Estimating quadratic variation using realised variance
Ole E. Barndorff-Nielsen
The Centre for Mathematical Physics and Stochastics (MaPhySto),
University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
oebn@mi.aau.dk
www.LevyProcess.org
http://home.imf.au.dk/oebn/
Neil Shephard
Nuffield College, Oxford OX1 1NF, UK
neil.shephard@nuf.ox.ac.uk
www.LevyProcess.org
www.nuff.ox.ac.uk/users/shephard/
First draft: September 2001
March 2002
Abstract
This paper looks at some recent work on estimating quadratic variation using realised
variance (RV) — that is sums of M squared returns. This econometrics has been motivated
by the advent of the common availability of high frequency financial return data. When
the underlying process is a semimartingale we recall the fundamental result that RV is a
consistent (as M →∞) estimator of quadratic variation (QV). We express concern that
without additional assumptions it seems difficult to give any measure of uncertainty of the
RV in this context. The position dramatically changes when we work with a rather general
SV model — which is a special case of the semimartingale model. Then QV is integrated
variance and we can derive the asymptotic distribution of the RV and its rate of convergence.
These results do not require us to specify a model for either the drift or volatility functions,
although we have to impose some weak regularity assumptions. We illustrate the use of the
limit theory on some exchange rate data and some stock data. We show that even with large
values of M the RV is sometimes a quite noisy estimator of integrated variance.
Keywords: Integrated variance; Power variation; Quadratic variation; Realised variance; Re-
alised volatility; Semimartingale; Volatility.
1 Introduction
In this paper we ask two questions about realised variance
1
(RV), that is the sum of M squared
returns.
• What does RV estimate?
• How precise is RV?
1
Sums of squared returns are often called realised volatility in econometrics. We will discuss later in this
introduction why we prefer to call this statistic realised variance.
1
The answer to the first question is straightforward and well known: it is a consistent estimator
(as M →∞) of the corresponding quadratic variation (QV), for all semimartingales. We will
see that this is potentially helpful for QV is quite often an econometrically revealing quantity in
special cases of semimartingales models. Unfortunately, although RV is a consistent estimator of
QV in general, we do not know anything about its precision or indeed even its rate of convergence
as M →∞. This is potentially troublesome, for it does not allow us to deal with the issue
that RV and QV are distinct. This obstacle can be overcome when we work within a stochastic
volatility (SV) framework, which is an important special case of semimartingales with continuous
sample paths. For such models we have been able to derive the rate of convergence and indeed
the asymptotic distribution. We will illustrate this theory in the context of some high frequency
exchange rate data and daily stock data, showing in particular that RV can sometimes be a very
noisy estimator of QV even when M is large.
In order to formalise some of these issues we begin with some definitions and notation.
We write the log-price as y
∗
(t), where t denotes time. Such a price series is usually synthesised
from quote or transaction data. Over small time intervals the details of this construction greatly
matters and has been studied extensively in the econometric literature on market microstructure
(e.g. see the interesting work of Andreou and Ghysels (2001) and Bai, Russell, and Tiao (2000)
in this context). For the moment we abstract from this issue. If we think of a fixed interval of
time of length > 0, then the returns over the n-th such interval are defined as
y
n
= y
∗
(n) − y
∗
((n − 1) ) ,n=1, 2, ....
During this interval, we can also compute M intra- returns. These are defined, for the n-th
period, as
y
j,n
= y
∗
(n − 1) +
j
M
− y
∗
(n − 1) +
(j − 1)
M
,j=1, 2, ..., M.
Then many financial economists have measured variability during this period using realised
variance,definedas
[y
∗
M
]
n
=
M
j=1
y
2
j,n
.
This term is often called the realised volatility in econometrics, although we will keep back that
name for
M
j=1
y
2
j,n
,
reflecting our use of volatility to mean standard deviations rather than variances
2
.Examplesof
the use of realised variances are given by, for example, Merton (1980), Poterba and Summers
2
The use of volatility to denote standard deviations rather than variances is standard in financial economics.
