École polytechnique · estimating the integrated volatility 3 5 concludes showing the performance...

ESTIMATING THE INTEGRATED VOLATILITY INSTOCHASTIC VOLATILITY MODELS WITH LEVY TYPE JUMPS

Cecilia ManciniUniversita di Firenze, Dipartimento di matematica per le decisioni,

[email protected]

Summary

We consider a stochastic volatility financial model with jumps, where the jump partis a Levy process. Using a threshold method we show that one can identify theinstants of jump, on the basis of a discrete record of high frequency observationsof the modelled process. By that we reach an estimator of the integrated volatilitywhich we show to be consistent even in the case where the volatility process and theleading Brownian motion of the model are dependent. We show that such estimatoris also asymptotically Gaussian when the jump part has finite activity. Moreoverwe exhibit an estimator of each (stochastic) size of the occurred jumps: in the caseof constant volatility we give the asymptotic distribution of the estimation error.We present the histogram produced by our estimator of the integrated volatilitywithin three different simulated models. 1

1 introduction

In order to price derivatives on a given stock or to manage the risk of the assets wehold in our portfolio we need to clarify which sources of randomness drive them. Inparticular we need to ”correctly” identify a model for them.There is a wide literature where a stock is modelled by a diffusion process as wellas a more recent literature where the underlying is modelled by a jump-diffusionprocess with possibly stochastic volatility. However more recently some Authors(for instance Madan, (2001), Carr et al. (2002)) proposed to use pure jump models.Clearly in order to reproduce the high variability of the prices such models need tocontain at least an infinite activity jump random measure. Then how to select themost plausible model? how to check whether the diffusion component is excluded?Usually for financial stocks, financial indexes or commodities we dispose of discreteobservations. The key problem is thus to separate the jump part and the diffusionpart given discrete observations. This is precisely the problem we address ourselvesto.Estimation of the integrated volatility is also useful per se, in order to understandthe so called volatility derivatives (Carr and Lee, 2004).We will show that when a model has stochastic volatility and a possibly infinite

1AMS 2000 subject classifications. Primary: 62G05, 62G20, 62M99; secondary: 7M05 37M10.Key words and phrases: time series, stochastic volatility, Levy jump process, threshold estimators,jump-indicator, quadratic variation, integrated volatility, integrated quarticity, asymptotic proper-ties, not equally spaced observations, simulations

1

activity jump component, we are able to extract the integrated volatility.Aıt-Sahalia (2003) shows how in a parametric model the maximum likelihood esti-mator for the diffusion coefficient is as much efficient in a jump diffusion model (withfinite or infinite jump activity) as in a diffusion model, and thus he concludes thecapability of the maximum likelihood approach to perfectly disentangle the diffusionfrom the jump part.Our approach is proper also for models where the coefficients are not parametricallyspecified.Barndorff-Nielsen and Shephard (2004a, 2004c) define and use the bipower variationprocess to estimate the integrated volatility and also the quadpower variation (aswell as a power variation) to construct a test for the presence of the jump part in theprocess generating given discrete observations. They need that the volatility processis independent of the leading Brownian motion (no leverage assumption) and thatthe jump process has finite activity.Barndorff-Nielsen and al. (2005) prove the consistency and a central limit result forthe realized bipower variation when the underlying process is a diffusion.Woerner (2003) uses the power variation to estimate the integrated volatility: sheneeds the same assumption of independence and she tackles also the case where theobservations are not equally spaced.Completely different non parametric approaches are the following. Aıt-Sahalia andLo (1998) make a mixed cross-section and historic data calibration estimate, basedon the Nadaraya-Watson kernel, of the European options price functions and of theimplicit state price conditional density. That can potentially give estimated op-tions price functions not being consistent with the absence of arbitrage assumption.Bandi and Nguyen (2001) and Johannes (2002) estimate the infinitesimal momentsof a model with stochastic volatility plus jumps, reaching aggregate informationabout the model (stochastic) coefficients.In this work we are generalizing our threshold estimators, based on the quadraticvariation (Mancini, 2001, and Mancini, 2004), of the integrated volatility and of thejump instants and amplitudes which are shown to be consistent even in a stochasticvolatility and infinite activity jump model with also eventual leverage. The goodperformance of such estimators is shown within three different simulated models.

