nonparametric tests for analyzing the fine structure of price...
TRANSCRIPT
Nonparametric tests for analyzing the fine structure of price
fluctuations
Rama CONT ∗& Cecilia MANCINI†
Columbia University Center for Financial Engineering
Financial Engineering Report No. 2007–13November 2007.
Abstract
We consider a semimartingale model where (the logarithm of) an asset price is modeled
as the sum of a Levy process and a general Brownian semimartingale. Using a nonparamet-
ric threshold estimator for the continuous component of the quadratic variation (”integrated
variance”), we design
• a test for the presence of a continuous component in the price process
• a test for establishing whether the jump component has finite or infinite variation
based on observations on a discrete time grid. Using simulations of stochastic models com-
monly used in finance, we confirm the performance of our tests and compare them with anal-
ogous tests constructed using multipower variation estimators of integrated variance. Finally,
we apply our tests to investigate the fine structure of the DM/USD exchange rate process and
of SPX futures prices. In both cases, our tests reveal the presence of a non-zero Brownian
component, combined with a finite variation jump component.
∗Center for Financial Engineering & IEOR Dept., Columbia University, New York. Email:
[email protected].†Dipartimento di Matematica per le Decisioni, Universita di Firenze, Italy. Email:
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Contents
1 Introduction 3
2 Definitions and notations 42.1 Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Threshold estimator of integrated variance 63.1 A consistent estimator for integrated variance . . . . . . . . . . . . . . . . . . . . . 63.2 Central limit theorem for the threshold estimator of integrated variance . . . . . . 7
4 Statistical tests 184.1 Test for the presence of a continuous martingale component . . . . . . . . . . . . . 184.2 Testing whether the jump component has finite variation . . . . . . . . . . . . . . 19
5 Numerical experiments 205.1 Testing for the finite variation of the jump component . . . . . . . . . . . . . . . . 205.2 Test for the presence of a Brownian component. Comparison with multipower vari-
ation estimators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Applications to financial time series 276.1 Deutschemark/USD exchange rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.1.1 Does the jump component have finite variation? . . . . . . . . . . . . . . . 286.1.2 Does the price follow a pure-jump process? . . . . . . . . . . . . . . . . . . 28
6.2 SPX index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2.1 Is the jump component of the asset price of finite variation? . . . . . . . . . 296.2.2 Does the price follow a pure-jump process? . . . . . . . . . . . . . . . . . . 30
7 Conclusions 31
8 Appendix 31
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1 Introduction
In order to correct for various empirical shortcomings of diffusion models used in finance, thenecessity of allowing for discontinuities in price behavior has been increasingly recognized in therecent years, both in theoretical studies and in market practice. A wide variety of stochastic modelswith jumps have been proposed for modeling asset price dynamics [11]. While some of these modelsconsist in adding jumps to a (diffusion) process with continuous sample paths [22, 17], some authorshave argued for the use of purely discontinuous processes which move only through jumps [19, 8].Even within the class of purely discontinuous models, one finds a variety of models with differentqualitative properties: finite/infinite jump intensity, finite/infinite variation. These qualitativeproperties may seem to be of a purely theoretical interest but they turn out to have an impact onproperties of option prices: the smooth pasting property for American options [4], the behaviorof European option prices at short maturities [9] and the existence of derivatives (sensitivities)for European and barrier options [12] have been shown to depend on the class –diffusion, jump-diffusion or pure jump– the model belongs to and on the fine properties of the jumps.
In absence of a consensus pointing to a specific parametric class of models for price processes,it is therefore of interest to dispose of nonparametric procedures to investigate these qualitativeproperties of price processes: presence of jumps, presence or not of a Brownian component, natureof the jumps (infinite/finite variation).
The issue of nonparametric detection of jumps has been addressed in the recent literature (seefor example [5, 2, 18]). Other results on the fine structure of semimartingales are given in [26] and[3]. We address here related, but different, issues: testing for the presence of a nonzero Browniancomponent and discriminating between finite/infinite variation jumps.
We consider a semimartingale model for an asset price where the jump component is a Levy pro-cess and the continuous part has stochastic volatility. We use a nonparametric threshold techniqueto estimate the integrated variance [21], based on discretely-observed prices. Without imposingrestrictive assumptions on the stochastic volatility dynamics, we obtain a central limit theorem forour estimator and use it to design
• a test for the presence of a continuous martingale component in the price process, whichallows to discriminate between pure-jump and jump-diffusion models and
• a test for establishing whether the jump component has finite or infinite variation
Using simulations of stochastic models commonly used in finance, we check the performance ofour tests and compare with analogous tests constructed using multipower variation estimators ofintegrated variance developed in [7, 25]. We then apply our test to time series of DM/USD exchangerates and SPX futures. In both cases we find that a non-zero Brownian martingale componentis present in the process generating each data set and it is combined with a finite variation jumpcomponent.
The article is structured as follows. Section 2 introduces the framework, section 3 introducesthe threshold estimator of integrated variance and presents our main theoretical result which iscentral limit theorem for this threshold estimator. Section 4 describes a statistical test for detectingwhether the jump component has finite or infinite variation and a test for detecting the presenceof a Brownian martingale component in a process. In section 5 we apply the tests to simulations ofsome commonly used models and we assess the number of observations and the thresholds whichlead to reliable results. In section 6 we apply the tests, with the previously selected parameters,
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to DM/USD exchange rate and SPX index time series and we present our conclusions in section7. Section 8 is an appendix containing some results used in our proofs.
2 Definitions and notations
Consider the logarithm Xt of the price of a financial asset, modeled as a nonanticipative processwith possibly discontinuous paths, defined on a (filtered) probability space (Ω, (Ft)t∈[0,T ], F , P).In this section we discuss some mathematical preliminaries and define the model setup for Xt.
2.1 Mathematical preliminaries
A Levy process is a process Lt with independent stationary increments. The rate of occurrence ofjumps of a given size is determined by the Levy measure
ν(A) :=1t
E[#{s ∈ [0, T ], ΔLs ∈ A}],
where ΔLs = Ls − Ls− is the size of the jump occurred at time s. The measure ν verifies∫|x|≤1
x2ν(dx) < +∞,
This leaves the possibility of an infinite mass accumulated around the origin i.e. small jumpsoccurring infinitely often in the case ν(R) = ∞ in which case we will say the Levy process hasinfinite activity. The Blumenthal-Getoor (BG) index α, defined as
α := inf{δ ≥ 0,
∫|x|≤1
xδν(dx) < +∞},
measures the activity of small jumps of the Levy process L. In general we have α ∈ [0, 2[. Forinstance a compound Poisson process has α = 0. However even some infinite activity Levy pro-cesses have α = 0, as, for instance, the Gamma process or the Variance Gamma process (VG).An α-stable process has Blumenthal-Getoor index equal to α. An infinite activity process withBlumenthal-Getoor index α < 1 has paths with finite variation, while if α > 1 then the samplepaths have infinite variation almost-surely. Note that when α = 1 we can have either finite orinfinite variation (see examples in [7]). The Normal Inverse Gaussian process (NIG) and the Gen-eralized Hyperbolic Levy motion (GHL) have infinite variation and α = 1. The tempered stableprocess [8, 11] allows for α ∈ [0, 2[.
Any Levy process may be decomposed into the sum of a Brownian motion σWt and a discon-tinuous component which may then be further decomposed as
Lt = σWt + bt + Jt = σWt + bt + J1t + J2t, (1)
whereJ1s :=
∫ s
0
∫|x|>1
xμ(dx, dt)
is the sum of the “large” jumps i.e. with absolute value larger than one, μ is a Poisson randommeasure on [0, T ]× R with intensity measure ν(dx)dt and
J2s :=∫ s
0
∫|x|≤1
x[μ(dx, dt) − ν(dx)dt]
4
is the compensated sum of jumps smaller than one, and μ(dx, dt) := μ(dx, dt) − ν(dx)dt is thecompensated Poisson random measure.J1 is a compound Poisson process and can be represented as
J1s =Ns∑�=1
γ�.
where N is a Poisson process with intensity ν(|x| > 1) and γ� are IID.
2.2 Model setup
We consider the framework where Xt is driven by a (standard) Brownian motion W and a Levyprocess L without Gaussian part:
Xt = x0 +∫ t
0
atdt +∫ t
0
σt dWt + Lt, t ∈]0, T ], (2)
where a, σ are progressively measurable processes guaranteeing that (2) has a unique strong solutionon [0, T ] which is adapted and right continuous with left limits [14], and L is a pure-jump Levyprocess with Levy measure ν. We will denote by X0 the “continuous component” of X
X0t =∫ t
0
(as + b)ds +∫ t
0
σsdWs,
and by X1 the continuous part plus the large jumps X1 = X0 + J1; so X = X0 + J = X1 + J2. Wewill occasionally use the notation a = a + b.
The notation f(h) ∼ g(h) means that f(h) = O(g(h)) and g(h) = O(f(h)) as h → 0. We usethe following assumptions in the sequel:
Assumption 1 (A1). ∫|x|≤ε
x2ν(dx) = O(ε2−α), as ε → 0, (3)
∫|x|≤ε
x4ν(dx) ∼ ε4−α, as ε → 0 and (4)
∫ε<|x|≤1
|x|jν(dx) = O(jc + (−1)jcεj−α), j = 0, 1. (5)
where α is the Blumenthal Getoor index of L.
Assumption A1 is satisfied if for instance ν has a density f(x) such that f(x) behaves as K(|x|)|x|1+α
when x → 0, where K is a real function with limx→0
K(x) ∈ IR−{0}, and α is the Blumenthal-Getoorindex of L.In particular A1 holds for any of the models commonly used in finance such as NIG, VarianceGamma, tempered stable, α-stable, GHL, etc.
Typically, we do not observe Xt continuously, but dispose of a discrete record {x0, Xt1 , ..., Xtn−1 ,
Xtn} of X on a time grid ti = ih, for a given resolution h = T/n. The goal of this paper is toprovide, given such a discrete set of observations, two nonparametric tests for
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• detecting the presence of a continuous component in the price process
• analyzing the qualitative nature of the jump component i.e. whether it has finite or infinitevariation.
