev, Diebold, and Labys (2001) and Barndor?-Nielsen and Shephard (2002),
but many other papers have made important contributions. This work is likely to have deep
impacts on the econometrics of asset allocation and risk management. One of our observations
will be that inferences based on these methods, computed from observed market prices and so
under the physical measure, are also valid as inferences under all equivalent measures. This puts
this subject also at the heart of the econometrics of derivative pricing.
?
Prepared for the invited symposium on Financial Econometrics, 9th World Congress of the Econometric
Society, London, 20th August 2005. We are grateful to Tim Bollerslev, Eric Ghysels, Peter Hansen, Jean Jacod,
Dmitry Kulikov, Jeremy Large, Asger Lunde, Andrew Patton, Mark Podolskij, Kevin Sheppard and Jun Yu for
comments on an earlier draft. Talks based on this paper were also given in 2005 as the Hiemstra Lecture at the
13th Annual conference of the Society of Non-linear Dynamics and Econometrics in London, the keynote address
at the 3rd Nordic Econometric Meeting in Helsinki and as a Special Invited Lecture at the 25th European Meeting
of Statisticians in Oslo. Ole Barndor?-Nielsen’s work is supported by CAF (www.caf.dk), which is funded by the
Danish Social Science Research Council. Neil Shephard’s research is supported by the UK’s ESRC through the
grant “High frequency ?nancial econometrics based upon power variation.”
1
One of the most challenging problems in this context is dealing with various forms of market
frictions, which obscure the e?cient price from the econometrician. Here we will characterise four
types of statistical models of frictions and discuss how econometricians have been attempting
to overcome them.
In section 2 we will set out the basis of the econometrics of arbitrage-free price processes,
focusing on the centrality of quadratic variation. In section 3 we will discuss central limit
theorems for estimators of the QV process, while in section 4 the role of jumps in QV will be
highlighted, with bipower and multipower variation being used to identify them and to test
the hypothesis that there are no jumps in the price process. In section 5 we write about the
econometrics of market frictions, while in section 6 we conclude.
2 Arbitrage-free, frictionless price processes
2.1 Semimartingales and quadratic variation
Given a complete record of transaction or quote prices it is natural to model prices in contin-
uous time (e.g. Engle (2000)). This matches with the vast continuous time ?nancial economic
arbitrage-free theory based on a frictionless market. In this section and the next, we will dis-
cuss how to make inferences on the degree of variation in such frictionless worlds. Section 5
will extend this by characterising the types of frictions seen in practice and discuss strategies
econometricians have been using to overcome these di?culties.
In its most general case the fundamental theory of asset prices says that a vector of log-prices
at time t,
Y
t
=
_
Y
1
t
, ..., Y
p
t
_
,
must obey a semimartingale process (written Y ? o/) on some ?ltered probability space
_
?, T, (T
t
)
t?0
, P
_
in a frictionless market. The semimartingale is de?ned as being a process
which can be written as
Y = A +M, (1)
where A is a local ?nite variation process (A ? T1
loc
) and M is a local martingale (M ? /
loc
).
Compact introductions to the economics and mathematics of semimartingales are given in Back
(1991) and Protter (2004), respectively.
The Y process can exhibit jumps. It is tempting to decompose Y = Y
ct
+ Y
d
, where Y
ct
and Y
d
are the purely continuous and discontinuous sample path components of Y . However,
technically this de?nition is not clear as the jumps of the Y process can be so active that they
2
cannot be summed up. Thus we will de?ne
Y
ct
= A
c
+M
c
,
where M
c
is the continuous part of the local martingale component of Y and A
c
is A minus the
sum of its jumps
1
. Likewise, the continuous sample path subsets of classes of processes such as
o/ and /, will be denoted by o/
c
and /
c
.
