Nonparametric Estimation of the Leverage Effect using Information

Commentaires

Transcription

Nonparametric Estimation of the Leverage Effect using Information
Nonparametric Estimation of the Leverage Effect using Information
from Derivatives Markets∗
Ilze Kalnina†
Dacheng Xiu‡
Université de Montréal
Chicago Booth
May 21, 2014
Abstract
We consider a new nonparametric estimator of the leverage effect, which uses the data on
stock prices as well as a certain volatility instrument, such as the CBOE volatility index (VIX)
or Black-Scholes implied volatility. The theoretical justification for the new estimator exploits
the relationship between the volatility instrument and the spot volatility, together with a certain
invariance property of the spot correlation. We derive the asymptotic distribution of the estimator. We compare this instrument-based estimator with the nonparametric price-only leverage
estimator, for which we also derive the asymptotic distribution. The instrument-based estimator has a faster rate of convergence. We demonstrate in simulations and empirically that the
instrument-based estimator is substantially more precise than the price-only estimator. Finally,
we use the new estimator to provide time series of monthly leverage effects of the S&P 500 index
from 2003 to 2012 using two different volatility instruments. We also find that the credit risk,
liquidity, and the debt-to-equity are important determinants of the leverage effect.
Keywords: semimartingale; spot correlation; VIX; implied volatility; high frequency data.
JEL Codes: G12, C22, C14.
∗
We benefited much from discussions with Torben Andersen, Oleg Bondarenko, Stéphane Bonhomme, Francis
Diebold, Kirill Evdokimov, Christian Hansen, Zhiguo He, Jean Jacod, Bryan Kelly, Jia Li, Yingying Li, Per Mykland,
Benoit Perron, Mark Podolskij, Jeffrey Russell, Olivier Scaillet, Kevin Sheppard, George Tauchen, and Dan Wang. We
thank seminar and conference participants at the University of Pennsylvania, University of Chicago, Duke University,
Purdue University, Boston University, University of Rochester, Queen’s University, CIREQ Econometrics Conference
2014, the High-Frequency Data and High-Frequency Trading Conference at Chicago, the Econometric Study Group
2013 conference at Bristol, the 9th GNYEC, the First Conference in Econometric Theory at Universidad de San Andrés,
and the fifth Annual Modeling High-Frequency Data in Finance Conference.
†
Département de sciences économiques, Université de Montréal. Address: C.P.6128, succ. Centre-Ville, Montréal,
QC, H3C 3J7, Canada. E-mail address: [email protected] Kalnina’s research was supported by Institut
de finance mathématique de Montréal, SSHRC, and FQRSC.
‡
Booth School of Business, University of Chicago. Address: 5807 S Woodlawn Avenue, Chicago, IL 60637, USA. Email address: [email protected] Xiu gratefully acknowledges financial support from the FMC Faculty
Scholar Fund at the University of Chicago Booth School of Business.
1
1
Introduction
One of the most important empirical stylized facts about the volatility is the leverage effect, which
refers to the generally negative correlation between an asset return and its volatility changes. The
term “leverage” dates back to an early influential economic hypothesis of Black (1976) that explains
this correlation using the debt-to-equity ratio, a common financial leverage measure. To study various
financial theories about the leverage effect, one would relate the leverage effect to macroeconomic
variables and firm characteristics, which are typically updated at a monthly or quarterly frequency.
We are interested in developing a statistical theory in the framework of semi-martingales, which
can allow us to use high frequency data to estimate the leverage effect at macroeconomic frequencies.
This approach mirrors the highly successful literature on the nonparametric estimation and forecasting of volatility at these frequencies by the realized volatility using high frequency data (see, e.g.,
Andersen, Bollerslev, Diebold, and Labys (2003)). However, the estimation of the leverage effect is
more difficult because the volatility is unobservable. Since Black (1976), a common approach is to
conduct preliminary estimation of the volatility over small windows, then to compute the correlation
between returns and the increments of the estimated volatility. Such estimators appear to be very
noisy in practice, at least when estimating monthly or quarterly leverage effect.
We consider a new estimator that replaces preliminary estimation of volatility with high frequency
observations of a volatility instrument, such as the VIX or Black-Scholes implied volatility.1 The
use of the VIX, for example, often leads to a bias, since the VIX is not equal to the unobserved
volatility. In general, the VIX is an unknown function of the volatility, and the shape of this function
is related to the risk premia, embedded in S&P 500 options. That said, it follows by the Itô’s lemma
that the spot correlation between returns and the VIX increments is equal to the spot correlation
between returns and the spot volatility increments, because the unknown function capturing the risk
premia cancels out. Hence, we consider an estimator of the leverage effect using an aggregation of
the spot correlation estimates between the returns and the increments of the volatility instrument,
and denote it the Instrument-based Realized Leverage (IRL) estimator.
We develop appropriate nonparametric asymptotic theory for the IRL estimator under general
conditions. We show that the naive estimator has an asymptotic bias, which we correct. We also
show that a crucial part of the estimation is the truncation of jumps in the price as well as the
volatility instrument. If jumps are not truncated, the resulting estimator is inconsistent. We derive
1
Since September 22, 2003, the VIX (the volatility index of CBOE) has been constructed by the CBOE using a
weighted portfolio of options:
(Zt /100)2 =
2ert,τ τ τ
eft,τ
Z
0
P (τ, x)
dx +
x2
Z
∞
eft,τ
C(τ, x) dx
x2
where P (τ, x) and C(τ, x) are put and call options with time-to-maturity τ and strike x, and ft,τ is the log price of
forward contracts, see, e.g., Carr and Wu (2009).
2
the asymptotic distribution of the estimator.
We study its performance in Monte Carlo simulations and estimate the leverage effect of the
index using high frequency data. To illustrate the precision of the IRL estimator in practice, we
calculate the standard error for the S&P 500 leverage effect using the VIX as a volatility instrument.
We use 30-minute data on the VIX and the S&P 500 futures from the Chicago Board of Exchange
(CBOE) over the period 2003-2012. The median standard errors are 0.046, 0.027, and 0.013 using
weekly, monthly, and quarterly time spans. Because the VIX is not available for all stocks, we also
consider an alternative volatility instrument, the Black-Scholes implied volatility. We use 30-minute
data on the S&P 500 futures and the S&P 500 options. The median standard errors of the IRL
estimator are 0.041, 0.023, and 0.014 over weekly, monthly, and quarterly time spans.
We compare the instrument-based estimator with the nonparametric estimator of the leverage
effect using prices only. We call it the Price-only Realized Leverage (P RL) estimator. It is the
high-frequency counterpart of the typical leverage estimator discussed above. Our paper appears to
be the first to provide the asymptotic theory for the P RL estimator.2 The naive price-only leverage
estimator is biased and we provide a bias correction. The rate of convergence of the P RL estimator
is the square root of the rate of convergence of the IRL estimator, hence the P RL estimator is
bound to be much less precise. We calculate median standard errors for the P RL estimator of the
S&P 500 leverage using 5-minutes data on the S&P 500 futures, and they are 0.971 using quarterly,
0.634 using yearly time span, and 0.367 when estimating the leverage effect over the entire 10 years
of data we have. Clearly, the IRL appears to be substantially more precise.
Our comparison between the two estimators involves a stark trade-off. On the one hand, the
IRL estimator requires the availability of a volatility instrument (Assumption 2). Sections 2 and 5.3
discuss the applicability of this assumption. On the other hand, the P RL estimator does not seem
to yield practically useful estimates of the leverage correlation for monthly or even quarterly data
spans. In our empirical application, we find that the standard errors are large and the estimator is
unstable, and this is confirmed by our Monte Carlo simulations. These findings appear to be in line
with the fact that Aı̈t-Sahalia, Fan, and Li (2013) use 4 years of 1-minute data on the S&P 500 to
estimate the leverage effect. In other words, the price-only approach seems to be better suited for
cross-sectional studies.
The increased precision of the IRL implies it is possible to investigate how the leverage effect
changes over time, which is important for financial applications. We provide a time series of monthly
leverage effects of the S&P 500 index using the two instruments, the VIX and the Black-Scholes
implied volatility of the S&P 500 options. We also conduct a time series regression with the estimated
leverage effect and important financial variables.
The use of the VIX as an instrument or proxy for volatility to estimate the leverage effect has
2
Several papers have considered parametric price-only leverage estimates, as well as components of the P RL. We
discuss these in detail below.
3
been previously considered by a few papers. These papers, however, have not studied the asymptotic
theory of their estimators. Our paper fills this gap and has several important implications for the
applied work. Most closely related paper is an empirical study by Andersen, Bondarenko, and
Gonzalez-Perez (2012). Although authors mainly focus on developing an alternative volatility proxy
“Corridor VIX”, among other things they propose using this Corridor VIX as the volatility instrument
to estimate the leverage effect. They do not develop any asymptotic theory for their estimator, and
our results usefully supplement their application. In particular, we point out that their estimator
may be inconsistent in the presence of jumps. While the Corridor VIX removes artificial jumps
introduced by the CBOE calculation of the VIX, other jumps are retained, creating a bias in the
spot correlation estimates. Even if the jumps are truncated away, we show that the naive estimator
of correlation like theirs, i.e., an average of the spot correlation estimates, has an asymptotic bias
that our estimator removes. Finally, we also derive the asymptotic distribution of the estimators and
hence are able to evaluate their precision.
Several papers are related to the P RL estimator. Aı̈t-Sahalia, Fan, and Li (2013) remark that
empirically, the price-only naive leverage estimator with high-frequency data gives estimates that
are close to zero, and call it “the leverage effect puzzle.” In the parametric framework of the Heston
model, they show that the naive estimator is biased, and explain the sources of the bias. Our analysis
of the P RL estimator complements theirs in being fully nonparametric. Bandi and Reno (2012),
Jacod and Rosenbaum (2012b), Mykland, Shephard, and Sheppard (2012), Wang and Mykland
(2012), Vetter (2012b), and Vetter (2012a) estimate quantities that are related to the leverage effect
such as the volatility of volatility and the covariance between returns and volatility.3
Many parametric models have been proposed to capture the leverage effect with daily data. For
example, the popular EGARCH model of Nelson (1991) is motivated by the inability of the standard
GARCH models to capture the leverage effect. See also Yu (2005) for a discussion on estimating the
leverage effect using parametric stochastic volatility models. Such models usually assume a constant
leverage effect, and their estimation either involves the Markov Chain Monte Carlo algorithm with
or without options, see, e.g., Jacquier, Polson, and Rossi (2007) and Eraker (2004), or moment- or
likelihood-based approach using either at-the-money implied volatility or the VIX as proxies, see,
e.g., Pan (2002) and Aı̈t-Sahalia and Kimmel (2007). These methods require parametric assumptions
and are computationally expensive.
Section 2 outlines the model assumptions. Section 3 contains the asymptotic theory for the
nonparametric estimator of the leverage effect using a volatility instrument (the IRL estimator).
Section 4 presents the P RL estimator and the associated asymptotic theory. Section 5 provides
additional discussion on potential extensions. Section 6 provides Monte Carlo evidence and Section
7 presents empirical findings. Section 8 concludes. The Appendix contains the proofs.
3
Discussion following Theorem 3 contains a more detailed description of the relationship to our results.
4
2
The Model
We work in a general nonparametric framework that allows for potential jumps in prices and volatility.4
Assumption 1. Suppose that the underlying X and its spot variance follow an Itô semimartingale,
t
Z
Xt = X0 +
bs ds +
0
t
Z
Z tZ
σs dWs +
0
σt2 = σ02 +
t
Z
Z
t
b̃s ds +
0
0
δ(s, z)µ(ds, dz),
(1)
δ̃(s, z)µ(ds, dz),
(2)
R
Z tZ
fs +
τs dW
0
0
R
where µ(ds, dz) is a Poisson random measure with an intensity measure ν(ds, dz) = ds ⊗ λ(dz).
fs ) = ρs ds. Moreover, |δ (ω, t ∧ τm (ω) , z) | ∧ 1 ≤ Γm (z) and |δ̃ (ω, t ∧ τm (ω) , z) | ∧ 1 ≤
E(dWs dW