We shall use the following notations, where Z and V are semimartingales:
• ΔiZ is the increment Zti − Zti−1
• ΔZt is the size Zt − Zt− of the jump at time t
• [Z]t = [Z]ct +∑
0≤s≤t ΔZ2s is the quadratic variation process associated to Z
• IV =∫ T
0σ2
udu is the integrated variance of X
• IQ =∫ T
0σ4
udu is the integrated quarticity of X
• c will denote a generic constant.
3 Threshold estimator of integrated variance
In this section we introduce an estimator for the integrated variance∫ t
0σ2
udu which is consistentin the presence of jumps and characterize its asymptotic behavior via a central limit theorem.
3.1 A consistent estimator for integrated variance
The threshold estimator of∫ t
0 σ2udu, given by
ˆIV h :=n∑
i=1
(ΔiX)2I{(ΔiX)2≤r(h)}, (6)
is based on the idea of summing the squared increments of X , those whose absolute value is smallerthan some threshold. It is well known that the sum
∑ni=1(ΔiX)2 converges to
[X ]T =∫ T
0
σ2t dt +
∫ T
0
∫IR−{0}
x2μ(dx ds).
The idea is to eliminate the term∫ T
0
∫IR−{0} x2μ(dx ds) due to jumps by an appropriate choice of
the threshold r(h). Paul Levy’s law for the modulus of continuity of the Brownian motion [24, p.10]:
P
(limh→0
supi∈{1,...,n}
|ΔiW |√2|h logh| ≤ 1
)= 1.
allows to choose a threshold that asymptotically eliminates the jumps as Δt → 0. Let us firstrecall the following result from [21]:
Theorem 3.1. Assume at and σt are cadlag (or caglad) processes. and consider a deterministicfunction r(.) such that
limh→0
r(h) = 0, and limh→0
h logh
r(h)= 0.
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1) If X has jumps of finite activity then there exists a random variable h with P(h > 0) = 1 suchthat for any h > 0,
I{ΔiN=0}(ω) = I{(ΔiX)2≤r(h)}(ω), i = 1, ..., n. on{h ≤ h} (7)
2) The following threshold estimator is a consistent estimator of the integrated variance as h → 0,
n∑i=1
(ΔiX)2I{(ΔiX)2≤r(h)}P→∫ T
0
σ2t dt.
Since a and σ are cadlag (or caglad), their paths are a.s. bounded on [0, T ] so
lim suph→0
supi |∫ ti
ti−1asds|√|h log h| ≤ C and lim sup
h→0
supi |∫ ti
ti−1σ2
sds|h
≤ M, (8)
where M, C are almost-surely finite-valued random variables. By (7), if J has finite activity thethresholding procedure allows us to detect whether a jump has occurred in ]ti−1, ti]. It follows [21]that under the assumptions of theorem 3.1,
supi
| ∫ ti
ti−1asds +
∫ ti
ti−1σsdWs|√
2h log 1h
≤ C(ω) + M(ω) a.s. (9)
Therefore, in the finite jump intensity case, for sufficiently small h, (ΔiX)2 > r(h) > 2h log 1h indi-
cates the presence of jumps in ]ti−1, ti]. When J has infinite jump activity,∑n
i=1(ΔiX)2I{(ΔiX)2≤r(h)}(ω)behaves like
∑ni=1(ΔiX)2I{ΔiN=0,|ΔiJ2|≤2
√r(h)}(ω) for small h (lemma 8.1). Moreover the jumps
contributing to the increments ΔiX such that (ΔiX)2 ≤ r(h) have size smaller than c√
r(h) [21,Lemma 7.3] and their contribution vanishes when h → 0.
A few remarks related to the above results are useful at this point:
1. The function r(h) = hβ satisfies the conditions on r(h) required in theorem 3.1 for anyβ ∈]0, 1[.
2. The assumption of equally spaced observations is not essential, in fact if hi := ti − ti−1, fori = 1..n, we obtain a similar result replacing r(h) above by r(maxi hi) [21].
3. In the case where J has finite activity then (7) also allows to separate the jump componentJ from the Brownian component.
3.2 Central limit theorem for the threshold estimator of integrated vari-
ance
Denote by
η2(ε) :=∫|x|≤ε
x2ν(dx).
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Let us remark that if limh→0
r(h) = 0, and limh→0
h log 1h
r(h) = 0 then by (A1), we have that as h → 0
η2(2√
r(h))
=∫|x|≤2
√r(h)
x2ν(dx) = O(r(h)1−α2 ),
∫|x|≤2
√r(h)
x3ν(dx) = O(r(h)3−α
2 ),
∫|x|≤2
√r(h)
x4ν(dx) ∼ r(h)2−α2 ,
∫2√
r(h)<|x|≤1ν(dx) = O(r(h)−α/2),
∫2√
r(h)<|x|≤1xν(dx) = O(c − cr(h)
1−α2 ),
(10)
where α is the Blumenthal-Getoor index of J . In the sequel we shall use the following lemmas.
Lemma 3.2. [21, Remark 7.3.] If limh→0
r(h) = 0, and limh→0
h log hr(h) = 0, then a.s. for small h we
have, for all i = 1..n, that if |ΔiJ2| ≤ 2√
r(h) then in fact ΔiJ2 contains only jumps smaller than2√
r(h) in absolute value, so for all i = 1..n we have
ΔiJ2 I{(ΔiJ2)2≤4r(h)} =∫ ti
ti−1
∫|x|≤2
√r(h)
xμ(dx, dt) −∫ ti
ti−1
∫2√
r(h)<|x|≤1
xν(dx)dt, i = 1..n
Lemma 3.3. If limh→0
r(h) = 0, and limh→0
h log hr(h) = 0, then:
i) There exists a strictly positive variable h such that for all i = 1..n,
Ih≥h I{(ΔiX0)2>r(h)} = 0 a.s., (11)
ii) for any fixed c > 0, as h → 0, nP{ΔiN = 0, (ΔiJ2)2 > cr(h)} → 0, (12)
iii) In the case r(h) = hβ , we have lim suph→0
hαβ2
n∑i=1
P{(ΔiX)2 > r(h)} ≤ c in probability
(13)
Proof. Equality (11) is a consequence of (9), while (12) is a consequence of the fact that N and J2
are independent and of the Chebyshev inequality:
nP{ΔiN = 0, (ΔiJ2)2 > cr(h)} ≤ nO(h)E[(ΔiJ2)2]
cr(h)= O
(h
r(h)
)
as h → 0.The proof of (13) is similar to that of proposition 5.9 in [3] but since we have different assumptionshere, we give a much simpler proof. It is sufficient we show that
P{(ΔiX)2 > r(h)} ≤ ch1−αβ2 . (14)
First we show that
P{|ΔiX | >√
r(h)} = P{|ΔiJ2| >√
r(h)/4} + O(h1−αβ/2) (15)
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so that for (14) it is sufficient to prove that
P{|ΔiJ2| >√
r(h)/4} ≤ ch1−αβ2 . (16)
To show (15) let us note that if |ΔiX | >√
r(h) then either ΔiJ1 = 0 or |ΔiJ2| >√
r(h)/4, sincea.s. for sufficiently small h,√
r(h) < |ΔiX | ≤ |ΔiX0| + |ΔiJ1| + |ΔiJ2| ≤√
r(h)/2 + |ΔiJ1| + |ΔiJ2|.Thus
P{|ΔiX >√
r(h)|} ≤ P{ΔiJ1 = 0} + P{|ΔiJ2| >√
r(h)/4},and since P{ΔiJ1 = 0} = O(h) = o(h1−αβ/2), (15) is verified.Secondly, define
Nt :=∑s≤t
I{|ΔJ2s|>√
r(h)/4},
and write
P{|ΔiJ2| >√
r(h)/4} = P{ΔiN = 0, |ΔiJ2| >√
r(h)/4} + P{ΔiN ≥ 1, |ΔiJ2| >√
r(h)/4}≤ P{ΔiN ≥ 1} + P{ΔiN = 0, |ΔiJ2| >
√r(h)/4}.
Note that Nt =∫ t
0
∫|x|>
√r(h)/4
μ(dx, dt) is a compound Poisson process having predictable com-
pensator given by∫ t
0
∫|x|>
√r(h)/4
ν(dx)dt = tν{|x| >√
r(h)/4}, where by our assumptions ν{|x| >√r(h)/4} = O(r(h)−α/2), so P{ΔiN ≥ 1} = O(hν{|x| >
√r(h)/4}) = O(h1−αβ/2) and thus the
first term above is dominated by h1−αβ/2 as we need.Finally on {ΔiN = 0} our J2 does not have jumps bigger than
√r(h)/4, so
ΔiJ2 =∫ ti
ti−1
∫|x|≤
√r(h)/4
xμ(dx, dt) − h
∫√
r(h)/4<|x|≤1
xν(dx),
therefore
P{ΔiN = 0, |ΔiJ2| >√
r(h)/4} ≤ P{ |ΔiJ2| >√
r(h)/4, |ΔJ2s| ≤√
r(h)/4 for all s ∈]ti−1, ti]}
≤ 4E[(ΔiJ2)2I{|ΔJ2s|≤
√r(h)/4 for all s∈]ti−1,ti]}
]r(h)
= O(hη2(
√r(h))
r(h)
)= O(h1−αβ/2),
and (16) is verified.
Denote
J(h)t :=
∫ t
0
∫|x|≤2
√r(h)
x μ(dx, dt) −∫ t
0
∫2√
r(h)<|x|≤1
xν(dx)dt, (17)
ΔiJ2m :=∫ ti
ti−1
∫|x|≤2
√r(h)
xμ(dx, dt), ΔiJ2c :=∫ ti
ti−1
∫2√
r(h)<|x|≤1
xν(dx)dt,
so that by lemma 3.2 for sufficiently small h, ∀i = 1..n,
ΔiJ2 I{(ΔiJ2)2≤4r(h)} = ΔiJ(h) = ΔiJ2m − ΔiJ2c.