Crucial to semimartingales, and to the economics of ?nancial risk, is the quadratic variation
(QV) process of (Y
, X
)
? o/. This can be de?ned as
[Y, X]
t
= p?lim
n??
t
j
?t
j=1
_
Y
t
j
?Y
t
j?1
_ _
X
t
j
?X
t
j?1
_
, (2)
(e.g. Protter (2004, p. 66–77)) for any deterministic sequence
2
of partitions 0 = t
0
< t
1
< ... <
t
n
= T with sup
j
¦t
j+1
? t
j
¦ ? 0 for n ? ?. The convergence is also locally uniform in time.
It can be shown that this probability limit exists for all semimartingales.
Throughout we employ the notation that
[Y ]
t
= [Y, Y ]
t
,
while we will sometimes refer to
_
[Y
l
]
t
as the quadratic volatility (QVol) process for Y
l
. It is
well known that
3
[Y ] = [Y
ct
] + [Y
d
], where [Y
d
]
t
=
0?u?t
?Y
u
?Y
u
(3)
with ?Y
t
= Y
t
?Y
t?
are the jumps in Y and noting that [A
ct
] = 0. In the probability literature
QV is usually de?ned in a di?erent, but equivalent, manner (see, for example, Protter (2004, p.
66))
[Y ]
t
= Y
t
Y
t
?2
_
t
0
Y
u?
dY
u
. (4)
2.2 Brownian semimartingales
In economics the most familiar semimartingale is the Brownian semimartingale (Y ? Bo/)
Y
t
=
_
t
0
a
u
du +
_
t
0
?
u
dW
u
, (5)
1
It is tempting to use the notation Y
c
for Y
ct
, but in the probability literature if Y ? SM then Y
c
= M
c
, so
Y
c
ignores A
c
.
2
The assumption that the times are deterministic can be relaxed to allow them to be any Riemann sequence
of adapted subdivisions. This is discussed in, for example, Jacod and Shiryaev (2003, p. 51). Economically this
is important for it means that we can also think of the limiting argument as the result of a joint process of Y and
a counting process N whose arrival times are the tj. So long as Y and N are adapted to at least their bivariate
natural ?ltration the limiting argument holds as the intensity of N increases o? to in?nity with n.
3
Although the sum of jumps of Y does not exist in general when Y ? SM, the sum of outer products of the
jumps always does exist. Hence [Y
d
] can be properly de?ned.
3
where a is a vector of predictable drifts, ? is a matrix volatility process whose elements are
c` adl` ag and W is a vector Brownian motion. The stochastic integral ?•W
t
, where f •g
t
is generic
notation for the process
_
t
0
f
u
dg
u
, is said to be a stochastic volatility process (? • W ? o1) —
e.g. the reviews in Ghysels, Harvey, and Renault (1996) and Shephard (2005). This vector
process has elements which are /
c
loc
. Doob (1953) showed that all continuous local martingales
with absolutely continuous quadratic variation can be written in the form of a SV process (see
Karatzas and Shreve (1991, p. 170–172))
4
. The drift
_
t
0
a
u
du has elements which are absolutely
continuous — an assumption which looks ad hoc, however arbitrage freeness plus the SV model
implies this property must hold (Karatzas and Shreve (1998, p. 3) and Andersen, Bollerslev,
Diebold, and Labys (2003, p. 583)). Hence Y ? Bo/ is a rather canonical model in the ?nance
theory of continuous sample path processes. Its use is bolstered by the facts that Ito calculus
for continuous sample path processes is relatively simple.
If Y ? Bo/ then
[Y ]
t
=
_
t
0
?
u
du
the integrated covariance process, while
dY
t
[T
t
? N (a
t
dt, ?
t
dt) , where ?
t
= ?
t
?
t
, (6)
where T
t
is the natural ?ltration – that is the information from the entire sample path of Y up
to time t. Thus a
t
dt and ?
t
dt have clear interpretations as the in?nitesimal predictive mean
and covariance of asset returns. This implies that A
t
=
_
t
0
E(dY
u
[T
u
) du while, centrally to our
interests,
d[Y ]
t
= Cov (dY
t
[T
t
) and [Y ]
t
=
_
t
0
Cov (dY
u
[T
u
) du.