Γm (z), for all (ω, t, z), where (τm ) is a localizing sequence of stopping times and, for some r ∈ (0, 1),
R
the function Γm on R satisfies R Γm (z)r λ (dz) < ∞, i.e., the jumps are summable. In addition, bt ,
b̃t , and τt are locally bounded, and ρt and τt are càdlàg. For any 0 ≤ s ≤ t, σs , σs− , τs , and τs−
are almost surely positive, |ρs | and |ρs− | are almost surely smaller than 1. Also, [X, X]cs 6= 0 and
[σ 2 , σ 2 ]cs 6= 0 hold almost surely, for each s.5
What is an appropriate measure of the leverage effect in this general framework? To motivate,
note that in the special case of the popular Heston model (see equations (20)-(21)), the leverage
effect is usually associated with the parameter ρ, which equals
2
ρ = lim Corr σs+∆
− σs2 , Xs+∆ − Xs .
∆→0
(3)
It also coincides with the correlation between the two Brownian motions of the Heston model, which
is constant over time. In the general case, the corresponding correlation is the spot leverage effect
ρs , and it varies over time. Then, a natural measure of the leverage effect over an interval [0, t] is
the corresponding scaled integral over time, i.e., the integrated leverage effect,
1
ILt =
t
Z
0
t
1
ρs ds =
t
Z
0
t
[σ 2 , X]c,0
s
q
q
ds,
[X, X]c,0
[σ 2 , σ 2 ]c,0
s
s
(4)
2
where prime denotes a derivative with respect to time, so that for example [X, X]c,0
s = σs . An
alternative leverage effect measure can be obtained as the continuous part of the quadratic covariation
4
This is a standard nonparametric framework for asset pricing , see, e.g. Delbaen and Schachermayer (1994), Carr
and Wu (2009), Bollerslev and Todorov (2011), and Aı̈t-Sahalia and Jacod (2012).
5
The superscript c denotes the continuous part of the process, and [X, X]s is the quadratic variation of X over [0, s].
5
between σ 2 and X, scaled appropriately,
Rt
ρs σs τs ds
[σ 2 , X]ct
p
qR 0
qR
=
V W ILt = p
.
t 2
t 2
[X, X]ct [σ 2 , σ 2 ]ct
σ ds
τ ds
0
s
0
(5)
s
This measure depends also on the path of the spot volatility and the volatility of volatility, hence
we name it the volatility-weighted integrated leverage effect (V W ILt ). Both measures provide a
quantitative characterization of the so-called leverage effect. Theoretically, spot leverage effect would
also be of interest, but it does not seem realistic to estimate it precisely in practice, so some kind
of averaging is needed. The above leverage measures correspond to two alternative averages with
different weights.
We next assume that we have access to what we call a volatility instrument.
Assumption 2. There exists an observable variable Zt , which is a monotone increasing and twice
differentiable function of σt2 and a smooth function of t, that is, Zt = f (t, σt2 ). In addition, [Z, Z]cs is
almost surely not vanishing and f 0 (s, x) > 0, for all 0 ≤ s ≤ t and x > 0.6 We call such a variable a
volatility instrument.
For all single-factor parametric models in the literature, including affine and non-affine models, we
can choose Zt = VIX2t , which satisfies this condition, see, e.g., Andersen, Bondarenko, and GonzalezPerez (2012), Li and Xiu (2013), and Todorov and Tauchen (2011). The difference between the
squared VIX, Zt , and the spot variance σt2 , is determined by the risk-neutral dynamics and related
to variance risk premia, see, e.g., Carr and Wu (2009) and Todorov (2010). As long as the risk-neutral
dynamics is driven by one state variable, Assumption 2 is satisfied. Certain multi-factor models also
admit the same relationship between the VIX and the spot variance. For instance, if additional
factors appear in the volatility of volatility or feature a stochastic leverage effect, instead of affecting
the risk neutral drift of the volatility process, then the VIX is still driven by a single volatility
factor. Even if there is another factor that affects the VIX, it is usually a slow mean-reverting factor
employed to capture the time-varying term structure of variance, see, e.g., Christoffersen, Heston,
and Jacobs (2009). Such a factor along with the term structure of variance should not vary much
within a short period of time, so that Assumption 2 may serve as first-order approximation for high
frequency data.
Besides the VIX, alternative volatility instruments can also be used. For example, for the S&P 500
index, we can also use the intraday VIX futures or VIX options. If the same one-factor assumption
holds for σt2 , VIX futures and VIX options are only driven by the same factor. As long as contracts are
liquid enough, they may serve as better instruments, as they are tradable. For individual equities,
VIX can be calculated.7 Alternatively, the Black-Scholes implied volatility with a fixed maturity
6
7
We use f 0 to denote the derivative of f with respect to the second argument.
In fact, the CBOE has calculated the VIX for a limited number of stocks since January 7, 2011.
6
and moneyness can be used as a volatility instrument.8 As long as the option prices satisfy the
homogeneity of degree 1, see, e.g., Joshi (2007) and Song and Xiu (2012), which holds for almost all
models considered in the empirical option pricing literature, Assumption 2 is guaranteed.
Assumption 2 implies that the spot correlation between the returns and the spot volatility increments equals the spot correlation between the returns and the increments of the instrument Zt ,
[Z, X]c,0
[σ 2 , X]c,0
s
s
q
q
=q
.
ρs ≡ q
c,0
c,0
c,0
c,0
2
2
[X, X]s [σ , σ ]s
[X, X]s [Z, Z]s
The last equality is true by Itô’s lemma for the general case with jumps, see, e.g., Proposition 8.19 of
Cont and Tankov (2004). This implies that the integrated leverage ILt is invariant with respect to
the functional transformation f . It is important as it eliminates the effect of risk premia embedded
in the function f , so that ILt can be estimated using a volatility instrument. In contrast, V W ILt
is not invariant with respect to the functional transformation f unless f is linear, so it cannot be
estimated using a volatility instrument.
While we do allow for general price and volatility jumps, they do not contribute to our definition
of leverage. We choose not to include them for the following reasons. First, the integrated leverage
ILt is based on the spot correlation, which is an intuitive generalization of the ρ parameter in the
Heston model. It also endows the ILt measure with invariance with respect to transformation f .
However, the spot correlation is not well defined without the exclusion of jumps.9 Second, Andersen,
Bondarenko, and Gonzalez-Perez (2012) point out that the VIX contains artificial jumps due to its
implementation by the CBOE. The above leverage measures using the VIX are robust to the artificial
jumps, because they only depend on the continuous part. Third, we find in the empirical study that
truncating off the jump component makes the estimates more stable. Finally, Bollerslev, Kreschmer,
Pigorsch, and Tauchen (2009) find that the leverage effect works primarily through the continuous
components.
We now develop a theory of the instrument-based estimation, which uses the above equality to
replace the preliminary estimation of the volatility with the high frequency observations of a volatility
instrument.
3
Leverage Estimation using a Volatility Instrument
The current section discusses the asymptotic theory for the instrument-based realized leverage (IRL),
which exploits additional information embedded in the instrument. It constitutes the main contribution of our paper. We use the VIX as an example, but the results apply to any volatility instrument.
8
Medvedev and Scaillet (2007) point out that the implied volatility of an at-the-money option with a small timeto-maturity is close to the unobserved volatility.
9
Only the continuous part of the quadratic variation is absolutely continuous and differentiable almost everywhere.
7
The asymptotic results of the IRL estimator draw extensively from the general theoretical framework
of Jacod and Protter (2012), Jacod and Rosenbaum (2012a), and Jacod and Rosenbaum (2012b).
Suppose we have a sample of equidistant observations on X and Z over the interval [0, t]. Denote
the distance between adjacent observations by ∆n . We truncate jumps in X and Z using thresholds
un and u0n that satisfy the conditions in Theorem 1 below.
The VIX is a popular implied volatility measure which is expected to be highly informative
about the current volatility state. Therefore, a simple estimator of the correlation between returns
and changes of volatility is the truncated realized correlation between Z and X,
c
\
[Z,
X]t
q
RCorr (Z, X) = q
c
c
\
\
[X, X]t [Z,
Z]t
[t/∆n ]
X
(∆ni Z)(∆ni X) · 1{|∆ni Z|≤u0n } 1{|∆ni X|≤un }
i=1
=v
u[t/∆n ]
uX
t
(∆n Z)2 1
i
0
{|∆n
i Z|≤un }
v
u[t/∆n ]
uX
t
(∆n X)2 · 1
i=1
i
,
(6)
{|∆n
i X|≤un }
i=1
where Zt is the squared VIXt , and the jump truncation levels un and u0n satisfy the conditions of
Theorem 1 below. We know this estimator is consistent for a certain quantity,
[Z, X]ct
p
p
.
RCorr (Z, X) → p
[Z, Z]ct [X, X]ct
To have any idea about how the above quantity is related to the leverage effect, one has to know the
relationship between Z and σ 2 , which is the reason our Assumption 2 is needed.
By Assumption 2 and using the Itô’s Lemma, we obtain
Rt 0
f s, σs2 [σ 2 , X]c,0
[Z, X]ct
s ds
0
p
p
= qR
,
p
c
c
t 0
[Z, Z]t [X, X]t
c
[f (s, σ 2 )]2 [σ 2 , σ 2 ]c,0
s ds [X, X]
s
0
(7)
t
which equals neither ILt nor V W ILt in general. Unless the function f is linear, the realized correlation RCorr (Z, X) estimates an amalgamation of the leverage effect V W ILt and the risk-neutral
parameters that incorporate risk premia. If f is linear, the probability limit of the realized correlation
RCorr (Z, X) coincides with the V W ILt measure.
Theorem 1 presents the asymptotic distribution of the RCorr (Z, X) estimator. Appendix A
presents the proof of the theorem, as well as a consistent estimator of the asymptotic variance of
RCorr (Z, X). We use the notation to denote asymptotic equivalence and L-s to denote stable
convergence in law, see for example Aldous and Eagleson (1978) and Jacod and Protter (2012).
0
$
Theorem 1. Suppose Assumptions 1 and 2 hold. Let un ∆$
n , un ∆n for $ ∈ (1/(4 − 2r), 1/2).
8
Then, for any fixed t and as ∆n → 0,
∆−1/2
n
Z
RCorr (Z, X) −
t
σs τs ρs f 0 s, σs2 ds
0
Z
0
Z
t
−1/2
σs2 ds
0
−1/2 !
q
t
2
L−s
0
2
f s, σs τs ds
→ Z VtRcorr ,
(8)
where VtRcorr is defined in (A.1), Z is a standard normal random variable defined on the extension
of the original probability space.
Investigation of the probability limit of RCorr(Z, X) reveals that f 0 cancels out if we calculate
ratios on local windows instead of the interval [0, t]. In particular, if one takes the limit t → 0
in equation (7), the f 0 cancels out and one obtains the spot leverage at time zero (ρ0 ). The spot
leverage at any point can be estimated using local realized correlation,
kn
X
∆ni+j Z
∆ni+j X · 1{|∆n Z |≤u0 } 1{|∆n X |≤un }
n
i+j
i+j
j=1
n
v
,
ρbi = v
u kn
u kn
uX n
uX n
2
2
t
∆i+j Z 1{|∆n Z |≤u0 } t
∆i+j X 1{|∆n X |≤un }
n
i+j
i+j
j=1
(9)
j=1
where kn is the number of observations in a small block. It is crucial to truncate jumps as the
−1/2
true spot correlation is only defined for the continuous components. Choose kn = θ∆n
θ ∈ (0, ∞). As the length of the local window kn ∆n shrinks to zero,
ρbn[s/∆n ]
for some
estimates consistently
the spot leverage ρs ,
f 0 s, σs2 [σ 2 , X]c,0
[Z, X]sc,0
s
q
q
=q
[Z, Z]c,0
[X, X]c,0
[f 0 (s, σs2 )]2 [σ 2 , σ 2 ]c,0
[X, X]c,0
s
s
s
s
p
ρbn[s/∆n ] → q
(10)
[σ 2 , X]c,0
s
q
=q
= ρs .
c,0
[σ 2 , σ 2 ]s [X, X]c,0
s
Notice how this relationship holds for any smooth function f . In particular, the spot leverage ρs
is invariant to functional transformations of σ 2 and X, a property that is analogous to the linear
invariant property of the standard correlation of two random variables. This property is important
in that it eliminates the impact of risk premia embedded in the pricing function f . A consistent
estimator of the integrated leverage ILt is thereby a sum of the estimators of the spot leverage
(appropriately scaled) over the interval of interest [0, t], that is,
IRLnaive
t
∆n
=
t
[t/∆n ]−kn
p
ρbni →
X
i=0
9
1
t
Z
t
ρs ds = ILt .
0
(11)
We denote this estimator the naive instrument-based realized leverage, IRLnaive . Although it is
consistent, it has a first-order asymptotic bias.
kn IRLnaive
t
− ILt
1
→
2t
p
Z
t
(ρ3s − ρs )ds,
(12)
0
which can be deduced from Theorem 2 below.
Our proposed estimator, the instrument-based realized leverage, IRL, corrects this asymptotic
bias as well as an additional finite sample bias,
∆n
IRLt =
t
[t/∆n ]−kn X
ρbni
i=0
k ∆ 1 n 3
n n
n
−
+
(b
ρi ) − ρbi
ρbn0 + ρbn[t/∆n ]−kn .
2kn
2t
(13)
The last term is a finite sample bias, which arises because the end-of-sample observations enter the
estimator differently than most other observations when overlapping blocks are used. The following
theorem presents the asymptotic distribution of the IRL estimator. The estimator of the asymptotic
variance is given in Appendix B.
$
0
Theorem 2. Suppose Assumptions 1 and 2 hold. Let un ∆$
n , un ∆n and with some ζ > 0,
5/(12 − 2r) ≤ $ < 1/2. Then, for any fixed t, and as ∆n → 0, kn → ∞ with kn2 ∆n → 0 and
kn3 ∆n → ∞, we have
L−s
∆−1/2
(IRLt − ILt ) → Z
n
q
VtIRL ,
where
VtIRL =
1
t2
Z
t
1 − 2ρ2s + ρ4s ds,
0
and Z is a standard normal random variable defined on the extension of the original probability space.
−1/2
When estimating the spot correlation, only the rate of the bandwidth ∆n
is compatible with
the optimal rate of convergence, see Alvarez, Panloup, Pontier, and Savy (2012). Interestingly,