ΔiJ2m is the martingale part of ΔiJ2 I{(ΔiJ2)2≤4r(h)}, that is, the compensated sum of jumpssmaller in absolute value than 2
√r(h), while ΔiJ2c is the compensator of the jumps larger than
2√
r(h).The following proposition gives us the limit in probability of the variance of the estimation
error ˆIV h − IV .
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Proposition 3.4. If a and σ are cadlag processes, and r(h) = hβ with 1 > β > 12−α/2 ∈]1/2, 1[
then as h → 0ˆIQh :=
∑i(ΔiX)4I{(ΔiX)2≤r(h)}
3h
P→∫ T
0
σ4t dt.
Proof of proposition 3.4.∑
i(ΔiX)4I{(ΔiX)2≤r(h)}3h =
∑i(ΔiX1)4I{(ΔiX1)2≤4r(h)}
3h +
∑i(ΔiX1)4(I{(ΔiX)2≤r(h)}−I{(ΔiX1)2≤4r(h)})
3h +∑4
k=1 c∑
i(ΔiX1)4−k(ΔiJ2)kI{(ΔiX)2≤r(h)}
3h :=∑3
j=1 Ij(h)
By proposition 3.4 in [21], I1(h) tends to∫ T
0 σ4t dt in probability. We show here that each one of
the other terms tends to zero in probability.
Let us consider I2(h): on {(ΔiX)2 ≤ r(h), (ΔiX1)2 > 4r(h)} we have
√r(h) > |ΔiX | > |ΔiX1| − |ΔiJ2| > 2
√r(h) − |ΔiJ2|
so |ΔiJ2| >√
r(h). Moreover if |ΔiX1| > 2√
r(h) we necessarily have ΔiN = 0, since
|ΔiX0| + |ΔiJ1| ≥ |ΔiX1| > 2√
r(h)
and by (9), for sufficiently small h, for all i = 1..n, |ΔiX0| ≤√
r(h) thus |ΔiJ1| > 2√
r(h) −|ΔiX0| ≥
√r(h). It follows that
P{∑
i(ΔiX1)4I{(ΔiX)2≤r(h),(ΔiX1)2>4r(h)}h
= 0}≤ nP{|ΔiJ2| >
√r(h), ΔiN = 0} → 0.
On the other hand on (ΔiX1)2 ≤ 4r(h)} we have that, for sufficiently small h, ΔiN = 0 for alli = 1..n, because
|ΔiJ1| − |ΔiX0| ≤ |ΔiX1| ≤ 2√
r(h)
so if ΔiN = 0 then for small h in fact ΔiN = 1 and ΔJ1s ≥ 1 by definition of J1, therefore if itwas ΔiN = 0 we would have 1 ≤ ΔiJ1 ≤ 2
√r(h) +
√r(h) = 3
√r(h), but this is impossible for
small h.I follows that
{(ΔiX)2 > r(h), (ΔiX1)2 ≤ 4r(h)} ⊂ {(ΔiX0 + ΔiJ2)2 > r(h)}
⊂ {(ΔiX0)2 >r(h)4
} ∪ {(ΔiJ2)2 >r(h)4
}.This implies, by (11) that as h → 0
∑i(ΔiX1)4I{(ΔiX)2>r(h),(ΔiX1)2≤4r(h)}
h≤∑
i(ΔiX0)4I{(ΔiJ2)2>r(h)/4}h
≤ Λh ln2 1h
∑i
I{(ΔiJ2)2>r(h)/4}P→ 0,
having used (16). We can conclude that I2(h) P→ 0, as h → 0.
10
Now we consider I3(h) :=∑4
k=1 I3,k(h), where each term
I3,k(h) :=∑
i(ΔiX1)4−k(ΔiJ2)kI{(ΔiX)2≤r(h)}3h
, k = 1..4
is decomposable as ∑i(ΔiX1)4−k(ΔiJ2)kI{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)}
3h+
13h
∑i
(ΔiX1)4−k(ΔiJ2)kI{(ΔiX)2≤r(h),(ΔiJ2)2>4r(h)}. (18)
On {(ΔiX)2 ≤ r(h), (ΔiJ2)2 > 4r(h)} we have that ΔiN = 0, since
2√
r(h) −√
r(h)4
− |ΔiJ1| < |ΔiJ2| − |ΔiJ1| − |ΔiX0| ≤ |ΔiX | ≤√
r(h)
and then |ΔiJ1| > 3√
r(h)/4. So the probability that the second term differs from zero is boundedby (12). As for the first term, on {(ΔiX)2 ≤ r(h), (ΔiJ2)2 ≤ 4r(h)} we have that, for sufficientlysmall h, ΔiN = 0, because
|ΔiJ1| − |ΔiX0| − |ΔiJ2| ≤ |ΔiX | ≤√
r(h)
and then |ΔiJ1| ≤√
r(h)+2√
r(h)+√
r(h) = 4√
r(h). So the first term in (18) is a.s. dominatedby ∑
i(ΔiX1)4−k(ΔiJ2)kI{ΔiN=0,(ΔiJ2)2≤4r(h)}3h
≤∑
i(ΔiX0)4−k(ΔiJ2)kI{(ΔiJ2)2≤4r(h)}3h
.
To show that this last term tends to zero in probability for each k = 1..4, we use the decompositionof ΔiJ2I{(ΔiJ2)2≤4r(h)} given in lemma 3.2 and we obtain that a.s.
∣∣∑i(ΔiX0)4−k(ΔiJ2)kI{(ΔiJ2)2≤4r(h)}
∣∣3h
≤
c
∑i |ΔiX0|4−k|ΔiJ2m|k
3h+ c
∑i |ΔiX0|4−k|ΔiJ2c|k
3h.
Now ∑i |ΔiX0|4−k|ΔiJ2c|k
3h≤ (h ln
1h
) 4−k2 nhk−1c|1 − r(h)
1−α2 |k ≤
chk/2(ln
1h
) 4−k2 + chk/2
(ln
1h
) 4−k2 r(h)k 1−α
2 =
o(1) + c
(h
r(h)ln
4−kk
1h
)k/2
r(h)k(1− α2 )
which tends to zero ∀k = 1, 2, 3, 4 by the assumptions on r(h).As for ∑
i(ΔiX0)4−k(ΔiJ2m)k
h, (19)
we need to deal separately with each k. Note that since a and σ are locally bounded on Ω× [0, T ],we can assume they are bounded without loss of generality, so E[(
∫ ti
ti−1σsdWs)2k] = O(hk) for
each k = 1...4, using e.g. the Burkholder inequality [23, p. 226], and a.s. (∫ ti
ti−1asds)2k = o(hk).
Therefore E[(ΔiX0)2k] = O(hk) for each k = 1..4. For k = 1 the expected value of (19) is bounded
11
by nh
√E[(ΔiX0)6]
√E(ΔiJ2m)2 =O(η(
√r(h))) and thus it tends to zero as h → 0.
As for k = 2, ∑i(ΔiX0)2(ΔiJ2m)2
h≤ h ln
1h
∑i(ΔiJ2m)2
h, (20)
whose expected value is given by
ln1h
η2(2√
r(h)) P→ 0,
as h → 0, by the assumptions made on r(h).Concerning k = 3, we have∑
i |ΔiX0||ΔiJ2m|3h
≤ c
h
∑i
(ΔiX0)2(ΔiJ2m)2 +c
h
∑i
(ΔiJ2m)4,
so that this step is reduced to the step with k = 4.Finally, for k = 4, we have
E
[∑i(ΔiJ2m)4
h
]≤ c
E[(ΔiJ2m)4]h2
=ch∫|x|≤
√r(h)
x4ν(dx) + 3cE2[(ΔiJ2m)2]
h2≤
ch−1r(h)2−α/2 + cr(h)2−α → 0.
Our next result provides an estimate for the quadratic variation due to the small jumps of X .It will be used to prove the central limit theorem for ˆIV h.
Theorem 3.5. Under assumption A1 and the assumptions of proposition 3.4, as h → 0∑i(ΔiJ2)2 I{(ΔiJ2)2≤4r(h)} − Tη2(2
√r(h))√
Tr(h)1−α/4�h
d→ N (0, 1), (21)
where �h =∫|x|≤2
√r(h)
x4ν(dx)/r(h)2−α/2 tends, as h → 0, to a constant depending on ν.
Proof. We apply the Lindeberg-Feller theorem (see the appendix) to the double array sequenceHni given by the normalized versions of the variables (ΔiJ2)2I{(ΔiJ2)2≤4r(h)} and n = T/h. Wehave
Eni := E[(ΔiJ2)2I{(ΔiJ2)2≤4r(h)}] = (22)
E
⎡⎣(∫ ti
ti−1
∫|x|≤2
√r(h)
xμ(dx, dt) − h
∫2√
r(h)<|x|≤1
xν(dx)
)2⎤⎦ =
hη2(2√
r(h)) +
(h
∫2√
r(h)<|x|≤1
xν(dx)
)2
.
For h → 0, Eni tends to zero as hη2(2√
r(h)). Using relations (10) we also obtain that
v2ni := var[(ΔiJ2)2I{(ΔiJ2)2≤4r(h)}] =
E[(ΔiJ2)4I{(ΔiJ2)2≤4r(h)}] − E2[(ΔiJ2)2I{(ΔiJ2)2≤4r(h)}] =
O(h
∫|x|≤2
√r(h)
x4ν(dx))
= O(hr(h)2−α/2).
12
Consider then
Hni :=(ΔiJ2)2I{(ΔiJ2)2≤4r(h)} − Eni√
n vni.
We are now going to verify that the Lindeberg condition
∀δ > 0n∑
i=1
E[H2niI|Hni|>δ] →n 0
is satisfied, since this implies thatn∑
i=1
Hnid→ N (0, 1),
that is (21).We have to compute the asymptotic behavior of
∑ni=1 E[H2
niI{|Hni|>δ}] = nE[H2n1I{|Hn1|>δ}]. We
show that for small h the set {|Hn1| > δ} is empty, so that nE[H2n1I{|Hn1|>δ}] is zero for h small
enough, as we need.