Thus A and [Y ] are the integrated in?nitesimal predictive mean and covariance of the asset
prices, respectively.
2.3 Jump processes
There is no plausible economic theory which says that prices must follow continuous sample path
processes. Indeed we will see later that statistically it is rather easy to reject this hypothesis even
for price processes drawn from very thickly traded markets. In this paper we will add a ?nite
activity jump process (this means there are a ?nite number of jumps in a ?xed time interval)
J
t
=
Nt
j=1
C
j
, adapted to the ?ltration generated by Y , to the Brownian semimartingale model.
4
An example of a continuous local martingale which has no SV representation is a time-change Brownian
motion where the time-change takes the form of the so-called “devil’s staircase,” which is continuous and non-
decreasing but not absolutely continuous (see, for example, Munroe (1953, Section 27)). This relates to the work
of, for example, Calvet and Fisher (2002) on multifractals.
4
This yields
Y
t
=
_
t
0
a
u
du +
_
t
0
?
u
dW
u
+
Nt
j=1
C
j
. (7)
Here N is a simple counting process and the C are the associated non-zero jumps (which we
assume have a covariance) which happen at times 0 = ?
0
< ?
1
< ?
2
< ... . It is helpful to
decompose J into J = J
A
+ J
M
, where, assuming J has an absolutely continuous intensity,
J
A
t
=
_
t
0
c
u
du, and c
t
= E(dJ
t
[T
t
). Then J
M
is the compensated jump process, so J
M
? /,
while J
A
? T1
ct
loc
. Thus Y has the decomposition as in (1), with A
t
=
_
t
0
(a
u
+c
u
) du and
M
t
=
_
t
0
?
u
dW
u
+
Nt
j=1
C
j
?
_
t
0
c
u
du.
It is easy to see that [Y
d
]
t
=
Nt
j=1
C
j
C
j
and so
[Y ]
t
=
_
t
0
?
u
du +
Nt
j=1
C
j
C
j
.
Again we note that E(dY
t
[T
t
) = (a
t
+c
t
) dt, but now,
Cov (?
t
dW
t
, dJ
t
[T
t
) = 0, (8)
so
Cov (dY
t
[T
t
) = ?
t
dt + Cov (dJ
t
[T
t
) ,= d[Y ]
t
.
This means that the QV process aggregates the components of the variation of prices and so is
not su?cient to learn the integrated covariance process.
To identify the components of the QV process we can use the bipower variation (BPV)
process introduced by Barndor?-Nielsen and Shephard (2006). So long as it exists, the p p
matrix BPV process ¦Y ¦ has l, k-th element
_
Y
l
, Y
k
_
=
1
4
__
Y
l
+Y
k
_
?
_
Y
l
?Y
k
__
, l, k, = 1, 2, ..., p, (9)
where, so long as the limit exists and the convergence is locally uniform in t,
5
_
Y
l
_
t
= p?lim
??0
t/?
j=1
¸
¸
¸Y
l
?(j?1)
?Y
l
?(j?2)
¸
¸
¸
¸
¸
¸Y
l
?j
?Y
l
?(j?1)
¸
¸
¸ . (10)
5
In order to simplify some of the later results we consistently ignore end e?ects in variation statistics. This can
be justi?ed in two ways, either by (a) setting Yt = 0 for t < 0, (b) letting Y start being a semimartingale at zero
at time C < 0 not at time 0. The latter seems realistic when dealing with markets open 24 hours a day, borrowing
returns from small periods of the previous day. It means that there is a modest degree of wash over from one
days variation statistics into the next day. There seems little econometric reasons why this should be a worry.
Assumption (b) can also be used in equity markets when combined with some form of stochastic imputation,
adding in arti?cal simulated returns for the missing period — see the related comments in Barndor?-Nielsen and
Shephard (2002).