Theorem 2 and its proof show that for the purpose of estimating the integrated correlation, a slower
rate for kn ensures the feasibility of the inference by diminishing asymptotic biases, while maintaining
−1/2
the same asymptotic variance as the case when kn is of order ∆n
. The final estimator is easy
to construct and has a simple expression for the asymptotic variance. Our simulations show it also
has a good finite sample performance, and empirically we find it tends to produce more stable time
series than other alternatives.
The IRL estimator assumes existence of a volatility instrument, which might not be available
for all stocks. We now describe nonparametric price-only leverage estimation.
10
4
Leverage Estimation using Price Observations alone
The current section describes the price-only realized leverage, a nonparametric estimator of the
leverage effect using price (X) observations alone, without using any volatility instrument, and
presents its asymptotic distribution.
We start with the high-frequency equivalent of the typical leverage estimator, i.e., a plug-in
correlation estimator. If observations on the spot variance σ 2 were available, the simplest estimator
of the leverage effect would be the truncated realized correlation between σ 2 and X, RCorr σ 2 , X .
It would estimate the volatility-weighted integrated leverage V W ILt . The unobservable spot variance
can be estimated with the truncated realized variance over a small block, that is,
2
σ
bi∆
n
kn
2
1 X
∆ni+j X 1{|∆n X |≤un } ,
=
i+j
kn ∆n
j=1
with the threshold un satisfying conditions of Theorem 3 below. We denote the resulting plug-in
estimator the naive price-only realized leverage,
c
P RLnaive
t
\
2 , X]
[σ
qt
,
=q
c
c
\
2
2
\
[σ , σ ]t [X, X]t
(14)
where
c
[t/∆n ]
\
[X,
X]t =
X
(∆ni X)2 1{|∆n X |≤un } ,
i
i=1
c
1
\
2 , X] =
[σ
t
kn
c
1
2, σ2] =
[σ\
t
kn
[t/∆n ]−2kn X
2
2
σ
b(i+k
−σ
bi∆
n
n )∆n
i=0
[t/∆n ]−2kn X
2
2
σ
b(i+k
−σ
bi∆
n
n )∆n
2
X(i+kn )∆n − Xi∆n · 1{|X(i+k )∆ −Xi∆ |≤vn } ,
n
n
n
,
i=0
and where the thresholds un and vn are specified below. Note that we use overlapping block estimates
of σ
b2 , i.e., we use an average of subsampled realized covariances and variances. It results in an
estimator that has smaller asymptotic variance than the non-overlapping version while retaining the
same probability limit. Unfortunately, this estimator is inconsistent,
p
P RLnaive
→
t
1 2
[σ , X]ct
r 2
p
[X, X]ct 23 [σ 2 , σ 2 ]ct +
6
θ2
Rt
,
(15)
4
0 σu du
as ∆n → 0, a result that can be deduced from Theorem 3. The above probability limit suggests the
11
following consistent estimator, which we call the price-only realized leverage,
bc
\
2 , X]
[σ
,
P RLt = q
qt
bc
c
\
2
2
\
[σ , σ ]t [X, X]t
(16)
where
bc
\
2 , X]
[σ
t
bc
2, σ2]
[σ\
t
c
\
2 , X] ,
= 2[σ
t
=
c
3 \
[σ 2 , σ 2 ]t −
2
(17)
[t/∆n ]−2kn
X
i=0
6
2
4
1−
σ
bi∆
.
n
kn2
kn
(18)
Theorem 3. Suppose Assumption 1 holds with δ̃ = 0 and r ∈ (0, 1/2). Moreover, τ is a continuous
Rt
Rt
Itô semimartingale, i.e. τt = τ0 + 0 b̄s ds + 0 σ̄s dW s , with b̄s locally bounded, and σ̄s locally bounded
fs . Suppose
and càdlàg. W s is another Brownian motion, which can be correlated with Ws and W
√
−1/2
$
kn = θ/ ∆n for some θ ∈ (0, ∞). Let kn ∆n , un ∆$
for $ ∈
n and vn (kn ∆n )
[7/(16 − 4r), 1/2). Then, for any fixed t and as ∆n → 0,
L−s
∆−1/4
(P RLt − V W ILt ) → Z
n
q
VtP RL ,
(19)
where VtP RL is defined in equation (C.23) and where Z is a standard normal random variable defined
on the extension of the original probability space.
The above theorem assumes no volatility jumps while allowing for price jumps.10 The expression
of the asymptotic variance of the P RL, VtP RL , is rather involved and is therefore delegated to the
appendix. Appendix C contains the proof of Theorem 3, as well as presents a consistent estimator
of VtP RL . An alternative way to estimate the asymptotic variance is by applying the subsampling
method of Kalnina (2011a) and Kalnina (2011b). Note that the rate of convergence of the P RL
estimator is only the square root of the rate of convergence of the IRL estimator.
The numerator of the P RL estimator is closely related to Wang and Mykland (2012) who estimate
2 σ , X t , although their results were obtained in a somewhat different setting.11 Specifically, they
assume no jumps and use non-overlapping blocks, which results in a larger asymptotic variance.
The bias-corrected estimator of the volatility of volatility is the same as in Jacod and Rosenbaum
(2012b) who only show consistency, and similar to Vetter (2012b) who assumes no jumps. The proof
of Theorem 3 derives the joint asymptotic distribution of these components needed to evaluate the
standard errors of the P RL estimator.
We remark here on the relationship of our Theorem 3 to Aı̈t-Sahalia, Fan, and Li (2013). They
10
Incorporating volatility jumps is possible, but substantially complicates the proof and further deteriorates the finite
sample performance of this estimator.
11
Mykland and Zhang (2009) contains interesting preliminary analysis used in Wang and Mykland (2012).
12
provide an insightful discussion on different biases arising due to the latent nature of the spot
volatility when estimating the leverage effect in the Heston model. For example, one can use their
results to interpret the factors 2/3 and 1/2 in equation (15) as smoothing biases. The additive bias
Rt
in equation (15), θ42 0 σs4 ds, is due to the estimation error of integrated volatilities over subsamples,
which resembles 4tE(σt4 ) in their bias correction. They provide a bias-corrected estimator for the
leverage, but do not derive its asymptotic properties.
The P RL estimator arises naturally when considering the popular realized correlation estimator.
It estimates consistently the volatility-weighted leverage effect measure V W ILt . The P RL estimator can be modified to estimate the integrated leverage effect ILt (see the difference between the
RCorr(Z, X) and IRLnaive estimators), but the resulting estimator is somewhat more involved as it
requires an additional bias correction. Besides, in the price-only estimation, there is no obvious reason to prefer one leverage measure over the other. In contrast, a naive attempt to estimate V W ILt
using a volatility instrument results in contamination with risk premia, and estimation of the ILt
measure by the IRL estimator solves the problem.
5
5.1
Discussion
Asynchronicity
One potential drawback of using an additional volatility instrument as compared to using return
data alone lies in the asynchronicity between observations of the underlying index and the volatility
instrument. In particular, the S&P 500 index and the VIX are updated approximately every 15
seconds but rarely in tandem. As a result, there might be a bias towards zero for the volatility
instrument-based estimators. This is the so-called Epps Effect, see, e.g., Zhang (2011), Hayashi and
Yoshida (2005), Barndorff-Nielsen, Hansen, Lunde, and Shephard (2011), Aı̈t-Sahalia, Fan, and Xiu
(2010), and Li, Mykland, Renault, Zhang, and Zheng (2013). To mitigate this concern, we follow a
common approach in the literature and sample all series, including the implied volatility, at lower
frequencies, which is determined by the signature plots (see Section 7.1). Also, we use high frequency
S&P 500 futures to construct the leverage effect estimators, which are known to be very liquid.
5.2
Market Microstructure Noise
The market microstructure noise is an important ingredient of high frequency transaction level
data. See Chen and Xu (2013) for a discussion regarding this noise in the option observations.
Volatility signature plots as discussed in Hansen and Lunde (2006) are very useful tools for identifying
frequencies that are contaminated by noise. Noise-robust estimators for the quadratic variation
−1/4
can have a rate of convergence at most ∆n
, see, e.g., Zhang, Mykland, and Aı̈t-Sahalia (2005),
Barndorff-Nielsen, Hansen, Lunde, and Shephard (2008), Reiß (2011), Fan and Wang (2007), Jacod,
13
Li, Mykland, Podolskij, and Vetter (2009), Kalnina and Linton (2008), and Xiu (2010). Wang and
−1/8
Mykland (2012) use the pre-averaging method to construct a noise-robust ∆n -convergent estimator
of σ 2 , X t . We remark that a pre-averaged version of the P RLt would still suffer the problem of
being too noisy over short intervals of time, as demonstrated by our simulations.12 Extensions of
our theory to a pre-averaged version of IRLt require an extension of Jacod and Rosenbaum (2012b)
to pre-averaging based estimators, which is well beyond the scope of this paper. Since we assume no
microstructure noise, our estimator cannot be calculated at the highest available frequency where this
assumption is clearly violated. Instead, we follow the common approach in the literature by sampling
at lower frequencies guided by the signature plots. When estimating the integrated variance, Bandi
and Russell (2008), and Aı̈t-Sahalia, Mykland, and Zhang (2005) study the choice of the optimal
frequency. The choice of the optimal sampling frequency for the estimation of the leverage effect is
clearly an interesting question that is left for future research.
5.3
Multi-Factor Volatility Models
What Assumption 2 effectively needs is that under the risk-neutral measure, the dynamics of the VIX
is only driven by the unobserved volatility, i.e., Zt = f t, σt2 . Although this does not necessarily
imply that the volatility process can only be driven by one factor,13 it does rule out a few popular
models in the empirical finance literature, see, e.g., Christoffersen, Heston, and Jacobs (2009). For
example, a common two-factor volatility process under the risk-neutral measure Q can be specified
2 + σ 2 , where for i = 1, 2,
as σt2 = σ1t
2t
2
2
dσit
= κi (θi − σit
)dt + γi σit dWitQ ,
Q
Q
) = 0, E(dWitQ dWtQ ) = ρit dt, where WtQ drives the risk-neutral dynamics of the
dW2t
and E(dW1t
Q
return, i.e., dXt = σt dWtQ + dLQ
t , with Lt a pure-jump Lévy process. Then we can derive the
2 + c σ 2 , where c , c , and c are constants, and usually c 6= c . In
squared VIX as Zt = c0 + c1 σ1t
2 2t
0
1
2
1
2
this case, the IRLt estimator converges to
1
t
t
Z
0
c1 γ1 ρ1s σ1s + c2 γ2 ρ2s σ2s
1
p
ds 6=
2
2
2
2
2
2
t
c1 γ1 σ1s + c2 γ2 σ2s
Z
0
t
γ1 ρ1s σ1s + γ2 ρ2s σ2s
p
ds = ILt .
2 + γ 2σ2
γ12 σ1s
2 2s
−1/8
12
In particular, a ∆n -convergent noise-robust P RL would be able to use frequencies like 1 second, but the right
panel of Table 2 (denoted “1 month, 1-second”) shows that even the faster converging realized leverage is quite noisy
with 1 second data over 1 month. So a noise-robust P RL estimator would also require longer time periods for precise
estimates.
13
For example, the following class of multi-factor volatility models also yields a one-factor relationship between the
VIX and the instantaneous variance:
dσt2 = κ(θ − σt2 )dt + dMtQ ,
where MtQ is a martingale that could be driven by multiple factors.
14
2 , is that it captures the term
Nevertheless, the typical justification for one of the two factors, say σ2t
2 will not contribute much to the quadratic (co-)variations, γ
structure of variance. In that case, σ2t
2
R
1 t
is much smaller than γ1 , so we obtain ILt ≈ t 0 ρ1s ds, and the IRL estimator is consistent.
6
Simulations
This section considers the finite sample properties of the instrument-based leverage estimators using
two different volatility instruments, the VIX and the implied volatility, followed by the finite sample
properties of the price-only estimators.
We simulate two models, the Heston and the log volatility (LogV) models. In the Heston model,
the log-price X satisfies
dXt = (µ0 + µ1 σt2 )dt + σt dWt + JX dNt ,
(20)
ft + Jσ2 dNt − βσ2 λt dt,
dσt2 = κ(ξ − σt2 )dt + γσt dW
(21)
ft are two Brownian Motions with correlation ρ, Nt is a Poisson process with statewhere Wt and W
2 ),
dependent intensity λt = λ0 + λ1 σt2 , JX is a random jump size of X satisfying JX ∼ N (µX , σX
and Jσ2 is a random jump size of σt2 satisfying Jσ2 ∼ exp(βσ2 ). We set the parameters to the values
typically used in the literature: κ = 5, γ = 0.35, ρ = −0.8, ξ = 0.09, µX = −0.02, σX = 0.05,
βσ2 = 0.01, λ0 = 30, λ1 = 60, µ0 = 0.05, and µ1 = 0.5. In this model, VIX2t = a + bσt2 where
the constants a and b depend on the parameters of the risk-neutral dynamics of X (see Li and Xiu
(2013)). We set them to VIX2t = 1002 · 0.01 + 0.8σt2 . Note that in the Heston model, ILt = V W ILt
and both measures equal ρ.
In the LogV model, the log-price X satisfies
dXt = (µ0 + µ1 σt2 )dt + σt dWt + JX dNt − λµX dt,
ft + JF dNt − µF λdt,
dFt = κFt dt + σdW
σt2 = exp (α + βFt ) ,
ft are two Brownian Motions with correlation ρ, Nt is a Poisson process with
where again Wt and W
2 ), and J ∼ N (µ , σ 2 ). We set the values of the parameters to α = −2.8,
intensity λ, JX ∼ N (µX , σX
F
F
F
β = 7.5, ρ = −0.75, µX = −0.02, σX = 0.05, µF = 0.02, σF = 0.02, σ = 0.6, κ = −4, λ = 25,
µ0 = 0.05, and µ1 = 0.5. In this model, VIX2t = a + bσt2c for some constants a, b, and c (see Li and
Xiu (2013)). We set VIX2t = 1002 · 0.01 + 0.75σt2×0.8 . In this model, IRL converges to ILt = ρ,
but the limit of RCorr (X, Z) is
R t 2c+1
σ
ds
f 0 s, σs2 [σ 2 , X]c,0
s ds
qR
= ρ qR 0 s qR
≤ ρ,
p
t 0
t 4c
t 2
2 )]2 [σ 2 , σ 2 ]c,0 ds [X, X]c
[f
(s,
σ
σ
ds
σ
ds
s
s
t
0
0 s
0 s
Rt
0
15
(22)
LLN: RCorr
LLN: IRL and IRL−naive
50
50
37.5
37.5
25
25
12.5
12.5
0
−0.85
−0.8 −0.75
0
−0.85
−0.7 −0.65
CLT: RCorr
0.4
0.2
0.2
−2
0
−0.7 −0.65
CLT: IRL and IRL−naive
0.4
0
−4
−0.8 −0.75
2
0
−4
4
−2
0
2
4