{|Hn1| > δ} = {|(Δ1J2)2I{(Δ1J2)2≤4r(h)} − En1| > δ√
n vn1},however for small h we have Eni = O(hr(h)1−
α2 ), so that a.s.
|(Δ1J2)2I{(Δ1J2)2≤4r(h)} − En1| ≤ 4r(h) + O(hr(h)1−α2 ),
and thus a.s. |(Δ1J2)2I{(Δ1J2)2≤4r(h)}−En1| is asymptotically smaller than δ√
n vn1 = O(r(h)1−α/4).
We are now ready to state our central limit result for the estimator ˆIV h.
Proposition 3.6. Assume A1 and that a and σ are cadlag, σ ≡ 0; choose r(h) = hβ withβ ∈]12 , 1[. Let α be the Blumenthal-Getoor index of J . Thena) if α < 1 and β > 1
2−α ∈ [1/2, 1[ we have
∑ni=1(ΔiX)2I{(ΔiX)2≤r(h)} −
∫ T
0σ2
t dt√23
∑i(ΔiX)4I{(ΔiX)2≤r(h)}
d→ N (0, 1) ;
b) if α ≥ 1 we have, for any β ∈]0, 1[,
∑ni=1(ΔiX)2I{(ΔiX)2≤r(h)} −
∫ T
0 σ2t dt√
23
∑i(ΔiX)4I{(ΔiX)2≤r(h)}
P→ +∞.
Proof. Note that if β > 12−α then β > 1
2−α/2 , so that the assumptions of proposition 3.4 are
satisfied. Since X = X1 + J2, we decompose∑n
i=1(ΔiX)2I{(ΔiX)2≤r(h)} −∫ T
0 σ2t dt
√2h
√∑i(ΔiX)4I{(ΔiX)2≤r(h)}
3h
=∑n
i=1(ΔiX1)2I{(ΔiX1)2≤4r(h)} − IV√23
∑i(ΔiX)4I{(ΔiX)2≤r(h)}
(23)
+√
2hIQ√23
∑i(ΔiX)4I{(ΔiX)2≤r(h)}
[∑ni=1(ΔiX1)2
(I{(ΔiX)2≤r(h)}−I{(ΔiX1)2≤4r(h)}
)√
2hIQ
+2∑n
i=1 ΔiX1ΔiJ2I{(ΔiX)2≤r(h)}√2hIQ
+∑n
i=1(ΔiJ2)2I{(ΔiX)2≤r(h)}√2hIQ
]:=∑4
j=1 Ij(h).
(24)
13
The term I1(h) is asymptotically Gaussian (see appendix). We note that we can assume w.l.g.that both a := a+ b and σ are bounded a.s. The term I2(h) simplifies as follows. If (ΔiX)2 ≤ r(h)and (ΔiX1)2 > 4r(h) then |ΔiJ2| >
√r(h) and ΔiN = 0, exactly as for I2(h) in the proof of
proposition 3.4. It follows that
P{∑n
i=1(ΔiX1)2I{(ΔiX)2≤r(h),(ΔiX1)2>4r(h)}√2hIQ
= 0}≤ nP{ΔiN = 0, |ΔiX1| > 2
√r(h)} → 0.
The main factor of the remaining part of I2(h) is∑ni=1(ΔiX1)2I{(ΔiX)2>r(h),(ΔiX1)2≤4r(h)}√
2hIQ.
We remark that on {|ΔiX1| ≤ 2√
r(h)} we have ΔiN = 0, thus (ΔiX1)2 = (ΔiX0)2. Moreover
∑ni=1
( ∫ ti
ti−1audu
)2I{(ΔiX)2>r(h),(ΔiX1)2≤4r(h)}√
2hIQ= O(
√h) → 0,
and, by (13)∑ni=1
∫ ti
ti−1audu
∫ ti
ti−1σudWuI{(ΔiX)2>r(h),(ΔiX1)2≤4r(h)}√
2hIQ≤ c
√h
√h ln
1h
n∑i=1
I{(ΔiX)2>r(h)}
= O(h1−αβ/2
√ln
1h
)→ 0.
Therefore in probability
limh→0
I2(h) = limh→0
−∑n
i=1
( ∫ ti
ti−1σudWu
)2I{(ΔiX)2>r(h),(ΔiX1)2≤4r(h)}√2hIQ
.
Now we show that term I3(h) in (24) has the same limit of
−∑n
i=1
∫ ti
ti−1σudWuΔiJ2I{(ΔiJ2)2≤4r(h),(ΔiX)2>r(h)}√
2hIQ.
First recall that ΔiX1 = ΔiX0 + ΔiJ1, so within the sum
1√h
n∑i=1
ΔiJ1ΔiJ2I{(ΔiX)2≤r(h)}
term i contributes only when ΔiN = 0, in which case we also have (ΔiJ2)2 > r(h), because forsufficiently small h if ΔiN = 0 then ΔiN = 1 and by construction |ΔJ1s| ≥ 1, so for small h√
r(h) > |ΔiX | > |ΔiJ1| − |ΔiJ2| ≥ 1 − |ΔiJ2| > 2√
r(h) − |ΔiJ2|
therefore |ΔiJ2| >√
r(h). That implies
P
{∑ni=1 ΔiJ1ΔiJ2I{(ΔiX)2≤r(h)}√
2hIQ= 0
}≤ nP{ΔiN = 0, |ΔiJ2| >
√r(h)} → 0.
Secondly, as for∑n
i=1 ΔiX0ΔiJ2I{(ΔiX)2≤r(h)}√h
, as in the the beginning of the proof of lemma 8.1, wehave ∑n
i=1 ΔiX0ΔiJ2I{(ΔiX)2≤r(h)}√h
=∑n
i=1 ΔiX0ΔiJ2I{(ΔiX)2≤r(h),(ΔiJ)2≤4r(h)}√h
, (25)
14
however since both P
{∑ni=1 ΔiX0ΔiJ2I{(ΔiX)2≤r(h),(ΔiJ)2≤4r(h),ΔiN �=0}√
h= 0}
and
P
{∑ni=1 ΔiX0ΔiJ2I{(ΔiX)2≤r(h),(Δi J2)2≤4r(h),ΔiN �=0}√
h= 0}
are dominated by nP{ΔiN = 0, (ΔiJ2)2 >
cr(h)} → 0, then in probability we have
lim∑n
i=1 ΔiX0ΔiJ2I{(ΔiX)2≤r(h),(ΔiJ)2≤4r(h)}√h
= lim∑n
i=1 ΔiX0ΔiJ2I{(ΔiX)2≤r(h),(ΔiJ)2≤4r(h),ΔiN=0}√h
= lim
∑ni=1 ΔiX0ΔiJ2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h),ΔiN=0}√
h= lim
∑ni=1 ΔiX0ΔiJ2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)}√
h.
Moreover by the Cauchy-Schwartz inequality, we have
∑ni=1
∫ ti
ti−1auduΔiJ2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)}√
h≤√∑n
i=1(∫ ti
ti−1audu)2
√h
√√√√ n∑i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h)}
≤ c
√√√√ n∑i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h)}, (26)
which tends to zero in probability, since in probability
limh
n∑i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h)} = limh
∫ T
0
∫|x|≤2
√r(h)
x2ν(dx) = limh
Tη2(√
r(h)) = 0.
On the other hand ∑ni=1
( ∫ ti
ti−1σudWu
)ΔiJ2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)}√
h
=
∑ni=1
( ∫ ti
ti−1σudWu
)ΔiJ2m√
h−∑n
i=1
( ∫ ti
ti−1σudWu
)ΔiJ2c√
h(27)
−h−1/2n∑
i=1
( ∫ ti
ti−1
σudWu
)ΔiJ2I{(ΔiJ2)2≤4r(h),(ΔiX)2>r(h)},
where the first two terms tend to zero in probability. In fact
Δ1J2c
∑ni=1
∫ ti
ti−1σudWu√
h=
√h
∫2√
r(h)<|x|≤1
xν(dx) ·∫ T
0
σt.dWt = Oa.s.(√
h(1 − r(h)1−α
2 )) a.s.→ 0,
while
Z :=
∑ni=1
∫ ti
ti−1σudWuΔiJ2m√
h
is such that Z2 P→ 0. In fact, since∫ ti
ti−1σudWu and ΔiJ2m are martingale differences, it is sufficient
to show that ∑ni=1(∫ ti
ti−1σudWu)2(ΔiJ2m)2
h
P→ 0,
which coincides with (20).
15
Putting together the simplified versions of each I2(h) and I3(h) and doing as in (25) for I4(h),we reach that ∑n
i=1(ΔiX)2I{(ΔiX)2≤r(h)} −∫ T
0σ2
t dt√
2h
√∑i(ΔiX)4I{(ΔiX)2≤r(h)}
3h
has the same limit in distribution as
N −∑n
i=1
( ∫ ti
ti−1σudWu
)2I{(ΔiX)2>r(h),(ΔiX1)2≤4r(h)}√2hIQ
(28)
−∑n
i=1
∫ ti
ti−1σudWuΔiJ2I{(ΔiJ2)2≤4r(h),(ΔiX)2>r(h)}√
2hIQ
+
∑ni=1(ΔiJ2)2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)}√
2hIQ,
where N denotes a random variable with standard Gaussian law. However note that the secondand third terms of (28) are negligible with respect to r(h)1−α/2
√2hIQ
, in fact
(2hIQ)−1/2∑n
i=1
( ∫ ti
ti−1σudWu
)2I{(ΔiX)2>r(h),(ΔiX1)2≤4r(h)}
(2hIQ)−1/2r(h)1−α/2
=
∑ni=1
( ∫ ti
ti−1σudWu
)2I{(ΔiX)2>r(h),(ΔiX1)2≤4r(h)}
r(h)1−α/2≤∑n
i=1 h ln 1hI{(ΔiX)2>r(h)}
r(h)1−α/2
and
E
[∑ni=1 h ln 1
hI{(ΔiX)2>r(h)}r(h)1−α/2
]≤ h1−β ln
1h→ 0,
and ∑ni=1
∫ ti
ti−1σudWuΔiJ2I{(ΔiJ2)2≤4r(h),(ΔiX)2>r(h)}
r(h)1−α/2
≤√
h ln1h
√r(h)∑n
i=1 I{(ΔiX)2>r(h)}r(h)1−α/2
and
E[√
h ln1h
√r(h)∑n
i=1 I{(ΔiX)2>r(h)}r(h)1−α/2
]= O
(√h ln
1h
h−αβ/2r(h)−1/2+α/2
)→ 0.