5
Here ¸x| is the ?oor function, which is the largest integer less than or equal to x. Combining
the results in Barndor?-Nielsen and Shephard (2006) and Barndor?-Nielsen, Graversen, Jacod,
Podolskij, and Shephard (2005) if Y is the form of (7) then, without any additional assumptions,
µ
?2
1
¦Y ¦
t
=
_
t
0
?
u
du,
where µ
r
= E[U[
r
, U ? N(0, 1) and r > 0, which means that
[Y ]
t
?µ
?2
1
¦Y ¦
t
=
Nt
j=1
C
j
C
j
.
At ?rst sight the robustness of BPV looks rather magical, but it is a consequence of the fact
that only a ?nite number of terms in the sum (10) are a?ected by jumps, while each return
which does not have a jump goes to zero in probability. Therefore, since the probability of
jumps in contiguous time intervals goes to zero as ? ? 0, those terms which do include jumps do
not impact the probability limit. The extension of this result to the case where J is an in?nite
activity jump process is discussed in Section 4.4.
2.4 Forecasting
Suppose Y obeys (7) and introduce the generic notation
y
t+s,t
= Y
t+s
?Y
t
= a
t+s,t
+m
t+s,t
, t, s > 0.
So long as the covariance exists,
Cov (y
t+s,t
[T
t
) = Cov (a
t+s,t
[T
t
) + Cov (m
t+s,t
[T
t
)
+Cov (a
t+s,t
, m
t+s,t
[T
t
) + Cov (m
t+s,t
, a
t+s,t
, [T
t
) .
Notice how complicated this expression is compared to the covariance in (6), which is due to the
fact that s is not necessarily dt and so a
t+s,t
is no longer known given T
t
— while
_
t+dt
t
a
u
du was.
However, in all likelihood for small s, a makes a rather modest contribution to the predictive
covariance of Y .
This suggests using the approximation that
Cov (y
t+s,t
[T
t
) · Cov (m
t+s,t
[T
t
) .
Now using (8) so
Cov (m
t+s,t
[T
t
) = E([Y ]
t+s
?[Y ]
t
[T
t
) ?E
_
__
t+s
t
c
u
du
___
t+s
t
c
u
du
_
[T
t
_
.
6
Hence if c or s is small then we might approximate
Cov (Y
t+s
?Y
t
[T
t
) · E([Y ]
t+s
?[Y ]
t
[T
t
)
= E([? • W]
t+s
?[? • W]
t
[T
t
) + E([J]
t+s
?[J]
t
[T
t
) .
Thus an interesting forecasting strategy for covariances is to forecast the increments of the QV
process or its components. As the QV process and its components are themselves estimable,
though with substantial possible error, this is feasible. This approach to forecasting has been
advocated in a series of in?uential papers by Andersen, Bollerslev, Diebold, and Labys (2001),
Andersen, Bollerslev, Diebold, and Ebens (2001) and Andersen, Bollerslev, Diebold, and Labys
(2003), while the important earlier paper by Andersen and Bollerslev (1998a) was stimulating
in the context of measuring the forecast performance of GARCH models. The use of forecast-
ing using estimates of the increments of the components of QV was introduced by Andersen,
Bollerslev, and Diebold (2003). We will return to it in section 3.9 when we have developed an
asymptotic theory for estimating the QV process and its components.
2.5 Realised QV & BPV
The QV process can be estimated in many di?erent ways. The most immediate is the realised
QV estimator
[Y
?
]
t
=
t/?
j=1
_
Y
j?
?Y
(j?1)?
_ _
Y
j?
?Y
(j?1)?
_
,
where ? > 0. This is the outer product of returns computed over a ?xed interval of time of
length ?. By construction, as ? ? 0, [Y
?
]
t
p
? [Y ]
t
. Likewise
_
Y
l
?
_
t
=
t/?
j=1
¸
¸
¸Y
l
?(j?1)
?Y
l
?(j?2)
¸
¸
¸
¸
¸
¸Y
l
?j
?Y
l
?(j?1)
¸
¸
¸ , l = 1, 2, ..., p, (11)
_
Y
l
?
, Y
k
?
_
=
1
4
__
Y
l
?
+Y
k
?
_
?
_
Y
l
?