Figure 1: Instrument-Based Leverage Effect Estimators in the Heston Model. The upper panel
provides the histograms of the instrument-based estimators, whereas the bottom panel plots the
standardized histograms. The gray areas show the histograms of the RCorr and IRL estimators.
The dashed lines show the histograms of the IRLnaive estimators. The solid lines denote the true
parameter values or the standard normal density. The parameters are ∆n = 1 minute, T = 5 days,
and θ = 0.25.
with the equality holding when c = 0.5. The above limit of RCorr (X, Z) is random and in general
neither equals to ILt nor V W ILt . Moreover, it is affected by the risk-neutral dynamics through
√
its dependence on c. Recall that kn is the number of observations in a block and θ = kn ∆n .
Throughout these experiments, we choose three different values of kn such
q that θ equals 0.25, 0.5,
c
0
\
Y ] /t∆0.47 , where Y is
and 0.75. We fix the truncation levels (values of un , u , and vn ) to be 3 [Y,
n
either the return of the stock or the change in the VIX, and [Y, Y
t
]ct
n
is estimated by bipower variation
(see Barndorff-Nielsen and Shephard (2004) and Aı̈t-Sahalia and Jacod (2012)).
In Figures 1 and 2, we provide histograms based on 10,000 repetitions of the three instrumentbased estimators using 1-minute returns spanning over a 5-day window. Figure 1 contains the
results for the Heston model and Figure 2 contains results for the log volatility model. In both
models, ILt = ρ, which equals −0.8 and −0.75, respectively. The first column in each figure reports
the centered estimators, P RLt − ILt , RCorr (X, Z) − ILt , and IRLt − ILt . RCorr (X, Z) is also
centered at ILt , even though in the logV model its probability limit is the left-hand side of equation
(22). The second column of each figure reports the estimators that are centered and scaled by the
rate of convergence and the estimated asymptotic variance. We see that IRLt and RCorr (X, Z) are
both well centered in the Heston model, but IRLt has a slightly smaller standard deviation. In the
log volatility model, RCorr (X, Z) deviates from ILt , whereas IRLt remains perfect.
16
LLN: RCorr
LLN: IRL and IRL−naive
50
50
37.5
37.5
25
25
12.5
12.5
0
−0.85
−0.8 −0.75
0
−0.85
−0.7 −0.65
CLT: RCorr
0.4
0.2
0.2
−2
0
−0.7 −0.65
CLT: IRL and IRL−naive
0.4
0
−4
−0.8 −0.75
2
0
−4
4
−2
0
2
4
Figure 2: Instrument-Based Leverage Effect Estimators in the LogV Model. The upper panel provides the histograms of the instrument-based estimators, whereas the bottom panel plots the standardized histograms. The gray areas show the histograms of the RCorr and IRL estimators. The
dashed lines show the histograms of the IRLnaive estimators. The solid lines denote the true parameter values or the standard normal density. The parameters are ∆n = 1 minute, T = 5 days, and
θ = 0.25.
Table 1 provides the median bias and interquartile range (IQR) of the three volatility instrumentbased estimators using 1-minute and 5-minute returns. For the Heston model (panel A), IRLt and
RCorr (X, Z) perform similarly. In the log volatility model (panel C), RCorr (X, Z) has a clear bias.
In any case, the IRL estimator performs the best.
We also conduct a simulation with an alternative volatility instrument, at-the-money implied
volatility with time-to-maturity equal to two months. For convenience, we simulate the standard
Heston model under the objective measure, as described in (20) and (21), but without any jumps.
Under the pricing measure, assuming the market price of volatility risk is proportional to the volatility
level, we have
f Q − ησt2 dt,
dσt2 = κ(ξ − σt2 )dt + γσt dW
t
(23)
where η = −1.5, corresponding to a negative variance risk premium. Based on this model, we
calculate the 2-month at-the-money implied volatility, and use it as our instrument. The results are
provided in the panel (B) of the Table 1. The bias of RCorr (X, Z) is very small in this particular
setting, although it is an inconsistent estimator of the ILt . The IRL estimator is very precise and
unbiased. We have also used out-of-the money implied volatility as an instrument, but the results
17
θ
0.250
1 week, 5-min
0.500
0.750
0.250
1 week, 1-min
0.500
0.750
(A) Heston Model: VIX-based estimators
RCorr: median bias
0.002
0.002
IRLnaive : median bias
0.077
0.147
IRL: median bias
0.002
0.002
RCorr: IQR
0.026
0.026
IRLnaive : IQR
0.025
0.023
IRL: IQR
0.026
0.026
0.002
0.218
0.002
0.026
0.021
0.026
0.001
0.034
0.001
0.011
0.011
0.011
0.001
0.065
0.001
0.011
0.010
0.011
0.001
0.097
0.001
0.011
0.010
0.011
(B) Heston Model: IMV-based estimators
RCorr: median bias
0.005
0.005
IRLnaive : median bias
0.074
0.139
IRL: median bias
0.003
0.003
RCorr: IQR
0.033
0.033
IRLnaive : IQR
0.030
0.028
IRL: IQR
0.031
0.032
0.005
0.205
0.003
0.033
0.025
0.032
0.003
0.033
0.001
0.014
0.013
0.013
0.003
0.062
0.001
0.014
0.013
0.014
0.003
0.092
0.001
0.014
0.012
0.014
(C) LogV Model: VIX-based estimators
RCorr: median bias
0.001
naive
IRL
: median bias
0.076
IRL: median bias
0.000
RCorr: IQR
0.025
naive
IRL
: IQR
0.024
IRL: IQR
0.025
0.001
0.216
0.000
0.025
0.020
0.025
0.000
0.034
0.000
0.011
0.011
0.011
0.000
0.065
0.000
0.011
0.010
0.011
0.000
0.097
0.000
0.011
0.010
0.011
0.001
0.145
0.000
0.025
0.022
0.025
Table 1: Instrument-Based Leverage Effect Estimators. The finite sample properties of RCorr(X, Z),
IRLnaive , and IRL. The IQR denotes the interquartile range.
18
are similar and hence are omitted.
In addition, we investigate the finite sample performance of the P RL estimator and the naive
P RL estimator in Table 2. We use the Heston model above but with no volatility jumps. The
results confirm the difficulty in using these price-only estimators to achieve reasonable finite sample
properties with data spanning only one quarter: the performance with 5-minute returns is very
weak. Its performance improves when using 10 years of data. Observe that with 5-minute returns
over 10 years, the P RL estimator delivers more precise results than with 1-second returns over 1
month, despite using more than twice the observations in the latter case. As we point out in Section 4,
increasing the time-span is more effective than sampling at higher frequency in the price-only setting.
Our experiment with 1-second returns effectively demonstrates that a version of the P RL estimator
that is robust to the market microstructure noise would not be able to deliver precise estimates over
the time span of one month (at least in the Heston model). Our 1-second experiment serves this
purpose because a noise-robust P RL would have a larger variance than the P RL estimator due to
the microstructure noise, but it would be able to use the higher frequencies such as 1-second.
LLN: PRL
CLT: PRL
5
0.4
3.75
2.5
0.2
1.25
0
−1.5
−1.2
−0.9
−0.6
−0.3
0
−4
0
−2
0
2
4
Figure 3: Price-Only Leverage Effect Estimators in the Heston Model. The left panel provides the
histograms of the P RL estimators, whereas the right panel plots the standardized histograms. The
gray areas show the histograms of the P RL estimators. The solid lines denote the true parameter
values or the standard normal density. The parameters are ∆n = 1 minute, T = 10 years, and θ = 1.
7
7.1
Empirical Results
Data and Preliminary Analysis
We use intraday time series of S&P 500 futures and the VIX from the Tick Data Inc., and the intraday
S&P 500 options from the Chicago Board Options Exchange (CBOE) over the time period 2003 2012. The VIX sample period starts from September 22, 2003, when the CBOE began disseminating
prices for the VIX with the new methodology, and ends on December 31, 2012. We extend the
intraday VIX series to January 1, 2003, by calculating the VIX using intraday options, based on the
19
θ
P RL: median bias
P RLnaive : median bias
P RL: IQR
P RLnaive : IQR
bc
2, σ2] < 0
P RL : % [σ\
t
P RL : % outside [−1, 1]
10 years, 5-min
0.75
1.00
1.25
0.08
0.07
0.07
0.65
0.61
0.58
0.22
0.18
0.17
0.04
0.04
0.05
1 quarter, 5-min
0.75
1.00
1.25
0.05
-0.01
-0.02
0.63
0.59
0.56
0.79
0.78
0.72
0.15
0.18
0.20
1 month, 1-sec
0.75
1.00
1.25
-0.03
-0.02
-0.01
0.61
0.57
0.53
0.35
0.29
0.26
0.07
0.08
0.08
0.0%
6.8%
32%
23%
2.3%
27%
0.0%
3.5%
0.0%
2.0%
22%
30%
14%
32%
0.4%
21%
0.0%
16%
Table 2: Price-Only Leverage Effect Estimators in the Heston Model. Finite sample properties of
P RL and P RLnaive for the Heston model with X jumps only. The IQR denotes the interquartile
bc
2, σ2]
range. % [σ\
< 0 denotes the percentage of experiments that yield negative volatility of
t
volatility estimates, whereas % outside [−1, 1] gives the percentage of correlation estimates that are
outside their natural range.
same method developed by the CBOE,14 so that the full sample period covers 10 years in total. We
obtain a time series of the S&P 500 future prices by rolling over the front contracts. In addition, we
use S&P 500 options to extract the intraday 5-minute time series of at-the-money two-month implied
volatility as an alternative volatility instrument. After removing non-trading days or half-trading
days, our sample contains 2438 days. Overnight returns are excluded from our data. Figure 4 plots
the time series of the intraday S&P 500 futures, the VIX, and the 2-month at-the-money implied
volatility from January 1, 2003 to December 31, 2012.
In our choice of the sampling frequencies, we take into account the potential issue of the microstructure noise contaminations and potential asynchronicity between the high frequency observations of different time series. In light of these aspects, we sample and synchronize S&P 500 futures,
the VIX, and implied volatility data on a range of frequencies from every 5-minute to every hour,
and use the signature plots in Figure 5 to guide our choice of an appropriate sampling frequency.
Both signature plots indicate potential biases at 5-minute frequencies that are decreasing with lower
frequencies, and the plots stabilize around the 30-minute frequency. We consider three different time
spans, but there is virtually no variation of the signature plots across the time spans, so the same
observations apply for the weekly, monthly, and quarterly estimates of the leverage effect.
7.2
Time Series of Estimates
We first compare the IRL estimates with two volatility instruments, the VIX and the implied volatility (IMV), along with the price-only realized leverage, the P RL, based on 10-years of S&P 500 future
prices alone.
We plot the two time series based on the IRL estimator in Figure 6, calculated using 30 minute
14
See the CBOE White paper on VIX via http://www.cboe.com/micro/vix/vixwhite.pdf.
20
S&P 500 Futures
VIX
80
1400
60
1200
1000
40
800
20
01/2003 06/2006 09/2009 12/2012
01/2003 06/2006 09/2009 12/2012
IMV
0.6
0.4
0.2
01/2003 06/2006 09/2009 12/2012
Figure 4: Time Series of the S&P 500 Future Prices, the VIX, and Implied Volatility. In this figure,
we plot the time series of the intraday S&P 500 futures, the VIX, and the 2-month at-the-money
implied volatility from January 1, 2003 to December 31, 2012.
VIX
−0.5
weekly
IMV
monthly
−0.5
quarterly
−0.6
−0.6
−0.7
−0.7
−0.8
−0.8
−0.9
5
15
25
35
45
−0.9
55
weekly
5
15
monthly
25
35
quarterly
45
55
Figure 5: Signature Plots of Leverage Effect Estimates. The leverage signature plots, i.e., the time
series averages of instrument-based leverage estimates, with minutes on the x-axis. The left panel is
based on the VIX, while the right panel uses the implied volatility. Estimates using weekly, monthly,
and quarterly time spans are denoted with blue solid line, black circles, and red stars, respectively.
21
frequency and using monthly time spans. The average for the VIX-based IRL is -0.78 and the average
for the IRL estimator based on the implied volatility is -0.74. Clearly, the leverage effect estimates of
IRL are negative and time-varying. The time series pattern of the two series is similar.15 Next, Table
3 provides leverage estimates over the time span of ten years (2003 - 2012) across different frequencies
and choices of the block size. We present the choice of the number of observations in a block in terms
√
of the parameter θ = kn ∆n . As can be seen, the values of θ or sampling frequencies (15-minute
and higher) do not have any effect for any of the two IRL estimators. On the other hand, the P RL
estimates are much smaller, and are very sensitive to the choices of θ and sampling frequencies. This
agrees with our reservations on the use of P RL in practice, despite it being constructed under weaker
assumptions.
0
Monthly IRL (IMV)
Monthly IRL (VIX)
Correlation
−0.2
−0.4
−0.6
−0.8
−1
01/2003
07/2005
01/2008
06/2010
12/2012
Figure 6: Time Series of Leverage Effect Estimates. Monthly leverage estimates of the S&P 500 for
the years 2003 - 2012 using 30-minute data, together with point-wise confidence intervals. The black
confidence band corresponds to the VIX-based IRL. The grey confidence band corresponds to the
IRL estimator based on the at-the-money implied volatility (IMV).
The time series plots suggest two observations. First, our estimates indicate that the Brownian
co-movement between the S&P 500 and its volatility is very pronounced and cannot be ignored. This
challenges the pure jump specification of volatility process as suggested by Todorov and Tauchen
(2011). Second, the correlation between the driving Brownian Motions of the price and the volatility
is clearly time-varying. We next investigate the potential contributing factors in this variation.
15
We do not present similar graphs for the P RL estimates as they are unstable and hence uninformative.
22
θ
0.25
0.5
0.75
5 min
15 min
30 min