Note that∑ni=1(ΔiJ2)2I{(ΔiJ2)2≤r(h)/16} + o(r(h)1−α/2) ≤∑n
i=1(ΔiJ2)2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)}
≤∑ni=1(ΔiJ2)2I{(ΔiJ2)2≤9r(h)/4} + o(r(h)1−α/2).
(29)
To show that let us first deal with∑n
i=1(ΔiJ2)2I{(ΔiX)2>r(h),(ΔiJ2)2≤4r(h)}.
On {(ΔiX)2 > r(h)} we have either |ΔiJ1| >√
r(h)/4 or |ΔiJ2| >√
r(h)/4, since
√r(h) < |ΔiX | ≤ |ΔiX0| + |ΔiJ1| + |ΔiJ2| ≤
√r(h)/2 + |ΔiJ1| + |ΔiJ2|.
16
However
E[∑n
i=1(ΔiJ2)2I{(ΔiJ2)2≤4r(h),ΔiN �=0}r(h)1−α/2
]= O(hη2(
√r(h))nh
r(h)1−α/2
)→ 0,
thereforen∑
i=1
(ΔiJ2)2I{(ΔiX)2>r(h),(ΔiJ2)2≤4r(h)} ≤ o(r(h)1−α/2) +n∑
i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h),(ΔiJ2)2>r(h)/16}
= o(r(h)1−α/2) +n∑
i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h)} −n∑
i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h),(ΔiJ2)2≤r(h)/16}
= o(r(h)1−α/2) +n∑
i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h)} −n∑
i=1
(ΔiJ2)2I{(ΔiJ2)2≤r(h)/16}. (30)
On the other hand, on {2√r(h) ≥ |ΔiJ2| > 32
√r(h)} we have |ΔiX | >
√r(h), since either
ΔiN = 0, however∑n
i=1(ΔiJ2)2I{(ΔiJ2)2≤4r(h),ΔiN �=0}r(h)1−α/2 → 0, or ΔiN = 0, and thus
|ΔiX | > |ΔiJ2| − |ΔiX0| >32
√r(h) − 1
2
√r(h) =
√r(h).
That implies thatn∑
i=1
(ΔiJ2)2I{(ΔiX)2>r(h),(ΔiJ2)2≤4r(h)} ≥ o(r(h)1−α/2) +n∑
i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h),(ΔiJ2)2>9r(h)/4}
= o(r(h)1−α/2) +n∑
i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h)} −n∑
i=1
(ΔiJ2)2I{(ΔiJ2)2≤9r(h)/4}. (31)
Combining now (30) and (31), since
n∑i=1
(ΔiJ2)2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)} =n∑
i=1
(ΔiJ2)2I{(ΔiJ2)2≤4r(h)}−n∑
i=1
(ΔiJ2)2I{(ΔiX)2>r(h),(ΔiJ2)2≤4r(h)}
we reach (29).
Therefore (28) can be written as
N +r(h)1−α/2
√2hIQ
[o(1) +
∑ni=1(ΔiJ2)2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)}
r(h)1−α/2
].
If α < 1 then
limh
∑ni=1(ΔiJ2)2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)}
r(h)1−α/2≤ lim
h
∑ni=1(ΔiJ2)2I{(ΔiJ2)2≤9r(h)/4}
r(h)1−α/2
= limh
N cr(h)α/4 + Tc = Tc
and r(h)1−α/2√
h→ 0, so that
ˆIV h − IV√h
→ N .
As soon as, on the contrary, α ≥ 1 then r(h)1−α/2√
h→ +∞ and
limh
∑ni=1(ΔiJ2)2I{(ΔiX)2≤r(h),(ΔiJ2)2≤4r(h)}
r(h)1−α/2≥ lim
h
∑ni=1(ΔiJ2)2I{(ΔiJ2)2≤r(h)/16}
r(h)1−α/2
17
= limh
N cr(h)α/4 + T c = T c,
and thusˆIV h − IV√
h→ +∞.
Remark. Jacod [15] has shown a related central limit theorem for a threshold estimator of IV
where J is a semimartingale under the assumption that σ is a semimartingale with absolutelycontinuous characteristics and having a certain stochastic integral representation.Central limit theorems for the power variation estimators of
∫ T
0σp
sds in presence of jumps can befound in [15, 25, 7] and for the multipower variation estimators, for the case α < 1, in [25, 7].
4 Statistical tests
4.1 Test for the presence of a continuous martingale component
We now use the above results to design a test to detect the presence of a continuous martingalecomponent
∫ t
0σtdWt given discretely recorded observations. Our test is feasible in the case when
J has Blumenthal-Getoor index α < 1 i.e. the jumps are of finite variation. In the next subsectionwe will describe a test to check whether α < 1 or not.The test proceeds as follows. First, we choose a coefficient β ∈ [1/2, 1[ close to 1. If we disposeof an estimate α of the Blumenthal Getoor index [26, 3] we choose β > 1
2−α ∈ [1/2, 1[. We thenchoose a threshold r(h) = hβ and define
ˆIQh :=13h
∑i
(ΔiX)4I{(ΔiX)2≤r(h)}
As shown in proposition 3.6, as h → 0 or, equivalently, as n = T/h → ∞,
ˆIV h − ∫ T
0σ2
t dt√2h ˆIQh
d→ N (0, 1) .
Note that if σ ≡ 0 then both the numerator and the denominator tend to zero. To handle the caseσ ≡ 0 we add to the data ΔiX an IID Gaussian noise with known variance v2:
ΔiXv := ΔiX + v
√hZi Zi
IID∼ N(0, 1)
Now the law of ΔiXv coincides with that of the increment of
Xv =∫ ·
0
asds +∫ t
0
σtdWt + vW v + J,
where W v is a standard Brownian motion independent from W . Now as n → ∞n∑
i=1
(ΔiXv)2 P→ [Xv]T =
∫ T
0
σ2sds + v2T + T
∫IR−{0}
x2ν(dx),
and I{(ΔiXv)2≤r(h)} rules out the contributions of the jumps of Xv, so that under the assumptionsof proposition 3.6, as h → 0
ˆIVv
h :=n∑
i=1
(ΔiXv)2I{(ΔiXv)2≤r(h)}
P→∫ T
0
σ2sds + v2T := IV v.
18
Moreover under the null hypothesis(H0) σ ≡ 0
we have ˆIVv
hP→ v2T,
ˆIQv
h :=13h
∑i
(ΔiXv)4I{(ΔiXv)2≤r(h)}
P→ v4T =: IQv
and
Uh :=ˆIV
v
h − v2T√2h ˆIQ
v
h
d→ N (0, 1), (32)
as h → 0: in particular P{|Uh| > 1.96} has to be close to 5%.Note that if on the contrary σ ≡ 0 we have that the limit in probability of ˆIV
v
h is strictly largerthan v2T and, by lemma 8.1, passing to a subsequence, a.s.
limh→0
h ˆIQv
h = c limh→0
∑i
(ΔiXv)4I{(ΔiXv)2≤r(h)} = c lim
h→0
∑i
(ΔiXv)4I{ΔiN=0,(ΔiJ2)2≤2r(h)}
≤ c limh→0
∑i
(ΔiX0 + ΔiJ2 + vΔiW
v)4
I{(ΔiJ2)2≤2r(h)}
≤ c limh→0
∑i
(ΔiX0)4 + c limh→0
∑i
(ΔiJ2)4I{(ΔiJ2)2≤2r(h)} + c limh→0
∑i
(vΔiWv)4.
Using that limh→0
∑i(ΔiJ2)4I{(ΔiJ2)2≤2r(h)} ≤ lim
h→02r(h)
∑i(ΔiJ2)2I{(ΔiJ2)2≤2r(h)} = 0 by (26),∑
i(ΔiX0)4/hP→ c∫ T
0σ4
sds and∑
i(vΔiWv)4/h
a.s.→ cv4, we have that as h → 0
h ˆIQv
hP→ 0.
Therefore under the alternative (H1) σ ≡ 0 the test statistic Uh diverges in probability to +∞, sothat
P{|Uh| > 1.96} >> 5%.
In the simulation tests, if P(|Uh| > 1.96) >> 0.05 we conclude that the test leads to a rejection of(H0) : σ ≡ 0.
4.2 Testing whether the jump component has finite variation
Proposition 3.6 allows us to construct a test for discriminating whether α < 1 or α ≥ 1, that is,
whether the jumps have finite variation. However we cannot use ( ˆIV h − IV )/√
2h ˆIQh directly,since we do not know the true value of σ to compute the integrated variance IV . We consider thesequence
HhT :=n∑
i=1
ΔiXI{(ΔiX)2>r(h)} = HnT := XT −n∑
i=1
ΔiXI{(ΔiX)2≤r(h)}.
When α < 1, analogously as in lemma 8.1with√
r(h) in place of r(h) as bound for the maxi=1..n |ani|,we reach that HhT behaves as
XT −n∑
i=1
(ΔiX0 + ΔiJ2)I{ΔiN=0,(ΔiJ2)2≤r(h)},
19
when h → 0. Moreover this last term has the same asymptotic behaviour as
XT −n∑
i=1
(ΔiX0 + ΔiJ2I{(ΔiJ2)2≤r(h)}) = XT − X0T +n∑
i=1
(ΔiJ2m − ΔiJ2c)
= JT +∫ T
0
∫|x|≤
√r(h)
xμ(dx, dt) − T
∫√
r(h)<|x|≤1
xν(dx),
by application of lemma 3.2. Now∫ T
0
∫|x|≤
√r(h)
xμ(dx, dt) L2→ 0, while∫√
r(h)<|x|≤1xν(dx) =
O(c − cr(h)(1−α)/2) tends to a constant (α < 1). Consider now n independent standard Gaussianrandom variables Zi and define
ΔiHv := ΔiXI{(ΔiX)2>r(h)} + v
√hZi.