?Y
k
?
__
and ¦Y
?
¦
p
? ¦Y ¦.
In practice, the presence of market frictions can potentially mean that this limiting argument
is not really available as an accurate guide to the behaviour of these statistics for small ?. Such
di?culties with limiting arguments, which are present in almost all areas of econometrics and
statistics, do not invalidate the use of asymptotics, for it is used to provide predictions about
?nite sample behaviour. Probability limits are, of course, coarse and we will respond to this by
re?ning our understanding by developing central limit theorems and hope they will make good
predictions when ? is moderately small. For very small ? these asymptotic predictions become
poor guides as frictions bite hard and this will be discussed in section 5.
7
In ?nancial econometrics the focus is often on the increments of the QV and realised QV
over set time intervals, like one day. Let us de?ne the daily QV
V
i
= [Y ]
hi
?[Y ]
h(i?1)
, i = 1, 2, ...
while it is estimated by the realised daily QV
´
V
i
= [Y
?
]
hi
?[Y
?
]
h(i?1)
, i = 1, 2, ....
Clearly
´
V
i
p
? V
i
as ? ? 0. The l-th diagonal element of
´
V
i
, written
´
V
l,l
i
is called the realised
variance
6
of asset l, while its square root is its realised volatility. The latter estimates the
_
V
l,l
i
,
the daily QVol process of asset l. The l, k-th element of
´
V
i
,
´
V
l,k
i
, is called the realised covariance
between assets l and k. O? these objects we can de?ne standard dependence measures, like
realised regression
´
?
l,k
i
=
´
V
l,k
i
´
V
k,k
i
p
? ?
l,k
i
=
V
l,k
i
V
k,k
i
,
which estimates the QV regression and the realised correlation
´?
l,k
i
=
´
V
l,k
i
_
´
V
l,l
i
´
V
k,k
i
p
? ?
l,k
i
=
V
l,k
i
_
V
l,l
i
V
k,k
i
,
which estimates the QV correlation. Similar daily objects can be calculated o? the realised BPV
process
´
B
i
= µ
?2
1
_
¦Y
?
¦
hi
?¦Y
?
¦
h(i?1)
_
, i = 1, 2, ...
which estimates
B
i
=
_
Y
ct
¸
hi
?
_
Y
ct
¸
h(i?1)
=
_
hi
h(i?1)
?
2
u
du, i = 1, 2, ...
Realised volatility has a very long history in ?nancial economics. It appears in, for exam-
ple, Rosenberg (1972), O?cer (1973), Merton (1980), French, Schwert, and Stambaugh (1987),
Schwert (1989) and Schwert (1998), with Merton (1980) making the implicit connection with the
case where ? ? 0 in the pure scaled Brownian motion plus drift case. Of course, in probability
theory QV was discussed as early as Wiener (1924) and L´evy (1937) and appears as a crucial
object in the development of the stochastic analysis of semimartingales which occurred in the
second half of the last century. For more general ?nancial processes a closer connection between
realised QV and QV, and its use for econometric purposes, was made in a series of independent
and concurrent papers by Comte and Renault (1998), Barndor?-Nielsen and Shephard (2001)
and Andersen, Bollerslev, Diebold, and Labys (2001). The realised regressions and correlations
6
Some authors call
V
l,l
i
the realised volatility, but throughout this paper we follow the tradition in ?nance of
using volatility to mean standard deviation type objects.
8
were de?ned and studied in detail by Andersen, Bollerslev, Diebold, and Labys (2003) and
Barndor?-Nielsen and Shephard (2004).
A major motivation for Barndor?-Nielsen and Shephard (2002) and Andersen, Bollerslev,
Diebold, and Labys (2001) was the fact that volatility in ?nancial markets is highly and unstably
diurnal within a day, responding to regularly timed macroeconomic news announcements, social
norms such as lunch times and sleeping or the opening of other markets. This makes estimating
lim
??0
_
[Y ]
t+?