An outline of the paper is as follows: in section 2 we introduce the frameworkand the notations; in section 3 we deal with the case in which the financial modelcontains a finite activity jump part: we show that by the threshold method we canidentify each instant of jump, which gives as a consequence threshold estimators ofthe integrated volatility, of the integrated quarticity and of each stochastic size ofthe occurred jumps. Using results of Barndorff-Nielsen and Shephard (2004a, 2004b)and Barndorff-Nielsen and al. (2005) we show the asymptotic normality of the esti-mator of the integrated volatility. Moreover we find the asymptotic distribution ofthe estimation error of the sizes of jump when the volatility is constant. Section 4 isdevoted to the case the underlying process contains an infinite activity jump part:we show that again the threshold estimator of the integrated volatility is consistent,even under leverage and even when the observations are not equally spaced. Section

2

Estimating the integrated volatility 3

5 concludes showing the performance of the estimator of the integrated volatilitywithin three different simulated models.

Acknowledgements I’m sincerely grateful to Rama Cont, who kindly hosted meat the Polytechnique of Paris for one month for this work. I thank him also forthe fact that he is always ready to discuss. He suggested me to use model 2 insection 5. I thank Sergio Vessella and Marcello Galeotti who supported me for sucha stay by MIUR grant number 2002013279 and Progetto Strategico. I sincerelythank Jean Jacod for the very important discussions during the ”Infinite activityjump processes” week at the Newton Institute in Cambridge.

2 The framework

On the space (Ω, (Ft)t∈[0,T ], F , P), let W be a standard Brownian motion and J bea pure jump Levy process. Let (Xt)t∈[0,T ] be a real process such that X0 ∈ IR and

dXt = atdt + σtdWt + dJt, t ∈ [0, T ], (1)

where a, σ are progressively measurable processes such to guarantee that (1) hasunique strong solution on [0, T ] being adapted and right continuous with left limits(se e.g. Ikeda and Watanabe, 1981). Suppose we dispose for each n of a discreterecord X0, Xt1 , ..., Xtn−1 , Xtn of observations of a realization of X, with ti = ih,for a given step h.

For instance X represents the logarithm of the price evolution of a financial stockor index.

Notations.For any process Z, let us denote by ∆iZ the increment Zti − Zti−1 and by ∆Zt thesize Zt − Zt− of the jump (eventually) occurred at time t.In the case J is a finite activity jump process, denote by (τj)j=1..NT

the instants ofits jumps; by τ (i) the first instant a jump occurs within ]ti−1, ti], if ∆iN ≥ 1; byγτj the size of the jump occurred at time τj ; by γ(i) the size γτ (i) of the first jumpwithin ]ti−1, ti], if ∆iN ≥ 1; and by γ

.= minj=1..NT|γτj | .

[Z] is the quadratic variation process associated to Z.[Zh]T is the estimator

∑ni=1(∆iZ)2 of the quadratic variation [Z]T .

< Zc > is the quadratic variation [Zc] of the continuous martingale part of Z.FZ denotes the sigma-algebra generated by the process Z.H.W is the process given by the stochastic integral

∫ ·0 HsdWs.

Yt =∫ t0 σ2

udu is te intgrated volatility of our underlying process X.Given a sequence xhh of real numbers, we write xh = O(h) to mean that limh→0

xhh =

c, c a constant.By c (low case) we will denote generically a constant.P lim means ”limit in probability”; d lim means ”limit in distribution”.


3 The case of finite activity jump component

3.1 Consistency

An important variable related to X and containing the integrated volatility is thequadratic variation at T

[X]T =∫ T

0σ2

t dt +∫ T

0

∫

IRx2µ(dx, dt), (2)

where µ is the Poisson random jump measure associated to the Levy process J (seee.g. Cont and Tankov, 2004 and Ikeda and Watanabe, 1981). In fact we easily reachthe value of [X]T using discrete observations, since P lim|π(t)|→0

∑j=1,...,m(Xsj −

Xsj−1)2 .= [X]t, t ∈ [0, T ], where π(t) is a finite partition t0 = 0, t1, ..., tm = t of

[0, t], and |π(t)| = minj |tj − tj−1|.We will consider in this section the case in which J is a compound Poisson process,the only kind of Levy process having finite jump activity on finite time horizons:

Jt =∫ t

0γsdNs =

Nt∑

j=1

γτj .