Under the null hypothesis of α < 1, ΔiHv is the increment of an approximation in law of
HvT := JT + cT + vW v
T .
As a consequence the quantity
ˆIVHv
h :=∑
i
(ΔiHv)2I{(ΔiHv)2≤r(h)}
will be an estimator of the integrated variance v2T of HvT , so under the null hypothesis (H0) α < 1
we can find β ∈]12 , 1[ such that
U(α)h :=
ˆIVHv
h − v2T√2h ˆIQ
Hv
h
d→ N (0, 1) ,
where ˆIQHv
h := 13h
∑i(ΔiH
v)4I{(ΔiHv)2≤r(h)} and r(h) = hβ. In particular P{|U (α)
h | > 1.96}
hasto be close to the value of 5%.If on the contrary α ≥ 1 then, by proposition 3.6, for any β ∈]0, 1[
U(α)h =
ˆIVHv
h − v2T√2h ˆIQ
Hv
h
P→ +∞,
so that if |U (α)h | > 1.96 we reject (H0) at a 95% confidence level.
Remark. From a practical point of view we first need to decide whether α < 1 or not. If α < 1we can apply the test for the presence of a Brownian component.
5 Numerical experiments
5.1 Testing for the finite variation of the jump component
We simulate n increments ΔiX of a process
X = σW + J,
20
where J is a Levy symmetric α−stable process, σ a positive constant and W a standard Brownianmotion. Then we determine ΔiH
v with given constant v = 0.0001 and compute U(α)h . We repeat
the same procedure H times, and we generate H values U(α)h(j), j = 1..H , given by
U(α)h(j) =
ˆIVHv
h − v2nh√2h ˆIQ
Hv
h
. (33)
Then we compute the percentage of the H statistics U(α)h(j) such that |U (α)
h(j)| > 1.96. We obtain thefollowing results, for σ = 0.2, time horizon T = n × h, threshold exponent β = 0.999, and steph = 1/(252× 84), corresponding to five minute observations, in annual unit of measure, in a sevenhours open market.
α 12−α n H P (|U (α)
h | > 1.96) (H0) α < 10.6 .71 1000 1000 0.035 acceped1.6 1000 1000 0.414 rejected
The table tells us that in fact the results we obtain from our test are reliable if we dispose ofn = 1000 observations and a time resolution of five minutes. In fact when the data generatingprocess has Blumenthal-Getoor index 0.6 the test leads us to accept the hypothesis (H0) α < 1,and thus allows us to recognize that α is less than one. On the contrary when the true process hasBlumenthal-Getoor index 1.6 the test tells us, as it should, to reject (H0). Figure 5.1 shows theempirical density of the values assumed by U
(α)h(j) in both cases α = 0.6 and α = 1.6.
−4 −3 −2 −1 0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
test for H0) α <1
−6 −5 −4 −3 −2 −1 0 1 2 3 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
test for H0) α <1
Figure 5.1. Test for α < 1 versus α ≥ 1. Empirical density of H values assumed by U(α)h(j).
dXt = σdWt + dJt, J symmetric α-stable, σ = 0.2, h = 1/(252 × 84), β = 0.999,
H = 1000, n = 1000, v = 0.0001. Left: stability index = 0.6; right: stability index = 1.6. The continuous
curve is a standard Gaussian density.
5.2 Test for the presence of a Brownian component. Comparison with
multipower variation estimators.
We simulate n increments ΔiX of a process
X = σW + J,
for different Levy processes J and constant or stochastic σ. For each model we implement thestatistic
Uh :=ˆIV
v
h − v2nh√2h ˆIQ
v
h
. (34)
21
We repeat H times, obtaining Uh(j), j = 1..H , and then we compute the percentage of the H
statistics such that |Uh(j)| > 1.96.
We now compare our results with those of an analogous test based on multipower variationestimators [5, 7, 25]. Define the bipower variation of X
Vr,s(X) :=n∑
i=2
|ΔiX |r|Δi−1X |s,
and more generally the multipower variation
Vr1,..,rk(X) :=
n∑i=k+1
|ΔiX |r1 |Δi−1X |r2 ...|Δi−kX |rk .
Woerner [25] has shown the convergence of such variations to the integral of proper powers of σ,under the following conditions: if X is a semimartingale without drift part, α < 1, the cadlagprocess σ is a.s. strictly positive, has paths regular enough and is independent of W , if r, s > 0,max(r, s) < 1 and r + s > α/(2 − α) then as h → 0
h1−r/2−s/2Vr,s(X) − μsμr
∫ T
0 σr+su du
√h
√CBPV
∫ T
0 σ2r+2su du
d→ N (0, 1),
where CBPV = μ2rμ2s +2μrμsμr+s −3μ2rμ
2s and μr = E[|Z|r], Z being a N (0, 1) random variable.
Moreover if ri > 0 for all i = 1..k, maxi ri < 1 and∑
i ri > α/(2 − α) then as h → 0
h1−∑ i ri/2 Vr1,..,rk(X) − μr1μr2 · · · μrk
∫ T
0σ∑
i riu du
√h
√CMPV
∫ T
0 σ2∑
i riu du
d→ N (0, 1),
where CMPV =k∏
p=1
μ2rp + 2k−1∑i=1
i∏p=1
μrp
k∏p=k−i+1
μrp
k−i∏p=1
μrp+rp+i − (2k − 1)k∏
p=1
μ2rp
.
Note that the integrals at denominators can be estimated using in turn the multipower variations.We have chosen to use the bipower variation V1/3,1/3(X) to estimate
∫ T
0 σ2/3u du, through
I 23 ,h := h
23 V 1
3 , 13(X)μ−2
13
,
and also the tripower variation V2/3,2/3,2/3(X) to estimate∫ T
0 σ2udu, through
I2,h := V 23 , 2
3 , 23(X)μ−3
23
.
Under (H0) σ ≡ 0 such estimators applied to Xv satisfy
UBPVh :=
Iv
23 ,h
− v2/3nh
μ−213
√hCBPV I
v
43 ,h
d→ N (0, 1), (35)
and
UTPVh :=
Iv
2,h − v2nh
μ−323
√hCTPV I
v
4,h
d→ N (0, 1), (36)
as h → 0, where CBPV is constructed using the powers r = s = 1/3 and CTPV is the constantCMPV constructed with the powers r1 = r2 = r3 = 2/3, and
Iv
43 ,h := h
13 V 2
3 , 23(Xv)μ−2
23
,
22
Iv
4,h := h−1V 45 , 4
5 , 45 , 4
5 , 45(Xv)μ−5
45
estimate respectively v43 nh and v4nh.
Under the alternative (H1) σ ≡ 0, we have that both UBPVh → +∞ and UTPV
h → +∞ as h → 0.So we compute the percentage of times that H statistics |UBPV
h(j) |, j = 1..H , assume values largerthan 1.96 and H statistics |UTPV
h(j) | assume values larger than 1.96. If such percentages are largerthan 0.05 then the multipower variations test leads us to reject (H0) σ ≡ 0.
We remark that if we implemented Uh, UBPVh and UTPV
h using v4nh, v43 nh and v4nh in place
of ˆIQv
h, Iv
43 ,h
and Iv
4,h respectively, the performance of each test would be much worst.
We denote
PThr := P (|Uh| > 1.96), PBPV := P (|UBPVh | > 1.96), PTPV := P (|UTPV
h | > 1.96).
Example 5.1. (Brownian motion plus compound Poisson process, BG index α = 0).We consider a jump diffusion process with finite activity compound Poisson jump part:
dXt = σdWt +NT∑i=1
Bi,
where Bi are i.i.d. with law N (0, 0.62), and N is a Poisson process with constant intensity λ = 5(the parameters are taken from [1]). With five minutes step h = 1/(84×252), and with v = 0.0001and β = 0.999 we obtain:
α n H σ PThr (H0) σ ≡ 0 PBPV (H0) σ ≡ 0 PTPV (H0) σ ≡ 00 1000 1000 0 0.0450 accept 0.2420 rejects 0.2730 rejects0 1000 1000 0.2 1 rejects 1 rejects 1 rejects
We find that the results given by the threshold test are reliable, since it correctly accepts (H0)when it is true and rejects (H0) when it is false. On the contrary the multipower tests give notcorrect results, since they reject (H0) even when it is true.
Figure 5.2 shows the empirical densities of the values assumed by each statistic Uh, UBPVh and
UTPVh when H = 1000, n = 1000 and h = 1/(252 × 84) and σ = 0, while Figure 5.3 shows
analogous empirical densities when σ = 0.2. The pictures lead us to make the same conclusions asthe previous table.
One reason why the BPV and TPV tests reject (H0) for samples as large as n = 1000 seems to bethat, contrary to the threshold estimator, the multipower variation estimators of Ik :=
∫ T
oσk
s ds arenot efficient estimators. In fact, to estimate I2, the standard error in (36) is μ−3
23
√CTPV = 1.75,
while the standard error given by (32) equals 1.41, the smallest value attainable (at least when X
has constant parameters, see [1]). For λ = 5 the step h = 1/(252× 84) seems to be not sufficientlysmall to allow UBPV
h and UTPVh to assume values which are typical of the standard Gaussian
law. If we implement the tests with λ = 2, or 1, then UBPVh and UTPV
h still reject (H0) but thepercentages P (|UBPV
h | > 1.96), P (|UTPVh | > 1.96) decrease. The same happens if we decrease h.