?[Y ]
t
_
/?
extremely di?cult. The very stimulating work of Genon-Catalot, Lar´edo, and Picard (1992),
Foster and Nelson (1996), Mykland and Zhang (2002) and Mykland and Zhang (2005) tries to
tackle this problem using a double asymptotics, as ? ? 0 and ? ? 0. However, in the last ?ve years
many econometrics researchers have mostly focused on naturally diurnally robust quantities like
the daily or weekly QV.
2.6 Changes in probability law, QV and BPV
The observed price process Y ? o/, governed by its data generating process or measure P,
is not uniquely interesting. In ?nancial economics the stochastic behaviour of Y under risk
neutral versions P
?
(i.e. so called equivalent martingale measures) are also important for they
determine the price of contingent assets based on Y . An interesting question is whether [Y ]
computed under P tells us anything about the behaviour of [Y ] under P
?
. To discuss this recall
that the notation P
?
0.
Building on the earlier CLT of Barndor?-Nielsen and Shephard (2006), Barndor?-Nielsen,
Graversen, Jacod, Podolskij, and Shephard (2005) have established a CLT which covers this
situation when Y ? Bo/. We will only present the univariate result, which has that as ? ? 0
so
?
?1/2
(¦Y
?
¦
t
?¦Y ¦
t
) ? µ
2
1
_
(2 +?)
_
t
0
?
2
u
dB
u
, (31)
where B ?? Y , the convergence is in law stable as a process and
? =
_
?
2
/4
_
+? ?5 · 0.6090.
This result, unlike Theorem 1, has some quite technical conditions associated with it in order to
control the degree to which the volatility process can jump; however we will not discuss those
issues here. Extending the result to cover the joint distribution of the estimators of the QV and
the BPV processes, they showed that
?
?1/2
_
µ
?2
1
¦Y
?
¦
t
?µ
?2
1
¦Y ¦
t
[Y
?
]
t
?[Y ]
t
_
L
? MN
__
0
0
_
,
_
(2 +?) 2
2 2
__
t
0
?
4
u
du
_
,
a Hausman (1978) type result as the estimator of the QV process is, of course, fully asymptoti-
cally e?cient when Y ? Bo/. Consequently
?
?1/2
_
[Y
?
]
t
?µ
?2
1
¦Y
?
¦
t
_
¸
?
_
t
0
?
4
u
du
L
? N (0, 1) , (32)
which can be used as the basis of a test of the null of no jumps.
4.2 Multipower variation
The “standard” estimator of integrated quarticity, given in (18), is not robust to jumps. One
way of overcoming this problem is to use a multipower variation (MPV) measure — introduced
by Barndor?-Nielsen and Shephard (2006). This is de?ned as
¦Y ¦
[r]
t
= p?lim
??0
?
(1?r
+
/2)
t/?
j=1
_
I
i=1
¸
¸
Y
?(j?i)
?Y
?(j?1?i)
¸
¸
r
i
_
,
where r
i
> 0, r = (r
1
, r
2
, ..., r
I
)
for all i and r
+
=
I
i=1
r
i
. The usual BPV process is the special
case ¦Y ¦
t
= ¦Y ¦
[1,1]
t
.
If Y obeys (7) and r
i
< 2 then
¦Y ¦
[r]
t
=
_
I
i=1
µ
r
i
_
_
t
0
?
r
+
u
du,
32
This process is approximated by the estimated MPV process
¦Y
?
¦
[r]
t
= ?
(1?r
+
/2)
t/?
j=1
_
I
i=1
¸
¸
Y
?(j?i)
?Y
?(j?1?i)
¸
¸
r
i
_
.
In particular the scaled realised tri and quadpower variation,
µ
?4
1
¦Y
?
¦
[1,1,1,1]
t
and µ
?3
4/3
¦Y
?
¦
[4/3,4/3,4/3]
t
,
respectively, both estimate
_
t
0
?