60 min
-0.688
-0.741
-0.766
-0.776
-0.685
-0.739
-0.765
-0.777
-0.684
-0.739
-0.765
-0.776
5 min
15 min
30 min
60 min
-0.668
-0.748
-0.782
-0.806
-0.667
-0.745
-0.780
-0.805
-0.666
-0.744
-0.780
-0.804
5 min
15 min
30 min
60 min
-0.395
-0.532
-0.421
-0.250
-0.467
-0.486
-0.323
-0.163
-0.444
-0.397
-0.412
-0.332
1
1.5
VIX-IRL
-0.684
-0.685
-0.739
-0.739
-0.764
-0.764
-0.775
-0.774
IMV-IRL
-0.665
-0.664
-0.744
-0.742
-0.779
-0.778
-0.803
-0.802
P RL
-0.444
-0.396
-0.355
-0.363
-0.479
-0.453
-0.422
-0.407
2
2.5
3
-0.685
-0.739
-0.763
-0.774
-0.684
-0.739
-0.762
-0.773
-0.684
-0.738
-0.762
-0.773
-0.662
-0.741
-0.777
-0.801
-0.662
-0.740
-0.776
-0.800
-0.661
-0.739
-0.776
-0.799
-0.405
-0.382
-0.413
-0.255
-0.404
-0.386
-0.331
-0.129
-0.406
-0.383
-0.256
-0.080
Table 3: Leverage Estimates across Frequencies and θ. The IRL estimates based on VIX, the IRL
estimates based on the implied volatility (IMV), as well as the P RL estimates. All three estimators
use the 10 year time span (2003-2012) for different sampling frequencies using
√ different block sizes.
Different columns represent different block lengths as determined by θ = kn ∆n .
Variable
TED
(1)
(2)
-0.078**
(0.034)
(3)
0.139**
(0.058)
LIQ
DER
LEVt−1
adj. R2
(4)
0.522***
(0.073)
0.276
0.497***
(0.072)
0.289
0.513***
(0.073)
0.282
-0.217*
(0.124)
0.524***
(0.071)
0.275
(5)
-0.039*
(0.021)
0.122**
(0.060)
-0.383*
(0.202)
0.504***
(0.070)
0.288
Table 4: Time Series Regression Results. Column (5) contains the results of the time series regression
(24) with the VIX-based IRL as the dependent variable. Columns (1)-(4) contain results of the
corresponding simple regressions. DER denotes the log total debt-to-total-equity ratio of the S&P
500 index, TED denotes the TED spread, and LIQ is a monthly liquidity measure. ε denotes
the corresponding AR(1) innovations. The reported standard errors are based on the Newey-West
procedure. All regressions include an intercept.
23
7.3
Regression Analysis
To explore the determinants of the variation of the leverage effect, we conduct the following regression
analysis. First, we select the following economic variables: a measure of the credit risk, the TED
spread (TED), calculated as the difference between the three-month LIBOR and the three-month TBill interest rate; a liquidity measure (LIQ), which is obtained from Pastor and Stambaugh (2003);
and the logarithm of the monthly total debt-to-total-equity ratio (DER) of the S&P 500 index,
downloaded from Bloomberg. Our sample consists of monthly leverage effects over 10 years, which
is a total of 118 observations.
We estimate the following AR(1) regression:
LEVt = β0 + β1 εT ED,t + β2 εLIQ,t + β3 εDER,t + β4 LEVt−1 + εt
(24)
where ε·,t denotes the corresponding AR(1) innovation of each covariate. From the regression results
in Table 4, we find that the credit and liquidity factors are relevant to the leverage effect and
their coefficients have signs consistent with the economic intuition (recall the dependent variable is
negative). They imply that the leverage effect is magnified in bad times, i.e., a 1 percentage drop of
stock price when credit risk is high and liquidity is low, may lead to a larger percentage increase of
risk. Moreover, the debt-to-equity is significant, which supports the financial leverage hypothesis of
Black (1976) that debt-to-equity is one of the determinants of the leverage effect. The latter finding
is robust with respect to various alternative specifications, which are omitted for brevity.
8
Conclusion
We study a new nonparametric estimator of the leverage effect, the instrument-based realized leverage, which gives reliable estimates over short intervals of time. The IRL estimator uses the data
from two sources: the stock prices as well as a volatility instrument such as Black-Scholes implied
volatility or the VIX. The two time series can be combined for two reasons. First, we assume a
nonparametric functional relationship between the stock price and the spot volatility. Second, the
spot correlation is invariant to smooth functional transformations of any of the two variables. We
prove the asymptotic theory of the IRL estimator and conduct simulation and empirical studies,
confirming its desirable properties. In contrast, should one be unwilling to make any assumptions
on the relationship between the spot volatility and some external time series from the derivative
markets, the spot volatility has to be estimated. We derive the asymptotic theory for the price-only
realized leverage, allowing us to evaluate its precision. We find that it has high standard errors when
evaluated at short intervals such as a month or a quarter, and it is very sensitive to the choice of
the block size and the sampling frequency. Nonparametric estimation of the leverage effect there-
24
fore entails a stark trade-off as to whether any information from the derivatives markets should be
used. We argue that the cost in the form of a mild additional assumption of a volatility instrument
is outweighed by the benefit of obtaining a precise yet nonparametric time series of leverage effect
estimates.
References
Aı̈t-Sahalia, Y., J. Fan, and Y. Li (2013): “The Leverage Effect Puzzle: Disentangling Sources
of Bias at High Frequency,” Journal of Financial Economics, forthcoming.
Aı̈t-Sahalia, Y., J. Fan, and D. Xiu (2010): “High-Frequency Covariance Estimates with Noisy
and Asynchronous Data,” Journal of the American Statistical Association, 105, 1504–1517.
Aı̈t-Sahalia, Y., and J. Jacod (2012): “Analyzing the Spectrum of Asset Returns: Jump and
Volatility Components in High Frequency Data,” Journal of Economic Literature, 50, 1005–1048.
Aı̈t-Sahalia, Y., and R. Kimmel (2007): “Maximum Likelihood Estimation of Stochastic Volatility Models,” Journal of Financial Economics, 83, 413–452.
Aı̈t-Sahalia, Y., P. A. Mykland, and L. Zhang (2005): “How Often to Sample a ContinuousTime Process in the Presence of Market Microstructure Noise,” Review of Financial Studies, 18,
351–416.
Aldous, D. G., and G. K. Eagleson (1978): “On Mixing and Stability of Limit Theorems.,”
Annals of Probability, 6, 325–331.
Alvarez, A., F. Panloup, M. Pontier, and N. Savy (2012): “Estimation of the Instantaneous
Volatility,” Statistical Inference and Stochastic Processes, 15, 27–59.
Andersen, T., O. Bondarenko, and M. T. Gonzalez-Perez (2012): “Uncovering Novel Features of Equity-Index Return Dynamics via Corridor Implied Volatility,” Discussion paper, Northwestern University.
Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys (2003): “Modeling and
Forecasting Realized Volatility,” Econometrica, 71, 579–625.
Bandi, F., and R. Reno (2012): “Time-Varying Leverage Effects,” Journal of Econometrics, 12,
94–113.
Bandi, F. M., and J. R. Russell (2008): “Microstructure Noise, Realized Volatility and Optimal
Sampling,” Review of Economic Studies, 75, 339–369.
25
Barndorff-Nielsen, O. E., P. R. Hansen, A. Lunde, and N. Shephard (2008): “Designing Realized Kernels to Measure Ex-post Variation of Equity Prices in the Presence of Noise,”
Econometrica, 76, 1481–1536.
Barndorff-Nielsen, O. E., P. R. Hansen, A. Lunde, and N. Shephard (2011): “Multivariate
Realised Kernels: Consistent Positive Semi-Definite Estimators of the Covariation of Equity Prices
with Noise and Non-Synchronous Trading,” Journal of Econometrics, 162, 149–169.
Barndorff-Nielsen, O. E., and N. Shephard (2004): “Power and bipower variation with
stochastic volatility and jumps,” Journal of Financial Econometrics, 2, 1–37.
Black (1976): “Studies of Stock Price Volatility Changes,” Proceedings of the 1976 Meetings of the
American Statistical Association, Business and Economic Statistics, 6, 177–181.
Bollerslev, T., U. Kreschmer, C. Pigorsch, and G. E. Tauchen (2009): “A Discrete-Time
Model for Daily S&P 500 Returns and Realized Variations: Jumps and Leverage Effects,” Journal
of Econometrics, 150, 151–166.
Bollerslev, T., and V. Todorov (2011): “Tails, Fears and Risk Premia,” Journal of Finance,
66, 2165–2211.
Carr, P., and L. Wu (2009): “Variance Risk Premiums,” Review of Financial Studies, 22, 1311–
1341.
Chen, S. X., and Z. Xu (2013): “On Implied Volatility for Options - Some Reasons to Smile and
More to Correct,” Journal of Econometrics, Forthcoming.
Christoffersen, P., S. Heston, and K. Jacobs (2009): “The Shape and Term Structure of the
Index Option Smirk: Why Multifactor Stochastic Volatility Models Work so Well.,” Management
Science, 55, 1914–1932.
Cont, R., and P. Tankov (2004): Financial Modelling with Jump Processes. Chapman &
Hall/CRC.
Delbaen, F., and W. Schachermayer (1994): “A General Version of the Fundamental Theorem
of Asset Pricing,” Mathematische Annalen, 300, 463–520.
Eraker, B. (2004): “Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and
Option Prices,” The Journal of Finance, 59, 1367–1404.
Fan, J., and Y. Wang (2007): “Multi-Scale Jump and Volatility Analysis for High-Frequency
Financial Data,” Journal of the American Statistical Association, 102, 1349–1362.
26
Hansen, P. R., and A. Lunde (2006): “Realized Variance and Market Microstructure Noise,”
Journal of Business and Economic Statistics, 24, 127–161.
Hayashi, T., and N. Yoshida (2005): “On Covariance Estimation of Non-synchronously Observed
Diffusion Processes,” Bernoulli, 11, 359–379.
Jacod, J., Y. Li, P. A. Mykland, M. Podolskij, and M. Vetter (2009): “Microstructure
Noise in the Continuous Case: The Pre-Averaging Approach,” Stochastic Processes and Their
Applications, 119, 2249–2276.
Jacod, J., and P. Protter (2012): Discretization of Processes. Springer.
Jacod, J., and M. Rosenbaum (2012a): “Estimation of Volatility Functionals: the Case of a
√
n
Window,” Discussion paper, Université de Paris-6.
Jacod, J., and M. Rosenbaum (2012b): “Quarticity and Other Functionals of Volatility: Efficient
Estimation,” Annals of Statistics, Forthcoming.
Jacquier, E., N. G. Polson, and P. E. Rossi (2007): “Bayesian analysis of stochastic volatility
models with fat-tails and correlated errors,” Journal of Econometrics, 127, 615–640.
Joshi, M. (2007): “Log-type models, Homogeneity of Option Prices and Convexity,” Discussion
paper, Melbourne University.
Kalnina, I. (2011a): “Nonparametric Tests of Time Variation in Betas,” Discussion paper, University of Montreal.
(2011b): “Subsampling High Frequency Data,” Journal of Econometrics, 161, 262–283.
Kalnina, I., and O. Linton (2008): “Estimating Quadratic Variation Consistently in the Presence
of Endogenous and Diurnal Measurement Error,” Journal of Econometrics, 147, 47–59.
Li, J., and D. Xiu (2013): “Spot Variance Regressions,” Discussion paper, Duke University and
University of Chicago.
Li, Y., P. A. Mykland, E. Renault, L. Zhang, and X. Zheng (2013): “Realized Volatility
When Sampling Times are Possibly Endogenous,” Econometric Theory, Forthcoming.
Medvedev, A., and O. Scaillet (2007): “Approximation and Calibration of Short-Term Implied
Volatilities Under Jump-Diffusion Stochastic Volatility,” Review of Financial Studies, 20(2), 427–
459.
27
Mykland, P. A., N. Shephard, and K. Sheppard (2012): “Efficient and feasible inference
for the components of financial variation using blocked multipower variation,” Economics Papers
2012-W02, Economics Group, Nuffield College, University of Oxford.
Mykland, P. A., and L. Zhang (2009): “Inference for Continuous Semimartingales Observed at
High Frequency,” Econometrica, 77(5), 1403–1445.
Nelson, D. B. (1991): “Conditional Heteroskedasticity in Asset Returns: A New Approach,”
Econometrica, 59, 347–370.
Pan, J. (2002): “The jump-risk premia implicit in options: Evidence from an integrated time-series
study,” Journal of Financial Economics, 63, 3–50.
Pastor, L., and R. F. Stambaugh (2003): “Liquidity Risk and Expected Stock Returns,” Journal
of Political Economy, 111, 642–685.
Reiß, M. (2011): “Asymptotic Equivalence for Inference on the Volatility from Noisy Observations,”
Annals of Statistics, 39(2), 772–802.
Song, Z., and D. Xiu (2012): “A Tale of Two Option Markets: State-Price Densities and Volatility
Risk,” Discussion paper, Federal Reserve Board and University of Chicago.
Todorov, V. (2010): “Variance Risk Premium Dynamics: The Role of Jumps,” Review of Financial
Studies, 23, 345–383.
Todorov, V., and G. E. Tauchen (2011): “Volatility Jumps,” Journal of Business & Economic
Statistics, 29(3), 356–371.
Vetter, M. (2012a): “Estimation of Correlation for Continuous Semimartingales,” Scandinavian
Journal of Statistics, 39, 757–771.
Vetter, M. (2012b): “Estimation of Integrated Volatility of Volatility with Applications to
Goodness-of-fit Testing,” Discussion paper, Ruhr-Universität Bochum.
Wang, D. C., and P. A. Mykland (2012): “The Estimation of Leverage Effect with High Frequency Data,” Discussion paper, The University of Chicago.
Xiu, D. (2010): “Quasi-Maximum Likelihood Estimation of Volatility with High Frequency Data,”
Journal of Econometrics, 159, 235–250.
Yu, J. (2005): “On Leverage in a Stochastic Volatility Model,” Journal of Econometrics, 127,
165–178.
28
Zhang, L. (2011): “Estimating Covariation: Epps Effect and Microstructure Noise,” Journal of
Econometrics, 160, 33–47.
Zhang, L., P. A. Mykland, and Y. Aı̈t-Sahalia (2005): “A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High-Frequency Data,” Journal of the American Statistical
Association, 100, 1394–1411.
Appendix
A
Proof of Theorem 1
By Itô’s Lemma, the Brownian part of dZt is given by
f 0 (t, σt2 )τt dW̃t .
Next we apply the central limit theorem below, which can be shown following Theorem 13.2.4 of
Jacod and Protter (2012).