Example 5.2. (Brownian motion + Variance Gamma, BG index α = 0). We now consider
Xt = σWt + cSt + ηW(2)St
, (37)
23
−5 0 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5Threshold test ratio
−5 0 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4BPV test ratio
−100 0 100 2000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4TPV test ratio
Figure 5.2. Empirical densities of the values assumed by each statistic Uh, UBPVh and UTPV
h when we
consider the model in example 5.1, with H = 1000 and n = 1000, h = 1/(252×84), σ = 0. The continuous
curve is the standard Gaussian density.
−20 0 20 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Threshold test ratio
−50 0 50 1000
0.2
0.4
0.6
0.8
1
1.2
1.4BPV test ratio
−20 0 20 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1TPV test ratio
Figure 5.3. Empirical densities of the values assumed by each statistic Uh, UBPVh and UTPV
h when we
consider the model in example 5.1, with H = 1000 and n = 1000, h = 1/(252 × 84), σ = 0.2. The
continuous curve is the standard Gaussian density.
where W , W (2) are independent standard Brownian motions, S is a Gamma subordinator whereSh has law Γ(h/b, b), b = V ar(S1) = 0.23, η = 0.2, c = −0.2. These are approximately thevalues given in Madan (2001) for a Variance Gamma model calibrated to the SPX index. Tosimulate the VG process we time–change a Brownian motion with drift [11, Chap. 5]. Withh = 1/(252×84), v = 0.0001, r(h) = h0.999, H = 1000 we obtain the following table which confirmsthat even for this model we can rely on our test.
α n σ PThr (H0) σ ≡ 0 PBPV (H0) σ ≡ 0 PTPV (H0) σ ≡ 00 1000 0 0.0280 accept 0.7070 reject 0.9000 reject0 1000 0.2 1 reject 1 reject 1 reject
Remark. Note that we have the same values for n and h as in the previous example, so that wecan rely on the test even if we do not know whether the underlying model has a finite activityjump component or a Variance Gamma jump component.
24
Figure 5.4 shows the empirical densities obtained computing the statistics Uh, UBPVh and UTPV
h
on H = 1000 simulated paths of model (37) and σ = 0. Analogously, Figure 5.5 is relative to model(37) with σ = 0.2. They confirm the reliability of our test procedure.
−5 0 50
1
2
3
4
5
6
7
8
9Threshold test ratio
−5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7BPV test ratio
−100 0 100 2000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4TPV test ratio
Figure 5.4. Empirical density of the test statistics Uh, UBPVh and UTPV
h for the Variance Gamma model
with Brownian component (Example 5.2) with H = 1000, h = 1/(252 × 84), n = 1000, σ = 0. The
continuous lines are the (asymptotic) standard Gaussian density. The continuous lines are the (asymptotic)
standard Gaussian density.
−20 0 20 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9Threshold test ratio
−50 0 50 1000
0.5
1
1.5BPV test ratio
−10 0 10 200
0.2
0.4
0.6
0.8
1
1.2
1.4TPV test ratio
Figure 5.5. Empirical density of the test statistics Uh, UBPVh and UTPV
h for the Variance Gamma model
with Brownian component (Example 5.2), with H = 1000, h = 1/(252 × 84), n = 1000, σ = 0.2. The
continuous lines are the (asymptotic) standard Gaussian density.
The threshold estimator behaves as expected since α < 1. More surprizing is the mediocreperformance of the tripower an bipower variation tests, since the assumptions of the central limittheorems (theorems 3.1 and 3.2 in [25]) are satisfied.
Example 5.3. (Brownian motion + α-stable jumps, BG index equal to α). Now wesimulate a process
X = σW + J,
25
with J a symmetric α-stable Levy process, with law at time 1 given by Sα(1, 0, 0). To simulatethe α-stable increments we use the algorithm described in [11, p.180]. For five minutes steph = 1/(252× 84) and β = 0.999 we obtain the following results confirming us that when α < 1 wecan rely on our test.
α n H σ PThr (H0) σ ≡ 0 PBPV (H0) σ ≡ 0 PTPV (H0) σ ≡ 00.3 1000 1000 0 0.0250 accept 0.2520 reject 0.4830 reject0.3 1000 1000 0.2 1 reject 1 reject 1 reject
The next table is relative to cases with α > 1 and it confirms us that we have not to rely on thetest results, for any choice of h, consistently to what we know from the theory: even when σ ≡ 0the statistic Uh diverges if α ≥ 1.
α n H σ PThr (H0) σ ≡ 0 PBPV (H0) σ ≡ 0 PTPV (H0) σ ≡ 01.2 1000 1000 0 1 reject 1 reject 1 reject1.2 1000 1000 0.2 1 reject 1 reject 1 reject
Example 5.4. (Diffusion plus NIG jumps, BG index α = 1). We simulate –using theprocedure described in [11, p.182]– the process
Xt = σWt + cSt + ηW(2)St
,
where S is an Inverse Gaussian subordinator [11, p. 116] having V ar(S1) = k = 0.23, η = 0.2, c =−0.2. We take h = 1/(252× 84), β = 0.999, v = 0.0001. The results below show that, as expected,our test for the presence of a Brownian component is not reliable in this borderline case whereα = 1.
α n H σ PThr (H0) σ ≡ 0 PBPV (H0) σ ≡ 0 PTPV (H0) σ ≡ 01 1000 1000 0 1 reject 1 reject 1 reject
Example 5.5. (Stochastic volatility plus Variance Gamma jumps, BG index α = 0).Let us now consider a process X having a stochastic volatility correlated to the Brownian motionleading X and with jump part given by a Variance Gamma process
dXt = (μ − σ2t /2)dt + σtdW
(1)t + dJt,
whereσt = eKt , dKt = −k(Kt − K)dt + ςdW
(2)t , d < W (1), W (2) >t= ρdt. (38)
We chose a negative correlation coefficient ρ = −0.7 and μ = 0, K0 = ln(0.3), k = 0.09, K =ln(0.25), ς = 0.05 so to ensure that a path of σ within [0, T ] varies most between 0.2 and 0.4.Moreover we simulate an independent Levy process Jt = cGt + ηW
(3)Gt
, where G is a subordinatorsuch that, for any h, (Gh)(P ) = Γ(h/b, b). As in example 5.2, the parameter b coincides withV ar(G1) and we chose b = 0.23, c = −0.2, η = 0.2.We generated n increments of this process, with step h = 1/(252 × 84), and applied the test forthe presence of a Brownian component with v = 0.0001 and threshold r(h) = h0.999. The resultsare shown in the following table and confirm the theoretical prediction: the test rejects (H0).
α n H σ PThr (H0) σ ≡ 0 PBPV (H0) σ ≡ 0 PTPV (H0) σ ≡ 00 1000 1000 stoch. 1 reject 1 reject 1 reject
26
6 Applications to financial time series
We apply our tests to explore which class of models plausibly explains given financial data. Weconsider two time series: the DM/USD exchange rate from 1-10-1991 to 29-11-1994 and the SPXfutures prices from 3-1-1994 to 18-12-1997.In both cases we have concluded that the price process can be represented as the sum of a Brownianmartingale component and a jump component with finite variation e.g. Blumenthal-Getoor indexα < 1.
6.1 Deutschemark/USD exchange rate
We consider the (widely studied) time series of the DM/$ exchange rates from 1991 to 1994 from theOlsen & Associated database. From this time series we construct a series of 5-minute returns: thissampling frequency avoids many microstructure effects seen at shorter time scales (e.g. seconds)while leaving us with a relatively large sample. We have a total of 64284 equally spaced 5 minutelog-returns, with h = 1
252×84 ≈ 4.7× 10−5. Figure 6.1 shows the differences ΔiX of the logarithmof the rates.
0 1 2 3 4 5 6 7
x 104
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Figure 6.1. Deutschemark/USD five minute log-returns, from October 1991 to November 1994.
27
We apply the threshold technique to extract the variations of the data generating process to beused to estimate IV . As in the simulation experiments, the threshold is the function r(h) = h0.999.
0 2 4 6
x 104
−8
−6
−4
−2
0
2
4
6
8x 10
−3Log returns of continuous component
0 2 4 6
x 104
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02Jumps in log price
Figure 6.2. X = log P . Left: plot of ΔiXI{(ΔiX)2≤r(h)}; right: plot of ΔiXI{(ΔiX)2>r(h)}; DM/USD
five minute returns, from October 1991 to November 1994.
6.1.1 Does the jump component have finite variation?
We apply the test of subsection 5.1 to our DM/USD exchange rate time series in order to study thejump component amount of activity. As in the simulation study, we use n = 1000 and v = 0.0001.In particular we divide the first 64000 data into 64 groups of 1000 observations and for each groupwe implement the statistic U
(α)h(j), j = 1..64. Then we check how many of these obtained values fall
into the interval [−1.96, 1.96]. Since, as we see in Figure 6.3, the values outside the interval are 4.7%, we accept (H0) α < 1.
6.1.2 Does the price follow a pure-jump process?
Since we have estimated α to be less than 1, we now can test the presence of a Brownian componentin the price process. We use the technique described in section 5.2 with n = 1000, as in thesimulation study. As a consequence H = 64. Since all absolute values we obtain of Uh are muchlarger than 1.96, as seen in Figure 6.4, we reject (H0) σ ≡ 0.
28
−4 −3 −2 −1 0 1 2 3 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
test for H0) α <1
Figure 6.3. Test for the finiteness of variation of the jumps in the DM-$ exchange rate. Empirical density
of the statistic U(α)h in (33) implemented on 64 groups of 1000 DM/USD exchange returns, h = 1/(252×84).
We accept (H0) α < 1 if and only if U(α)h behaves as a standard Gaussian random variable. Since only the
4.7 % of the realizations |U (α)h(j)|, j = 1..64, are outside the interval [−1.96, 1.96], we accept (H0) α < 1.
The continuous line is the (asymptotic) standard Gaussian density.
We remark that our results are consistent with the results of Barndorff-Nielsen and Shephard[5]. In fact, assuming a finite activity jump component (Blumenthal-Getoor index α = 0 < 1) fora time series of the DM-$ exchange rate, they detected the presence of jumps. Here the presenceof a non zero Brownian component allows to explain the variability of such rates.We can conclude, for instance, that a Variance Gamma model, with no Brownian component,would be inadequate for the DM-$ time series for the period from October 1991 to November1994.