4
u
du consistently in the presence of jumps. Hence either of these
objects can be used to replace the integrated quarticity in (32), so producing a non-parametric
test for the presence of jumps in the interval [0, t]. The test is conditionally consistent, meaning
if there is a jump, it will detected and has asymptotically the correct size. Extensive small
sample studies are reported in Huang and Tauchen (2005), who favour ratio versions of the
statistic like
?
?1/2
_
µ
?2
1
¦Y
?
¦
t
[Y
?
]
t
?1
_
¸
¸
¸
_
?
¦Y
?
¦
[1,1,1,1]
t
(¦Y
?
¦
t
)
2
L
? N (0, 1) ,
which has pretty reasonable ?nite sample properties. They also show that this test tends to
under reject the null of no jumps in the presence of some forms of market frictions.
It is clearly possible to carry out jump testing on separate days or weeks. Such tests are
asymptotically independent over these non-overlapping periods under the null hypothesis.
To illustrate this methodology we will apply the jump test to the DM/Dollar rate, asking
if the hypothesis of a continuous sample path is consistent with the data we have. Our focus
will mostly be on Friday January 15th 1988, although we will also give results for neighbouring
days to provide some context. In Figure 5 we plot 100 times the change during the week of the
discretised Y
?
, so a one unit uptick represents a 1% change, for a variety of values of n = 1/?,
as well as giving the ratio jump statistics
´
B
i
/
´
V
i
with their corresponding 99% critical values.
In Figure 5 there is a large uptick in the D-mark against the Dollar, with a movement of
nearly two percent in a ?ve minute period. This occurred on the Friday and was a response to
the news of a large fall in the U.S. balance of payment de?cit, which led to a large strengthening
of the Dollar. The data for January 15th had a large
´
V
i
but a much smaller
´
B
i
. Hence the
statistics are attributing a large component of
´
V
i
to the jump, with the adjusted ratio statistic
being larger than the corresponding 99% critical value. When ? is large the statistic is on the
borderline of being signi?cant, while the situation becomes much clearer as ? becomes small.
This illustration is typical of results presented in Barndor?-Nielsen and Shephard (2006) which
showed that many of the large jumps in this exchange rate correspond to macroeconomic news
33
0
1
2
3
4
Change in Y
?
during week
using ?=20 minutes
Mon Fri Thurs Wed Tues
0.50
0.75
1.00
Ratio jump statistic and 99% critical values
Mon Tues Wed Thurs
Fri
(µ
1
)
?2 ^
B
i
/
^
V
i
99% critical values
0
1
2
3
4
Change in Y
?
during week
using ?=5 minutes
Mon Tues Wed
Thurs Fri
0.25
0.50
0.75
1.00
Ratio jump statistic and 99% critical values
Mon Tues
Wed
Thurs
Fri
(µ
1
)
?2 ^
B
i
/
^
V
i
99% critical values
Figure 5: Left hand side: change in Y
?
during a week, centred at 0 on Monday 11th January
and running until Friday of that week. Drawn every 20 and 5 minutes. An up tick of 1 indicates
strengthening of the Dollar by 1%. Right hand side shows an index plot of
´
B
i
/
´
V
i
, which should
be around 1 if there are no jumps. Test is one sided, with criticial values also drawn as a line.
announcements. This is consistent with the recent economics literature documenting signi?cant
intraday announcement e?ects, e.g. Andersen, Bollerslev, Diebold, and Vega (2003).
4.3 Grids
It is clear that the martingale based CLT for irregularly spaced data for the estimator of the
QV process can be extended to cover the BPV case. We de?ne
¦Y
Gn
¦
t
=
t
j
?t
j=1
¸
¸
Y
t
j?1
?Y
t
j?2
¸
¸
¸
¸
Y
t
j
?Y
t
j?1
¸
¸
p
? ¦Y ¦
t
.
Using the same notation as before, we would expect the following result to hold, due to the fact
that H
G
is assumed to be continuous,
?
?1/2
_
µ
?2
1
¦Y
Gn
¦
t
?µ
?2
1
¦Y ¦
t
[Y
?
]
t
?[Y ]
t
_
L
? MN
__
0
0
__
(2 +?) 2
2 2
__
t
0
_
?H
G
u
?u
_
?