c
\
[X,
X]t − [X, X]ct

1  \c
c  L−s
√ 
[Z,
X]
−
[Z,
X]
t  → Yt
t
∆
c
c
\
[Z,
Z] − [Z, Z]

t
t
where Yt is defined on an extension of the original probability space, which conditional on F, is a
Gaussian process with covariance matrix:
E(Yt Yt> |F)
 R
2
Rt 2 2 0
Rt
t
2ρs σs f (s, σs2 )τs ds
2ρs σs3 f 0 (s, σs2 )τs ds
2σs4 ds
0
0
0
 Rt
0
0
3
Rt
Rt
2
2 2
2 2
3 0
2
=
 R0 2ρs σs f (s, σs )τs ds
0 2ρs σs f (s, σs )τs ds
0 (1 + ρs )σs τs f (s, σs ) ds
0
3
4
Rt
Rt 0
2
t
2 2
0
2
2
2
0 2ρs σs f (s, σs )τs ds
0 2ρs σs f (s, σs )τs ds
0 2 f (s, σs )τs ds


,

By the “Delta” method, the asymptotic variance of RCorr(X, Z) is
VtRcorr
Z
Z
3
0
2
1 2 −3 −1 t
1 2 −2 −2 t
2 2
2
= B3,t B1,t B2,t
(1 + ρs )σs f s, σs τs ds + B3,t B1,t B2,t
2ρs σs f 0 s, σs2 τs ds
4
2
0
0
Z t
Z t
1
4
−1 −1
−1 −3
2
+ B1,t
B2,t
2σs4 ds + B3,t
B1,t
B2,t
2 f 0 s, σs2 τs ds
4
0
0
Z t
−2 −1
− B3,t B1,t
B2,t
2ρs σs3 f 0 s, σs2 τs ds
0
Z t
2
−1 −2
− B3,t B1,t B2,t
2ρ2s σs2 f 0 t, σs2 τs ds,
0
29
t
Z
σs2 ds,
B1,t =
t
Z
B2,t =
f
0
s, σs2
τs
2
Z
ds,
B3,t =
0
0
t
σs τs ρs f 0 s, σs2 ds,
(A.1)
0
and its estimator can be constructed using Theorem 9.2.1 of Jacod and Protter (2012).
B
Proof of Theorem 2
Let ct be the covariation of the continuous parts of X and Z. By the localization procedure (see
Lemma 4.4.9 in Jacod and Protter (2012)), we can assume that kct k ≤ A0 and λ(ct ) > A−1
0 , where
λ(·) = min(c11 , c22 ), for any t.
Next, introduce the “correlation” function g(·) and its regularized version gA (·), for any A > 0,
as follows,
c12
g(c) = √ √ 1{c11 >0,c22 >0}
c11 c22
(
and gA (c) =
g(c)
if λ(c) > (2A)−1
0
if λ(c) ≤ (4A)−1 .
(B.2)
Notice that gA (·) is a C ∞ function which satisfies the polynomial growth condition in Jacod and
Protter (2012), namely, |∂ j gA (c)| ≤ K(1 + kck3−j ), for j = 0, 1, 2, 3, where ∂ j denotes the jth
derivatives.
Our spot leverage estimator is ρbni = g(b
ci∆n ), where
b
ci∆n =
b
c11
c12
i∆n b
i∆n
!
b
c21
c22
i∆n b
i∆n
and
b
c11
i∆n
kn
2
1 X
∆ni+j Z · 1{|∆n Z |≤u0 } ,
=
n
i+j
kn ∆n
j=1
b
c12
i∆n
kn
1 X
=
∆ni+j Z ∆ni+j X · 1{|∆n Z |≤u0 } 1{|∆n X |≤u0 } ,
n
n
i+j
i+j
kn ∆n
j=1
b
c22
i∆n
kn
2
1 X
=
∆ni+j X · 1{|∆n X |≤u0 } .
n
i+j
kn ∆n
j=1
We aim to apply Theorem 3.2 in Jacod and Rosenbaum (2012b), which concludes the following.
For any C 3 function g that satisfies the above polynomial growth condition, we have

Z t
[t/∆n ]−kn

X
1
−1/2
∆n
∆n
g(b
ci∆n ) −
g(cs )ds −

2kn
0
i=0
q
L−s
→
VtIRL Zt ,
30
2
X
j,k,l,m=1
2
∂jk,lm
g(b
ci∆n )(b
cjl
ckm
cjm
ckl
i∆n + b
i∆n )
i∆n b
i∆n b