See, for example, the literature on volatility and variance swaps, which are derivatives written on realised volatility
2
(1986), Schwert (1989), Richardson and Stock (1989), Schwert (1990), Taylor and Xu (1997) and
Christensen and Prabhala (1998). An elegant survey of the literature on this topic, including
a discussion of its economic importance, is given by Andersen, Bollerslev, and Diebold (2002).
See also the recent important contribution by Meddahi (2002).
From a formal econometric viewpoint we consider [y
∗
M
]
n
as an estimator, allowing us to study
its finite sample behaviour for fixed M or its asymptotic properties as M →∞. Unfortunately,
although we know RV converges to QV in probability, this result lacks a theory of measurement
error which makes it hard to use this estimator. It would seem additional assumptions are
needed. One set of assumptions is to say that y
∗
is Brownian motion with drift which is
deformed by a subordinator (that is a process with non-negative, independent and stationary
increments). We study QV in this context in Section 2 of this paper. Such models are frequently
used in finance in order to derive derivative pricing formulas. An alternative is to assume a rather
general stochastic volatility (SV) model. The latter framework is the mainstay of the discussion
we give here.
The SV model we work with has a very flexible form. We assume
y
∗
(t)=α(t)+
t
0
σ(s)dw(s),t≥ 0,
where α, the drift, and σ>0, the spot volatility, obey some weak assumptions outlined in
Section 3. In particular the spot volatility can have, for example, deterministic diurnal effects,
jumps, long memory, no unconditional mean or be non-stationary. No knowledge of the form of
the stochastic processes which govern α and σ are needed. SV models are a fundamental special
type of semimartingale; in particular most semimartingales which possess continuous sample
paths can be represented as SV models.
In these models, assuming σ and α are jointly independent from w, returns
y
n
|α
n
,σ
[2]
n
∼ N(α
n
,σ
[2]
n
), (1)
where
α
n
= α (n) − α ((n − 1) )(2)
and
σ
[2]
n
= σ
2∗
(n) − σ
2∗
((n − 1) ) , where σ
2∗
(t)=
t
0
σ
2
(s)ds. (3)
We call σ
2
(t) the spot variance and σ
2∗
(t) the integrated variance. Importantly, for all SV
models σ
2∗
exactly equals QV
σ
2∗
(t)=[y
∗
](t), (4)
or variance, which includes Demeterfi, Derman, Kamal, and Zou (1999), Howison, Rafailidis, and Rasmussen
(2000) and Chriss and Morokoff (1999). We have choosen to follow this nomenclature rather than the one more
familiar in econometrics.
3
and so
σ
[2]
n
=[y
∗
](n) − [y
∗
]((n − 1) )=[y
∗
]
[2]
n
.
Thus QV reveals exactly the actual variance σ
[2]
n
in SV models. It does this without knowledge
of the actual processes which govern α or σ.
10 20 30 100 200
1
2
Day 1
Realised var
Upper 97.5%
Lower 2.5%
10 20 30 100 200
1
2
Day 2
Realised var
Upper 97.5%
Lower 2.5%
10 20 30 100 200
1
2
3
Day 3
Realised var
Upper 97.5%
Lower 2.5%
10 20 30 100 200
1
2
3
Day 4
Realised var
Upper 97.5%
Lower 2.5%
10 20 30 100 200
1
2
Day 5
Realised var
Upper 97.5%
Lower 2.5%
10 20 30 100 200
1
2
Day 6
Realised var
Upper 97.5%
Lower 2.5%
10 20 30 100 200
1
2
Day 7
Realised var
Upper 97.5%
Lower 2.5%
10 20 30 100 200
1
2
Day 8
Realised var
Upper 97.5%
Lower 2.5%
10 20 30 100 200
1
2
Day 9
Realised var
Upper 97.5%
Lower 2.5%
Figure 1: First 9 days. RV is plotted against M, with the smallest M being 8, the largest 288.