6.2 SPX index
We consider the time series of the S&P500 futures prices from 3-1-1994 to 18-12-1997. We have atotal of 78497 equally spaced 5 minutes data, with h = 1
252×84 ≈ 4.7×10−5. The left part of Figure6.5 shows the differences ΔiX of the logarithm of the prices. We apply the threshold techniqueto extract the variations of the data generating process to be used to estimate IV , r(h) = h0.999
(right part of Figure 6.5).
6.2.1 Is the jump component of the asset price of finite variation?
We apply the test of section 5.1 to our SPX prices time series in order to study the jump componentamount of activity. Using n = 1000 and v = 0.0001, we divide the first 78000 data into 78 groupsof 1000 observations and for each group we implement the statistic U
(α)h(j), j = 1..64. Then we check
how many of these obtained values fall into the interval [−1.96, 1.96]. Since, as we see in Figure6.6, the values outside the interval are 5.1 %, we accept (H0) α < 1.
29
−4 −2 0 2 4 6 8 10 12 14 160
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
test for H0) α <1
Figure 6.4. Test for the presence of a Brownian component in the DM-$ exchange rate. Empirical density
of the statistic Uh in (34) implemented on 64 sequences of 1000 5-min returns. All values Uh(j) , j = 1..64,
are outside the interval [-1.96, 1.96]: we reject (H0) σ ≡ 0. The continuous line is the (asymptotic)
standard Gaussian density.
0 1 2 3 4 5 6 7 8
x 104
−0.03
−0.025
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02SXP five minutes returns
from 3−1−1994 to 18−12−1997
Del
tai X
0 2 4 6
x 104
−8
−6
−4
−2
0
2
4
6
8x 10
−3 continuous component
0 2 4 6
x 104
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015jumps in log prices
Figure 6.5. Left: SPX five minutes log-returns from January 1994 to December 1997. Center: plot of
ΔiXI{(ΔiX)2≤r(h)}, i = 1..n. Right: plot of ΔiXI{(ΔiX)2>r(h)}, i = 1..n.
6.2.2 Does the price follow a pure-jump process?
Since our test indicates α < 1, we now can test for the presence of a Brownian component in theprice process. We use n = 1000 and H = 78. Since all absolute values we obtain of |Uh| are muchlarger than 1.96, as we can see in Figure 6.7, we reject (H0) σ ≡ 0.
For example, a Variance Gamma model with no Brownian component would be inadequate forthe SPX futures prices time series.
We note that our findings contradict the conclusion of Carr et al. [8] who, assuming a temperedstable model plus a Brownian motion for the (log) SPX index from 1994 to 1998, conclude towardsa pure jump model using a parametric estimation methods. Under less restrictive assumptionson the structure of the process and using our non-parametric test, we find a non–zero Browniancomponent in the index.
30
−4 −3 −2 −1 0 1 2 3 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
test for H0) α <1
Figure 6.6. Testing the finiteness of variation of the jumps in SPX prices. Empirical density of the
statistic U(α)h in (33) implemented on 78 sequences of 1000 increments, h = 1/(252 × 84). 5.1 % of the
realizations |U (α)
h(j)|, j = 1..78, are outside the interval [-1.96, 1.96]: we accept (H0) α < 1. The continuous
line is the (asymptotic) standard Gaussian density.
7 Conclusions
We have shown a central limit theorem for a nonparametric threshold estimator of the continuouscomponent of the quadratic variation of a semimartingale whose jump component is a Levy process.Using this theoretical result we have proposed
• a test for the presence of a continuous martingale (“Brownian”) component
• a test for establishing whether the jumps of the process has finite or infinite variation
based on observations on a discrete time grid. Using simulations of stochastic models commonlyused in finance, we have shown that our test are reliable for realistic data sizes and comparefavorably with analogous tests constructed based on multipower variation estimators of integratedvariance. Applied to time series of the DM/USD exchange rate and to SPX futures, our testsreveal the presence of a non-zero Brownian component, combined with a finite variation jumpcomponent. Our empirical results –which are somewhat different from previous ones reported inthe literature based on parametric estimation methods– point to the sufficiency of “jump-diffusion”models which model the (log)-price as the sum of a Brownian martingale and a jump componentof finite variation.
8 Appendix
Lemma 8.1. If r(h) →h 0 and if n = T/h, supi=1..n |ahi| = 0(r(h)) as h → 0, then as h → 0∑i
|ahi|I{(ΔiX)2≤r(h)} −∑
i
|ahi|I{(ΔiJ2)2≤4r(h),ΔiN=0}P→ 0.
Proof. On {(ΔiX)2 ≤ r(h)} we have |ΔiJ | − |ΔiX0| ≤ |ΔiX | ≤√r(h) and thus, by (9), for small
31
−5 0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
test for H0) α <1
Figure 6.7. Test for the presence of a Brownian martingale component in the SPX prices. Empirical
density of the statistic Uh in (34) implemented on 78 groups of 1000 observed increments. Alll observations
|Uhj |, j = 1..78 are outside the interval [-1.96, 1.96]: we reject (H0) σ ≡ 0. The continuous line is the
(asymptotic) standard Gaussian density.
h, |ΔiJ | ≤ 2√
r(h), so that a.s.
limh→0
∑i
|ahi|I{(ΔiX)2≤r(h)} ≤ limh→0
∑i
|ahi|I{(ΔiJ)2≤4r(h)}.
However ∑i
|ahi|I{(ΔiJ)2≤4r(h),ΔiN �=0} ≤ supi
|ahi|NTa.s.→ 0, (39)
as h → 0, and thus a.s.
limh→0
∑i
|ahi|I{(ΔiX)2≤r(h)} ≤ limh→0
∑i
|ahi|I{(ΔiJ)2≤4r(h),ΔiN=0} =
limh→0
∑i
|ahi|I{(ΔiJ2)2≤4r(h),ΔiN=0}.
Now we show that on the other hand the positive quantity
limh→0
∑i
|ahi|(I{(ΔiJ)2≤4r(h),ΔiN=0} − I{(ΔiX)2≤r(h)})
vanishes a.s. as h → 0. In fact
{(ΔiJ)2 ≤ 4r(h), ΔiN = 0} − {(ΔiX)2 ≤ r(h)} =
{(ΔiJ)2 ≤ 4r(h), ΔiN = 0, (ΔiX)2 > r(h)} ⊂{|ΔiJ | ≤ 2
√r(h), ΔiN = 0, |ΔiX0| + |ΔiJ2| >
√r(h)}⊂{
|ΔiX0|>√
r(h)/2}∪{|ΔiJ2| ≤ 2
√r(h), |ΔiJ2|>
√r(h)/2
}.
Since, by (11), for sufficiently small h∑i
|ahi|I{|ΔiX0|>√
r(h)/2} = 0,
we a.s. havelimh→0
∑i
|ahi|(I{(ΔiJ)2≤4r(h),ΔiN=0} − I{(ΔiX)2≤r(h)}) ≤
32
limh→0
∑i
|ahi|I{|ΔiJ2|≤2√
r(h),|ΔiJ2|>√
r(h)/2},
however as h → 0E[∑
i
|ahi|I{|ΔiJ2|≤2√
r(h),|ΔiJ2|>√
r(h)/2}] ≤
O(r(h))nP{|ΔiJ2| ≤ 2√
r(h), |ΔiJ2| >√
r(h)/2} ≤O(r(h))nP{|ΔiJ2|I{|ΔiJ2|≤2
√r(h)} >
√r(h)/2} ≤
O(r(h))nE[(Δi J2)2I{|ΔiJ2|≤2
√r(h)}]
r(h)= O(r(h))n
hη2(2√
r(h))r(h)
→ 0.
Theorem 8.2 (Lindeberg-Feller, see Chung 1974, p.205). Assume that for each n the randomvariables on row n, Hn1, Hn2, ...Hnn, are independent, V ar[Hnj ] < ∞ for each n and j, andE[Hnj ] = 0,
∑nj=1 V ar[Hnj ] = 1. Let Fnj be the distribution function of Hnj, for each n and j.
Set Sn :=∑n
i=1 Hni.In order that, as n → ∞, the two conclusions below both hold:
(i) Snd→ N (0, 1) (40)
(ii) ∀ε > 0 limn→∞ max
i=1..nP (|Hnj | > ε) = 0 (41)
it is necessary and sufficient that for each η > 0 we have, as n → ∞,n∑
j=1
∫|x|>η
x2dFnj(x) → 0.
Proposition 8.3. [[21], proposition 3.4] If
1) J =∑Nt
j=1 γj is a finite activity jump process where N is a non-explosive counting processand the random variables γj satisfy, ∀t ∈ [0, T ], P{ΔNt = 0, γNt = 0} = 02) a and σ are cadlag processes3) r(h) is a deterministic function of the lag h between the observations, such that lim
h→0r(h) =
0, and limh→0
(h log 1h)/r(h) = 0,
then we have that as h → 0
ˆIQh :=13h
n∑i=1
(ΔiX)4I{(ΔiX)2≤r(h)}P→∫ T
0
σ4t dt.
Theorem 8.4. [[21], theorem 3.5] Suppose that
1) J =∑Nt
j=1 γj is a finite activity jump process where N is a non-explosive counting processand the random variables γj satisfy, ∀t ∈ [0, T ], P{ΔNt = 0, γNt = 0} = 02) a and σ ≡ 0 are cadlag processes3) r(h) is a deterministic function of the lag h between the observations, such that lim
h→0r(h) =
0, and limh→0
(h log 1h)/r(h) = 0,
then we have ( ˆIV h − IV )/√
2h ˆIQhd→ N (0, 1) .
33
Acknowledgements. This work has benefited from support by the European Science Founda-tion program Advanced Mathematical methods in Finance. C. Mancini thanks Ecole Polytechniqueand Columbia University’s IEOR Department for their hospitality. We thank Jean Jacod forimportant comments.
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