where cjl
s is the covariation of the continuous part of a multivariate Itô semimartingale, Zt is a
standard Gaussian variable, and VtIRL is the conditional variance given by
VtIRL (g)
2
X
=
Z
t
km
jm kl
∂jk g(cs )∂lm g(cs )(cjl
s cs + cs cs )ds.
j,k,l,m=1 0
b A and G,
b which are based on gA (·) and g(·), respectively.
Denote the bias-corrected estimators by G
b but it does establish the asymptotic distribution
The above theorem does not directly apply to G,
b A . Note that on ΩA0 = λ(cs ) > A−1 , for any s ∈ [0, t] , we have
of G
t
0
Z
t
t
Z
gA0 (cs )ds =
and VtIRL (gA0 ) = VtIRL (g).
g(cs )ds
0
0
b it remains to show that
Therefore, in order to establish the asymptotic distribution for G,
∆n−1/2
[t/∆n ]−kn o p
n
X
→ 0 and
∆n
ci∆n )
g(b
ci∆n ) − gA0 (b
(B.3)
i=0
(
∆n−1/2 ∆n
[t/∆n ]−kn
X
2
X
i=0
j,k,l,m=1
)
2
g(b
ci∆n ) − gA0 (b
∂jk,lm
ci∆n ) (b
cjl
ckm
cjm
ckl
i∆n + b
i∆n )
i∆n b
i∆n b
Note that the final correction term does not affect the asymptotic distribution, i.e.,
g(b
c([t/∆n ]−kn )∆n )) =
1/2
op (∆n )
and
kn ∆n
c0 )
2t (gA0 (b
+ gA0 (b
c([t/∆n ]−kn )∆n )) =
p
→ 0.
(B.4)
k n ∆n
c0 )
2t (g(b
1/2
op (∆n ).
By the Lipchitz continuity of λ(·), there exists some > 0 such that for any i,
kb
ci∆n − ci∆n k ≤ implies |λ(b
ci∆n ) − λ(ci∆n )| ≤ (2A0 )−1 ,
0
which implies λ(b
ci∆n ) > (2A0 )−1 on ΩA
t . Hence, we have
[t/∆n ]−kn
∆n1/2
X
[t/∆n ]−kn
|g(b
ci∆n ) − gA0 (b
ci∆n )|
X
≤∆1/2
n
(|g(b
ci∆n )| + |gA0 (b
ci∆n )|) 1
i=0
i=0
[t/∆n ]−kn
X
≤2∆n1/2
i=0
1
kb
ci∆n −ci∆n k>
On the other hand, for any 0 > 0,
[t/∆n ]−kn
∆1/2
n
X
i=0
1
kb
ci∆n −ci∆n
31
k>0
p
→
0,
.
|λ(b
ci∆n )|≤(2A0 )−1
+
since
[t/∆n ]−kn
E 1
X
∆1/2
n
kb
ci∆n −ci∆n
i=0
≤∆1/2
n
K
0 3
k>0
[t/∆n ]−kn
= ∆n1/2
X
P (kb
ci∆n − ci∆n k > 0 )
i=0
[t/∆n ]−kn n
X
3 o
3
0
ci∆n − ci∆n ci∆n − b
c0i∆n + E b
E b
i=0
K ≤ 0 3 an ∆(6−r)$−3+1/2
+ kn−3/2 ∆n−1/2 + kn ∆1/2
→ 0,
n
n
where the last inequality follows by (4.8) and (4.11) of Jacod and Rosenbaum (2012b), the fact that
$ ≥ 5/(12 − 2r), and that an → 0 and kn2 ∆n → 0 as ∆n → 0. Hence, by (2.2.35) of Jacod and
Protter (2012), we have
[t/∆n ]−kn
∆1/2
n
X
p
|g(b
ci∆n ) − gA0 (b
ci∆n )| → 0,
i=0
and (B.3) holds. To show (B.4), note that
2
X
2
g(b
ci∆n )(b
cjl
ckm
cjm
ckl
ρni )3 − ρbni
∂jk,lm
i∆n + b
i∆n ) = (b
i∆n b
i∆n b
j,k,l,m=1
and that
[t/∆X
n ]−kn
2
X
jm
jl
kl
km
2
1/2 (b
ci∆n )
ci∆n ) − gA0 (b
ci∆n + b
ci∆n b
∂jk,lm g(b
ci∆n ) 1
ci∆n b
∆n −1
λ(b
ci∆n )≤(2A0 )
i=0 j,k,l,m=1
[t/∆n ]−kn 2
X X
jl
jm kl
1/2
n 3
n
2
km
≤∆n
ρi ) − ρbi −
∂jk,lm gA0 (b
ci∆n )(b
ci∆n b
ci∆n + b
ci∆n b
ci∆n ) 1
(b
λ(b
ci∆n )≤(2A0 )−1
i=0
j,k,l,m=1
≤K∆n1/2
[t/∆n ]−kn X
1 + (1 + kb
ci∆n k) kb
ci∆n k2 1
kb
ci∆n −ci∆n k>
i=0
,
and that for any 0 > 0, by triangle inequality kb
ci∆n k ≤ kb
ci∆n − ci∆n k + kci∆n k, and that
1
ci∆n − ci∆n kj
≤ kb
, for j = 0, 1, 2, 3,
kb
ci∆n −ci∆n k>0
0 j
we have
[t/∆n ]−kn
∆1/2
n
X
E
1 + (1 + kb
ci∆n k) kb
ci∆n k2 1
kb
ci∆n −ci∆n k>0
i=0
32
≤K∆1/2
n
[t/∆n ]−kn X
3
0
3 E b
ci∆n − b
c0i∆n + E b
ci∆n − ci∆n .
i=0
Hence, (B.4) follows from the same argument as above.
b By direct calculations, we have
As a result, the CLT holds for G.
2
X
Z
t
km
∂jk g(cs )∂lm g(cs )(cjl
s cs
+
kl
cjm
s cs )ds
Z
=
j,k,l,m=1 0
t
(1 − 2ρ2s + ρ4s )ds.
0
This establishes the asymptotic distribution of the IRL estimator.
To estimate the asymptotic variance, we define
[t/∆n ]−kn
∆n
VbtIRL = 2
t
X
1 − 2b
ρ2i + ρb4i .
(B.5)
i=0
Then, by Theorem 9.4.1 of Jacod and Protter (2012) and the same localization argument, we have
p 1
VbtIRL → 2
t
C
Z
t
(1 − 2ρ2s + ρ4s )ds.
0
Proof of Theorem 3
We now prove Theorem 3. After that, we provide an estimator of the asymptotic variance. Recall
that
Z
t
Xt =X0 +
σt2
=σ02
Z
t
bs ds +
Z
0
t
+
Z tZ
σs dWs +
Z
0
t
b̃s ds +
δ(s, z)µ(ds, dz)
0
R
fs
τs dW
0
0
where µ(ds, dz) is a Poisson random measure with intensity measure ν(ds, dz) = ds ⊗ λ(dz). The
fs is ρs . bs and b̃s are locally bounded, and τs is also càdlàg. Moreover,
correlation between Ws and W
|δ (ω, t ∧ τm (ω) , z)| ∧ 1 ≤ Γm (z) for all (ω, t, z), where (τm ) is a localizing sequence of stopping times
R
and, for some r ∈ (0, 1), each function Γm on R satisfies R Γm (z)r λ (dz) < ∞. From time to time,
p
fs = ρs dWs + 1 − ρ2 dW̄s , where Ws and W̄s are independent Brownian motions.
we need to use dW
s
As usual, by localization, we can assume |bs | ≤ K, |Xs | ≤ K, |σs | ≤ K, |b̃s | ≤ K, and |τs | ≤ K for
0 ≤ s ≤ t.
Recall that we have the following notation:
Xt0
Z
=
t
Z
bs ds +
0
σs dWs ,
0
33
t
(C.6)
so that Xt = X0 + Xt0 +
P
s≤t ∆Xs .
And for i = 0, i, . . . , [t/∆n ] − kn ,
02
σ
bi∆
=
n
kn
1 X
(∆ni+j X 0 )2 ,
k n ∆n
(C.7)
j=1
0
0
where ∆ni X 0 = Xi∆
− X(i−1)∆
. We also define
n
n
0
0, X 0] =
\
[X
t
[t/∆n ]−kn X
0
0
− Xi∆
X(i+k
n
n )∆n
2
,
(C.8)
0
0
−
X
,
X(i+k
i∆
n
n )∆n
(C.9)
i=1
0
2
\
0 , σ 02 ] =
[X
t
kn
0
1
02 , σ 02 ] =
[σ\
t
kn
[t/∆n ]−2kn X
02
02
−σ
bi∆
σ
b(i+k
n
n )∆n
i=1
[t/∆n ]−2kn X
i=0
2
3 02
1
02
σ
b(i+kn )∆n − σ
bi∆
−6
n
2
kn
2
1−
kn
04
σ
bi∆
n
.
(C.10)
d·] is used to emphasis that the estimator does not have truncation.
The 0 outside [·,
Our strategy is to derive the joint CLT for the above estimators, and show that their asymptotic
variances are the same as the estimators defined in (16), (17) and (18). We then apply the Delta
0
0, X 0]
\
method to find the asymptotic variance of (16). From the analysis we will see below, [X
t
converges at a faster rate, hence its estimation will not affect the asymptotic variance of (16) at the
second order. Hence we only need the joint distribution of (C.9) and (C.10).
We at first make sure that the truncated estimators introduced in equations (16), (17) and (18)
0
0 , X 0 ] is well
\
will have the same asymptotic properties as (C.8), (C.9) and (C.10). The result for [X
0
t
c
0 , X 0 ] = [X
0 , X 0 ] on a set whose probability goes to 1 as n → ∞, see page 386
\
\
known. Note that [X
t
t
in Jacod and Protter (2012) for more details. Also note that
√
c
1 \c
0, X 0]
\
[X, X]t − [X
t → 0, u.c.p.
∆n
In fact, as long as 1/(4 − 2r) ≤ $ < 1, the above statement is true, according to Equation (13.2.27)
in Jacod and Protter (2012).
For the remaining two estimators, we start with the following inequalities, which can be shown
by the Cauchy-Schwarz and triangle inequalities:
2
4
02
2
04 2
02
2 2
02 bi∆n − σ
bi∆
≤
σ
b
−
σ
b
+
2
σ
b
−
σ
+
σ
σ
b
−
σ
b
σ
i∆n
i∆n
i∆n
i∆n
i∆n ,
i∆n
i∆n
n
2 2 2
2
02
02
σ
bi∆n − σ
b(i+kn )∆n − σ
bi∆n b(i+kn )∆n − σ
34
2 2 2
2
02 02
+
σ
b
−
σ
b
b(i+k
≤2 σ
b(i+kn )∆n − σ
i∆n
i∆n n )∆n
2
2
02 02
02 02
−
σ
b
b(i+k
+
σ
b
−
σ
b
σ
b
+ 2 σ
b(i+kn )∆n − σ
i∆n .
i∆n
i∆n
(i+kn )∆n
n )∆n
Also, using Equations (4.3) and (4.8) in Jacod and Rosenbaum (2012b), and the convexity of xp for
p ≥ 1, as well as Burkholder-Gundy and Hölder inequalities, we have for some sequence an going to
0, and for p, q ≥ 1:
q
2
02 ,
Eσ
bi∆n − σ
bi∆
≤ Kan ∆(2q−r)$+1−q
n
n
p
p
02
02
2
2
02 2
2
02 −
σ
+
σ
−
σ
b
+
σ
−
σ
bi∆
≤
E
σ
b
Eσ
b(i+kn )∆n − σ
i∆n
i∆n
i∆n (i+kn )∆n
(i+kn )∆n
(i+kn )∆n
n
≤ K(kn ∆n )p/2 .
Using Hölder inequality again, we have
0
bc
\
\
02
02
2
2
E [σ , σ ]t − [σ , σ ]t ≤ Kan ∆(4−r)$−3/2
,
n
Therefore, as long as $ ≥ 7/(16 − 4r),
∆−1/4
n
0
bc
\
\
02
02
2
2
[σ , σ ]t − [σ , σ ]t → 0, u.c.p.
Similarly, we have
02
2
0
0
2
02
X
−
X
b(i+kn )∆n − σ
bi∆
X
−
X
−
σ
b
−
σ
b
σ
i∆n
(i+kn )∆n
i∆n i∆n
(i+kn )∆n
(i+kn )∆n
n
· 1{|X(i+k )∆ −Xi∆ |≤vn }
n
n
n
0
02
02 0
≤ σ
b(i+kn )∆n − σ
bi∆n X(i+kn )∆n − Xi∆n − X(i+k
−
X
i∆n 1{|X(i+kn )∆n −Xi∆n |≤vn }
n )∆n
2
2
02
02
−
σ
b
−
σ
b
+ X(i+kn )∆n − Xi∆n σ
b(i+kn )∆n − σ
bi∆
i∆n 1{|X(i+k )∆ −Xi∆ |≤vn } .
(i+kn )∆n
n
n
n
n
Moreover, using (4.3) in Jacod and Rosenbaum (2012b) and the following inequality (which can be
shown by simple calculations)
|y|q 1{|x+y|≤ε} ≤ K
|x|q+η
εη
+ |y|q ∧ εq , for some η > 0,
we have
q
00
00 E X(i+k
−
X
1{|X(i+k )∆ −Xi∆ |≤vn }
i∆
)∆
n
n
n
n
n
n
q+4/(1−4$)
q
0
00
0
00 q
≤K vn−4/(1−4$) E X(i+k
−
X
+
E
X
−
X
∧
v
i∆n
i∆n
n
(i+kn )∆n
n )∆n
35
≤K (kn ∆n )q/2+(2−4$)/(1−4$) + Kan (kn ∆n )q$ (kn ∆n )1−$r .
Using Hölder inequality, we obtain
0
02
0
02 −
X
bi∆n X(i+kn )∆n − Xi∆n − X(i+k
E σ
b(i+kn )∆n − σ
1{|X(i+k )∆ −Xi∆ |≤vn }
i∆
)∆
n
n
n
n
n n
1/2
1/2+(2−4$)/(q−4q$)
1/q
$
(1−$r)/q
≤K(kn ∆n )
(kn ∆n )
+ Kan (kn ∆n ) (kn ∆n )
.
Also, based on estimates of (4.3) and (4.8) in Jacod and Rosenbaum (2012b), we have
o
n
2
02
2
02
1
−
σ
b
bi∆
−
σ
b
b(i+kn )∆n − σ
E X(i+kn )∆n − Xi∆n σ
i∆n {|X(i+kn )∆n −Xi∆n |≤vn }
(i+kn )∆n
n
o
n 02
2
2 02
bi∆n − σ
bi∆
b(i+k
b(i+kn )∆n − σ
≤vn E
+ σ
σ
n
n )∆n
≤K(kn ∆n )$ an ∆(2−r)$
.
n
Therefore, given $ ∈ [7/(16 − 4r), 1/2), we also have
∆−1/4
n
bc
bc
\
\
2
0
02
[X, σ ]t − [X , σ ]t → 0, u.c.p.
Using the same argument as on page 386 in Jacod and Protter (2012), we have
0
bc
\
\
0 , σ 02 ]
2 ] − [X
∆−1/4
[X,
σ
n
t → 0, u.c.p.
t
Now that we have shown that it is sufficient to find a joint CLT of (C.9) and (C.10), we proceed
with the following definitions. Let
Ai∆n
Bi∆n
kn Z (i+j)∆n
2 X
0
=
(Xs0 − X(i+j−1)∆
)dXs0
n
kn ∆n
(i+j−1)∆
n
j=1
Z (i+kn )∆n
1
=
σs2 ds.
kn ∆n i∆n
Using Itô’s formula, we have
02
σ
bi∆
= Ai∆n + Bi∆n .
n
Next we follow the same strategy used in Vetter (2012b) to find the joint CLT. Let p ∈ N be arbitrary.
We set for l = 1, 2, . . . , Jn (p) with Jn (p) = [([t/∆n ] − 2kn )/(p + 2)/kn ] the number of big blocks, and
the block end points
al (p) = (l − 1)(p + 2)kn ,
36
bl (p) = al (p) + pkn .
Define
kn
2 X
n
,
(C.11)
σα2 l (p)∆n Hi+j
kn ∆n
j=1
Z (i+kn )∆n
q
1
=
ταl (p)∆n ραl (p)∆n (Ws − Wi∆n ) + 1 − (ραl (p)∆n )2 (W̄s − W̄i∆n ) ds,
kn ∆n i∆n
Ãi∆n =
B̃i∆n
(C.12)
where
Hin
Z
=
i∆n
1
(Ws − W(i−1)∆n )dWs = ((∆in W )2 − ∆n ).
2
(i−1)∆n
Therefore, we have
Ã(i+kn )∆n − Ãi∆n
B̃(i+kn )∆n − B̃i∆n
kn
1 X
2
n
=
σα2 l (p)∆n (∆i+j+k
W )2 − (∆i+j
,
n
n W)
kn ∆n
j=1
Z (i+kn )∆n
1
τal (p)∆n ρal (p)∆n (Ws+kn ∆n − Ws )
=
kn ∆n i∆n
q
+ 1 − (ρal (p)∆n )2 (W̄s+kn ∆n − W̄s ) ds.
We can then define
Jn (p)
Utn,p =
X
Uln,p ,
(C.13)
i=1
bl (p)−1
Uln,p =
2
3 (Ã(i+kn )∆n − Ãi∆n ) + (B̃(i+kn )∆n − B̃i∆n )
2kn
i=al (p)
6
4
2
− pkn ∆n
σ
+ τal (p)∆n .
kn2 ∆n al (p)∆n
X
(C.14)
(C.15)
It has been shown by Vetter (2012b) that
0
02 , σ 02 ] − [σ 2 , σ 2 ]c − U n,p | = 0,
lim lim sup ∆n−1/4 E|[σ\
t
t
t
p→∞ ∆n →0
(C.16)
and the asymptotic variance of Utn,p has been calculated:
Jn (p)
Jn (p) 1 X
48p
151pkn2 ∆2n
1 X n
n,p 2
8
4
2
4
√
E
(U ) =
σal (p)∆n 2 + 12σal (p)∆n τal (p)∆n p∆n + τal (p)
kn ∆n
kn
70
∆n l=1 al (p) l
l=1
Z t
Z t
Z t
48
12
151θ
→ 3
σ 8 ds +
σ 4 τ 2 ds +
τ 4 ds, as ∆n → 0, p → ∞.
(C.17)
θ 0 s
θ 0 s s
70 0 s
37
0
\
0 , σ 02 ] . Define
Now we proceed with a similar strategy to calculate the asymptotic variance of [X
t
Jn (p)
Vtn,p
=
X
Vln,p , where
(C.18)
l=1
Vln,p
bl (p)−1 2 e
ei∆n + B
e(i+k )∆ − B
ei∆n σα (p)∆ W(i+k )∆ − Wi∆n
A(i+kn )∆n − A
n
n
n
n
n
l
kn
i=al (p)
)
− ∆n τal (p)∆n σal (p)∆n ρal (p)∆n
.
(C.19)
=
X
Notice that the terms with |i − m| > 2kn are vanished, and when i = m, the contribution is of a
smaller order, so we have
Eal (p) (Vln,p )2
bl (p)−1
=
i−1
X
8 e
e
e
e
A
−
A
+
B
−
B
σα2 l (p)∆n W(i+kn )∆n
i∆n
i∆n
(i+kn )∆n
(i+kn )∆n
kn2
i=al (p) m=i−2kn
em∆n
e(m+k )∆ − A
em∆n + B
e(m+k )∆ − B
−Wi∆n ) × A
W(m+kn )∆n − Wm∆n
n
n
n
n
2 o
−2∆2n τal (p)∆n σal (p)∆n ρal (p)∆n
+ smaller terms.
X
Eal (p)
The following calculations are used from time to time in the calculations below, which we provide
here for convenience. For i − 2kn ≤ m ≤ i, we can obtain:
kn
1
X
X
(−1)δ1 +δ3 1{1≤i−m+j1 +(δ1 −δ3 )kn ≤kn }
j1 =1 δ1 ,δ3 =0
=(2kn + 3m − 3i)1{m−i>−kn } − (2kn + m − i)1{m−i<−kn }
(C.20)
and
Z
(i+kn )∆n
Z
(m+kn )∆n
E(W̃s1 +kn ∆n − W̃s1 )(W̃s2 +kn ∆n − W̃s2 )ds1 ds2
i∆n
Z (i+kn )∆n
=
m∆n
Z (m+kn )∆n
(kn ∆n
m∆n
(m+2kn −i)3 3
∆n
6
3 −6(i−m)2 k +3(i−m)3
4kn
n
∆3n
6
− |s1 − s2 |) ∨ 0ds1 ds2
i∆n
(
=
m < i − kn
m > i − kn
.
(C.21)
We now analyze the terms that contribute to the asymptotic variance one by one. To start with,
n
o
Eal (p) (Ã(i+kn )∆n − Ãi∆n )(Ã(m+kn )∆n − Ãm∆n )(W(i+kn )∆n − Wi∆n )(W(m+kn )∆n − Wm∆n )
38
=
kn X
kn
4σa4l (p)∆n X
kn2 ∆2n
Eal (p) {(Hi+j1 +kn − Hi+j1 )(Hm+j2 +kn − Hm+j2 )
j1 =1 j2 =1
×(W(i+kn )∆n − Wi∆n )(W(m+kn )∆n − Wm∆n )
=
kn X
kn X
1 X
1
4σa4l (p)∆n X
kn2 ∆2n
(−1)δ1 +δ2 Eal (p) {(Hi+j1 +δ1 kn )(Hm+j2 +δ2 kn )
j1 =1 j2 =1 δ1 =0 δ2 =0
×(W(i+kn )∆n − Wi∆n )(W(m+kn )∆n − Wm∆n )
=
kn X
kn X
1 X
1
2σa4l (p)∆n ∆3n X
kn2 ∆2n
(−1)δ1 +δ2 1{i+j1 +δ1 kn =m+j2 +δ2 kn } × ((m + kn − i) ∨ 0)
j1 =1 j2 =1 δ1 =0 δ2 =0
4
2σa (p)∆
= 2l 2 n ∆3n (2kn + 3m − 3i)1{m−i>−kn } − (2kn + m
k n ∆n
2σa4 (p)∆
= 2l 2 n ∆3n (2kn + 3m − 3i)(m + kn − i)1{m−i>−kn } .
k n ∆n
− i)1{m−i<=−kn } × ((m + kn − i) ∨ 0)
Secondly, by tedious calculations, we have
n
o
Eal (p) (B̃(i+kn )∆n − B̃i∆n )(B̃(m+kn )∆n − B̃m∆n )(W(i+kn )∆n − Wi∆n )(W(m+kn )∆n − Wm∆n )
(
τa2l (p)∆n 1
4kn3 − 6(i − m)2 kn + 3(i − m)3 (m + kn − i)∆4n × 1{m>i−kn }
= 2 2
k n ∆n
6
o
n1
1
(m + kn − i)(kn + i − m)∆2n + (i − m)(2kn + i − m)∆2n ρ2al (p) × 1{m>i−kn }
×
2
2 )
1
1
+ (2m − 2i + kn )kn ∆2n + kn4 ∆4n ρ2al (p) .
2
4
Third, it is easy to observe that
n
o
Eal (p) (Ã(i+kn )∆n − Ãi∆n )(B̃(m+kn )∆n − B̃m∆n )(W(i+kn )∆n − Wi∆n )(W(m+kn )∆n − Wm∆n )
n
o
=Eal (p) (Ã(m+kn )∆n − Ãm∆n )(B̃(i+kn )∆n − B̃i∆n )(W(i+kn )∆n − Wi∆n )(W(m+kn )∆n − Wm∆n )
=0.
Overall, we have
Eal (p) (Vln,p )2
bl (p)−1
=
X
8σa2l (p)
i−1
X
i=al (p) m=i−2kn
+
τa2l (p)∆n
kn2 ∆2n
(
kn2
(
2σa4l (p)∆n
∆n
(2kn + 3m − 3i)(m + kn − i)1{m−i>−kn }
kn2
1
4kn3 − 6(i − m)2 kn + 3(i − m)3 (m + kn − i)∆4n × 1{m>i−kn }
6
39
n
o
1
+ (m − i + kn )2 ∆4n × (m + kn − i)(kn + i − m) + (i − m)(2kn − i + m) ρ2al (p)
4
)))
× 1{m>i−kn }
+ smaller terms
∆n 11
13
+ kn σa2l (p) τa2l (p) ∆2n + τa2l (p) ρ2al (p) σa2l (p) kn ∆2n + smaller terms.
=pkn 8σa6l (τ )
kn
5
15
Therefore, as ∆n → 0 and p → ∞,
Jn (p)
13
∆n 11
1
1 X
n,p 2
√ E(Vt ) = √
+ kn σa2l (p) τa2l (p) ∆2n + τa2l (p) ρ2al (p) σa2l (p) kn ∆2n
pkn 8σa6l (τ )
kn
5
15
∆n
∆n l=1
Z t
Z
Z
11θ t 2 2
13θ t 2 2 2
P 8
→
σs6 ds +
σs τs ds +
τ ρ σ ds.
θ 0
5 0
15 0 s s s
We are now left with the asymptotic covariance between the two estimators. Similarly, the
contributing terms are
Jn (p)
X
Coval (p) Vln,p , Uln,p .
l=1
Since both U and V are exactly conditionally centered we have