Also plotted are the 95% intervals. Code: se
realised.ox.
The above theory means that [y
∗
M
]
n
consistently estimates σ
[2]
n
, just using the theory of
semimartingales. Barndorff-Nielsen and Shephard (2002a) have shown that [y
∗
M
]
n
converges to
σ
[2]
n
at rate
√
M and have additionally derived the asymptotic distribution of the estimator:
M
j=1
y
2
j,n
−
n
(n−1)
σ
2
(s)ds
2
3
M
j=1
y
4
j,n
L
→ N(0, 1), (5)
as M →∞, thus providing a measure of the precision of this estimator. Their preferred form
of the result, due to its superior finite sample behaviour (see Barndorff-Nielsen and Shephard
(2001a)), is that as M →∞then
log
M
j=1
y
2
j,n
− log
σ
[2]
n
2
3
M
j=1
y
4
j,n
{
M
j=1
y
2
j,n
}
2
L
→ N(0, 1). (6)
4
This is a mixed Gaussian limit theory, that is the denominator is itself random. Of course this
theory can be used to provide approximations for realised volatility as well as realised variance.
The distribution of realised volatilies can also be approximated indirectly via (5) using the delta
method which gives
M
j=1
y
2
j,n
−
n
(n−1)
σ
2
(s)ds
2
12
M
j=1
y
4
j,n
M
j=1
y
2
j,n
L
→ N(0, 1). (7)
The log-based approximation (6) is likely to be preferred in practice when we construct confi-
dence intervals for realised volatility.
To illustrate this result we have used the same return data employed by Andersen, Boller-
slev, Diebold, and Labys (2001a) in their empirical study of the properties of realised variance,
although we have made slightly different adjustments to deal with some missing data (in the
context of this paper the effect of these differences are tiny, but were made here to be consistent
with our other work on this dataset). Full details of this are given in Barndorff-Nielsen and
Shephard (2002a). The data was kindly supplied to us by the Olsen group in Zurich. This
United States Dollar/ German Deutsche Mark series covers the ten year period from 1st De-
cember 1986 until 30th November 1996. It records every five minutes the most recent quote to
appear on the Reuters screen. Throughout we take to represent a day and so have up to 288
five minute returns to work with each day. This constrains our choice of M to taking the values
288, 144, 96, 72, 48, 36, 32, 24, 18, 16, 12, 9, 8, 6, 4, 3, 2, 1.
In Figure 1 we record, for a variety of values of M , RV and its 95% confidence intervals
(based on (6)) for the first 9 days of the dataset. This is the first time such graphs have been
produced. The result suggests that the confidence intervals do indeed narrow considerably with
M. However, even with M = 288 the intervals are sometimes quite wide. The implication is
that RV is a consistent but quite noisy estimator of σ
[2]
n
, especially when volatility is high. We
will return to this issue in more detail in Section 4.
The structure of this paper is as follows. In Section 2 we discuss the definition of quadratic
variation in the context of semimartingales. We will see that realised variance, by definition,
converges in probability to QV and so RV is a consistent estimator of QV. We give some exam-
ples where QV does not reveal the conditional variance of returns in a subordinated Brownian
motion model. Our conclusion is that more detailed assumptions are needed in order to pro-
vide a coherent analysis of RV. We also discuss the advantage of working with the conditional
expectation of RV and the convergence of it to the conditional expectation of quadratic varia-
tion. This follows some recent work by Andersen, Bollerslev, Diebold, and Labys (2001b). In
Section 3 we move on to consider the properties of RV in the context of SV models. We develop
5
![Figure 2: Daily RV, [y∗M ]n, drawn against n for the first 50 days of the sample. Also drawn as vertical bars are the 95% intervals based on the log transformation. Throughout M = 144 . Code: se realised.ox.](/figures/figure-2-daily-rv-y-m-n-drawn-against-n-for-the-first-50-3egphlzh.webp)