4
u
du
_
.
34
The integrated moderated quarticity can be estimated using µ
?4
1
¦Y
?
¦
[1,1,1,1]
t
, or a grid version,
which again implies that the usual feasible CLT continues to hold for irregularly spaced data.
This is the expected result from the analysis of power variation provided by Barndor?-Nielsen
and Shephard (2005c).
Potentially there are modest e?ciency gains to be had by computing the estimators of BPV
on multiple grids and then averaging them. The extension along these lines is straightforward
and will not be detailed here.
4.4 In?nite activity jumps
The probability limit of realised BPV is robust to ?nite activity jumps. A natural question to
ask is: (i) is the CLT also robust to jumps, (ii) is the probability limit also una?ected by in?nite
activity jumps, that is jump processes with an in?nite number of jumps in any ?nite period of
time. Both issues are studied by Barndor?-Nielsen, Shephard, and Winkel (2004) in the case
where the jumps are of L´evy type, while Woerner (2004) looks at the probability limit for more
general jump processes.
Barndor?-Nielsen, Shephard, and Winkel (2004) ?nd that the CLT for BPV is a?ected by
?nite activity jumps, but this is not true of tripower and high order measures of variation. The
reason for the robustness of tripower results is quite technical and we will not discuss it here.
However, it potentially means that inference under the assumption of jumps can be carried out
using tripower variation, which seems an exciting possibility. Both Barndor?-Nielsen, Shephard,
and Winkel (2004) and Woerner (2004) give results which prove that the probability limit of
realised BPV is una?ected by some types of in?nite activity jump processes. More work is
needed on this topic to make these result de?nitive. It is somewhat related to the parametric
study of A¨?t-Sahalia (2004). He shows that maximum likelihood estimation can disentangle a
homoskedastic di?usive component from a purely discontinuous in?nite activity L´evy component
of prices. Outside the likelihood framework, the paper also studies the optimal combinations
of moment functions for the generalized method of moment estimation of homoskedastic jump-
di?usions. Further insights can be found by looking at likelihood inference for L´evy processes,
which is studied by A¨?t-Sahalia and Jacod (2005a) and A¨?t-Sahalia and Jacod (2005b).
4.5 Testing the null of no continuous component
In some stimulating recent papers, Carr, Geman, Madan, and Yor (2003) and Carr and Wu
(2004), have argued that it is attractive to build SV models out of pure jump processes, with
no Brownian aspect. This is somewhat related to the material we discuss in section 5.6. It is
clearly important to be able to test this hypothesis, seeing if pure discreteness is consistent with
35
observed prices.
Barndor?-Nielsen, Shephard, and Winkel (2004) showed that
?
?1/2
_
¦Y
?
¦
[2/3,2/3,2/3]
t
?[Y
ct
]
t
_
has a mixed Gaussian limit and is robust to jumps. But this result is only valid if ? > 0, which
rules out its use for testing for pure discreteness. However, we can arti?cially add a scaled
Brownian motion, U = ?B, to the observed price process and then test if
?
?1/2
_
¦Y
?
+U
?
¦
[2/3,2/3,2/3]
t
??
2
t
_
is statistically signi?cantly greater than zero. In principle this would be a consistent non-
parametric test of the maintained hypothesis of Peter Carr and his coauthors.
4.6 Alternative methods for identifying jumps
Mancini (2001), Mancini (2004) and Mancini (2003) has developed robust estimators of
_
Y
ct
¸
in the presence of ?nite activity jumps. Her approach is to use truncation
t/?
j=1
_
Y
j?
?Y
(j?1)?
_
2
I(
¸
¸
Y
j?
?Y
(j?1)?
¸
¸
< r
?
), (33)
where I (.) is an indicator function. The crucial function r
?
has to have the property that
_
? log ?
?1
r
?1
?
? 0. It is motivated by the modulus of continuity of Brownian motion paths that
almost surely
lim
??0
sup
0?s,t?T
|t?s|