bl (p)−1
Coval (p) Vln,p , Uln,p = Eal (p) 
X
i−1
X

n,p n,p 
n,p
Ui,l
+ smaller terms,
+ Vm,l
Vi,ln,p Um,l
i=al (p) m=i−2kn
where
n,p
Ui,l
Vi,ln,p
2
3 6
4
2
=
(Ã(i+kn )∆n − Ãi∆n ) + (B̃(i+kn )∆n − B̃i∆n ) − ∆n
σ
+ τal (p)∆n ,
2kn
kn2 ∆n al (p)∆n
2 e
ei∆n + B
e(i+k )∆ − B
ei∆n σα (p)∆ W(i+k )∆ − Wi∆n
=
A(i+kn )∆n − A
n
n
n
n
n
l
kn
− ∆n τal (p)∆n σal (p)∆n ρal (p)∆n .
Again, we expand the paretheses and analyze these terms one by one. It turns out that there are
only two contributing terms. The first term is
6
e
e
e
e
A1 = 2 Eal (p) A(i+kn )∆n − Ai∆n σαl (p)∆n W(i+kn )∆n − Wi∆n × A(m+kn )∆n − Am∆n
kn
e(m+k )∆ − B
em∆n + A
e(m+k )∆ − A
em∆n σα (p)∆ W(m+k )∆ − Wm∆n
× B
n
n
n
n
n
n
n
l
e(i+k )∆ − A
ei∆n B
e(i+k )∆ − B
ei∆n
× A
n
n
n
n
o
12σα5 l (p)∆n ραl (p)∆n ταl (p)∆n ∆n n
(2k
+
3m
−
3i)1
−
(2k
+
m
−
i)1
=
n
n
{m−i>−k
}
{m−i≤−k
}
n
n
kn5
40
1
(m + kn − i)2 o
(m + kn − i)(kn + i − m) + (i − m)(2kn + m − i) +
1{m+kn >i} ,
2
2
2
n1
and the other term is
(
)
3σal (p)
2
e
e
e
e
A2 =Eal (p)
(B(i+kn )∆n − Bi∆n ) W(i+kn )∆n − Wi∆n (B(m+kn )∆n − Bm∆n )
kn2
(
)
3σal (p)
2
e(m+k )∆ − B
em∆n ) W(m+k )∆ − Wm∆n (B
e(i+k )∆ − B
ei∆n )
+ Eal (p)
(B
n
n
n
n
n
n
kn2
σX
2 2
− ∆2n τa2l (p) cσX
al (p)∆n − ∆n τal (p) cal (p)∆n
=
6ρal (p) σal (p) τa3l (p)
4kn3 − 6(i − m)2 kn + 3(i − m)3
kn5
6
n1
1
(m + kn − i)2 o
(m + kn − i)(kn + i − m) + (i − m)(2kn + m − i) +
×
1{m+kn >i} .
2
2
2
∆2n ×
Overall, we have
Jn (p)
Coval (p) Vln,p , Uln,p
X
l=1
bl (p)−1
=
X
i−1
X
i=al (p) m=i−2kn
(
o
12σα5 l (p)∆n ραl (p) ταl (p)∆n ∆n n
(2k
+
3m
−
3i)1
n
{m−i>−kn }
kn5
1
(m + kn − i)2 o
(m + kn − i)(kn + i − m) + (i − m)(2kn + m − i) +
1{m+kn >i}
2
2
2
6ρal (p) σal (p) τa3l (p)
4k 3 − 6(i − m)2 kn + 3(i − m)3
∆2n × n
+
5
kn
6
)
n1
1
(m + kn − i)2 o
×
(m + kn − i)(kn + i − m) + (i − m)(2kn + m − i) +
1{m+kn >i}
2
2
2
n1
13 σ 5 ρa (p) τa (p)∆n ∆n 73
l
al (p) l
=
+ ρal (p) σal (p) .τa3l (p) kn ∆2n × pkn
2
kn
30
Therefore, the asymptotic covariance is
√
Z t
Z
Jn (p)
1 X
73θ t
n,p
n,p p 13
5
Coval (p) Vl , Ul
σs ρs τs ds +
ρs σs τs3 ds.
→
2θ
30
∆n l=1
0
0
Using Lemma 6.1 of Vetter (2012b), we can show that for any fixed p ∈ N,
Jl (p)
1 X
p3
Eal (p) (Uln,p )4 ≤ C 3
→ 0,
∆n
kn ∆n
Jl (p)
1 X
p3
Eal (p) (Uln,p )4 ≤ C 3
→ 0.
∆n
k n ∆n
l=1
l=1
41
Also, following the argument on page 28 of Vetter (2012b), we can show that
Jn (p)
p
1 X
Eal (p) Vln,p (Nal+1 (p)∆n − Nal (p)∆n ) → 0,
∆n
l=1
where N is either W or W̄ or a bounded martingale orthogonal to them. Combining this with (2.16)
in Vetter (2012b) and Theorem 2.2.15 of Jacod and Protter (2012), we have shown that (Utn,p , Vtn,p )
converges jointly stably in law for any p ∈ N. Furthermore, it can be shown following the proof of
(C.16) in Vetter (2012b) that
0
\
0 , σ 02 ] − [X, σ 2 ]c − V n,p | = 0.
lim lim sup ∆n−1/4 E|[X
t
t
t
p→∞ ∆n →0
(C.22)
Therefore, recalling (2.6), (2.7), and (2.8) of Vetter (2012b), we have shown that
0
c 
2
\
0
02
[X , σ ]t − X, σ t
L−s

θb − θ := ∆−1/4
0
2 2 c  → M N (0, Φ) ,
n
02 , σ 02 ] − σ , σ
[σ\

∆−1/4
n
t
t
where
Φ(1,1)
Φ(2,2)
Φ(1,2)
Z
Z
Z
13θ t 2 2 2
11θ t 2 2
8 t 6
σ ds +
σ τ ρ ds +
σ τ ds,
=
θ 0 s
15 0 s s s
5 0 s s
Z
Z
Z
48 t 8
12 t 4 2
151θ t 4
= 3
σs ds +
σs τs ds +
τ ds,
θ 0
θ 0
70 0 s
Z
Z
73θ t
13 t 5
σ ρs τs ds +
ρs σs τs3 ds.
=
2θ 0 s
30 0
c
\
Note that [X,
X]t converges faster and can therefore be treated like a constant, denoted IV. The
b where
last step is to apply the Delta method, as we want the distribution of g(θ),
θ1
g (θ) = √ √ .
θ2 IV
We have
∇g (θ) =
∂g (θ) /∂θ1
∂g (θ) /∂θ2
−1/2
!
=
θ2
IV−1/2
−3/2
− 21 θ1 θ2
IV−1/2
!
,
b is given by
so by the Delta method, the asymptotic variance of g(θ)
[∇g (θ)]> Φ∇g (θ) = Φ(1,1) [∇g (θ)]2(1) + 2Φ(1,2) [∇g (θ)](1) [∇g (θ)](2) + Φ(2,2) [∇g (θ)]2(2) .
42
Therefore,
VtP RL
Z
=
t
σs2 ds
−1 "
t
Z
Φ(1,1)
0
−1
Z t
−2
Z t
τs2 ds
− Φ(2,2)
σs2 τs2 ρ2s ds
τs2 ds
0
1
+ Φ(1,2)
4
Z
0
0
2 Z t
−3 #
t
2
σs τs ρs ds
τs ds
.
0
(C.23)
0
This concludes the proof of Theorem 3. We now present an estimator of the asymptotic variance
Vt . Define
4
13
26 0,0,2,2
θGn
+ θGn0,1,0,2 + θGn1,1,0,0 ,
15
15
15
453 0,0,0,4 1 486 1,2,0,0
1 1038 0,4,0,0
=
θG
−
G
− 3
G
,
280 n
θ 35 n
θ 35 n
73
1
=θ G0,0,1,3
−
219θ3 − 65 Gn0,3,1,0 ,
n
20
15θ
Φ̂(1,1) =
Φ̂(2,2)
Φ̂(1,2)
where (with the convention 00 = 1)
[t/∆n ]−2kn
Gj,l,s,m
n
=∆n
X
(kn ∆n )−
s+m
2
2j
σ̂ 2l X(i+kn )∆n − Xi∆n
τ̂i∆
n i∆n
s
i=0
1s{|X
(i+kn )∆n −Xi∆n
|≤vn }
m
2
2
× σ̂(i+k
−
σ̂
.
i∆
)∆
n
n
n
We claim that
c −1
"
\
VbtP RL = [X,
X]t
bc −2
bc −1
bc
\
\
\
2
2
2
2
2
b
b
Φ(1,1) [σ , σ ]t
− Φ(2,2) [σ , X]t [σ , σ ]t
#
bc −3
bc 2
1b
\
2, σ2]
2 , X]
+ Φ(1,2) [σ
[σ\
t
t
4
is a consistent estimator of VtP RL .
b (2,2) is close to identical to the estimator of the asymptotic
First, note that the estimator Φ
variance of Vetter (2012b) (except for jump truncation). Also, the terms G0,0,2,2 and G0,1,0,2 in the
b (1,1) are similar to the asymptotic variance estimator of Wang and Mykland (2012)
expression of Φ
(except here we use overlapping blocks and jump truncation). The validity of VbtP RL follows from
the results below:
p
G0,0,1,3
→
n
G0,3,1,0
n
Z
t
τs σs5 ρs ds
6θ
0
Z t
p 3
→
τs σs5 ρs ds,
2 0
43
2
+
3
Z
t
τs3 σs ρs ds,
(C.24)
0
(C.25)
p
G1,1,0,0
→
n
Z
t
σs2 τs2 ds,
Z
Z t
1 t 2 2
p 4
→ 2
σs τs 3ρ2s + 4 ds,
σs6 ds +
θ 0
6 0
Z t
Z
2 t 2 2
p 4
6
σ ds +
σ τ ds.
→ 2
θ 0 s
3 0 s s
(C.26)
0
G0,0,2,2
n
G0,1,0,2
n
(C.27)
(C.28)
We prove here the convergence of the first term Gn0,0,1,3 . The proof for the other terms follows
from the same idea. Introduce
3
2
2
1{|X(i+k )∆ −Xi∆ |≤vn } ,
−
σ̂
gi = (kn ∆n )−2 X(i+kn )∆n − Xi∆n σ̂(i+k
i∆n
n )∆n
n
n
n
3
−2
02
0
02
0
0
gi = (kn ∆n )
X(i+kn )∆n − Xi∆n σ̂(i+kn )∆n − σ̂i∆n ,
02
where Xt0 and σ̂i∆
are defined in (C.6) and (C.7), respectively, so that
n
[t/∆n ]−2kn
Gn0,0,1,3
= ∆n
X
gi .
i=0
We also introduce
[t/∆n ]−2kn
G0n
X
= ∆n
gi0 .
i=0
First, applying the same argument before, we have
Gn0,0,1,3 − G0n → 0, u.c.p.
What remains to show is
p
G0n → 12
θ2
t
Z
t
τs σs5 ρs ds +
0
2
3
Z
t
τs3 σs ρs ds.
0
We use the strategy similar to the proof of Theorem 2.5 in Vetter (2012b). Define
kn
2
2 1 X
2
n
n
W
−
∆
W
,
σi∆
∆
i+j
i+j+kn
n
kn ∆n
j=1
Z (i+kn )∆n
1
=
τi∆n (ρi∆n (Ws+kn ∆n − Ws )
kn ∆n i∆n
q
+ 1 − ρ2i∆n W̄s+kn ∆n − W̄s ds.
A(i+kn )∆n − Ai∆n =
B (i+kn )∆n − B i∆n
44
Using equation (2.9) and Lemma 6.1 of Vetter (2012b) with p = 1, we obtain
3
02
02
−
σ̂
(kn ∆n )−3/2 σ̂(i+k
i∆n
n )∆n
p
3
= (kn ∆n )−3/2 A(i+kn )∆n − Ai∆n + B (i+kn )∆n − B i∆n
+ Op
k n ∆n .
Also, a classical result (which can be obtained by Burkholder inequality) is
p
−1/2
0
0
−
X
=
(k
∆
)
σ
W
−
W
+
O
(kn ∆n )−1/2 X(i+k
k
∆
.
n
n
p
n
n
i∆
i∆
(i+kn )∆n
n
n
i∆n
n )∆n
Therefore,
[t/∆n ]−2kn
G0n
=∆n
X
(kn ∆n )−2 σi∆n W(i+kn )∆n − Wi∆n
A(i+kn )∆n − Ai∆n
i=0
+ B (i+kn )∆n − B i∆n
3
+ Op
p
kn ∆n .
By conditional independence,

∆n
2
[t/∆n ]−2kn
X
gi0 − Ei gi0  = Op (kn ∆n ) .
i=0
The contributing terms in Ei (gi0 ) are
h
2
i
(kn ∆n )−2 Ei σi∆n W(i+kn )∆n − Wi∆n 3 A(i+kn )∆n − Ai∆n
B (i+kn )∆n − B i∆n
h
3 i
+ (kn ∆n )−2 Ei σi∆n W(i+kn )∆n − Wi∆n B (i+kn )∆n − B i∆n
,
2
5 ρ
3 σ
giving finally Ei (gi0 ) = 12 θt τi∆n σi∆
+ 32 τi∆
+op (1), which implies the equation (C.24).
ρ
n i∆n
n i∆n i∆n
45