In that case (2) becomes

[X]T =∫ T

0σ2

t dt +Nt∑

j=1

γ2τj

,

and the quadratic variation gives us only an aggregate information regarding boththe integrated volatility and the jump sizes. In order to identify

∫ T0 σ2

t dt form [X]Tthe key point will be to capture the time intervals ]ti−1, ti] where J has not jumped.The following theorem will give us an instrument to identify such intervals.

Theorem 3.1 Suppose that Jt =∫ t0 γsdNs is a compound Poisson process and

1) a.s. lim suph→0

supi |∫ ti

ti−1asds|

√h log 1

h

≤ C(ω) < ∞

2) σ is s.t.∫ T0 σ2

sds < ∞ and lim suph→0

supi |∫ ti

ti−1σ2

sds|h ≤ M(ω) < ∞;

3) r(h) is a deterministic function, of the step h between the observations, s.t.

limh→0

r(h) = 0, and limh→0

h log 1h

r(h) = 0,

then for P-almost all ω ∃h(ω) s.t. ∀h ≤ h(ω) we have

∀i = 1, ..., n, I∆iN=0(ω) = I(∆iX)2≤r(h)(ω). (3)

Proof. We need the following preliminary facts.


• The Paul Levy law for the modulus of continuity of Brownian motion’s paths(see e.g. on Baldi (1984), p.45, Sato (2001), p.10) gives us that:

a.s. limh→0

sup0 ≤ s1, s2 ≤ T|s2 − s1| = h

|Ws1 −Ws2 |√2h log 1

h

= 1

anda.s. lim

h→0sup

0 ≤ s1 ≤ T|s2 − s1| < h

|Ws1 −Ws2 |√2(s2 − s1) log 1

(s2−s1)

= 1. (4)

This implies that

a.s. limh→0

supi∈1,...,n

|∆iW |√2h log 1

h

≤ 1.

• Let us extend σ on t ∈]T,∞[ so that σ is s.t. ∀t > 0∫ t0 σ2

sds < ∞ and∫ +∞0 σ2

sds = +∞ a.s.. In this way the stochastic integral σ.W is in turn a timechanged Brownian motion (see Baldi, p.163, 166 or Ikeda-Watanabe, p.85):∫ ξt0 σsdWs

.= Bt is a Brownian motion, where, defining Yt =∫ t0 σ2

udu, (ξt)t is thepseudo-inverse of (Yt)t, ξt = infv : Yv > t. Moreover ∆i (σ.W ) = BYti

−BYti−1.

• We havea.s. sup

i∈1,...,n

|∆iσ.W |√2h log 1

h

≤

supi

|BYti−BYti−1

|√

2∆iY log 1∆iY

supi

√2∆iY log 1

∆iY√2Mh log 1

Mh

supi

√2Mh log 1

Mh√2h log 1

h

.

As h → 0, since a.s. M(ω) < ∞, the a.s. limit of the first factor is, by (4), boundedby 1; since the function x ln 1

x is increasing, the second term is a.s. bounded by 1;finally the third one tends a.s. to

√M(ω). That is, for small h,

a.s. supi∈1,...,n


h

≤√

M(ω) + 1 := M(ω).

It follows that, for small h,

supi

| ∫ titi−1

asds +∫ titi−1

σsdWs|√2h log 1

h

≤ supi

| ∫ titi−1

asds|√

2h log 1h

+supi

| ∫ titi−1

σsdWs|√2h log 1

h

≤ C(ω)+M(ω),

having used also assumption 1).

Proof of the theorem. Our first step is to see that a.s., for small h, ∀i, I∆iN=0 ≤I(∆iX)2≤r(h), and the second step is to see that also the other inequality holds, that


is a.s., for small h, ∀i, I∆iN=0 ≥ I(∆iX)2≤r(h), and so our claim holds true.

1) For each ω set J0,h = i ∈ 1, ..., n : ∆iN = 0: we are going to show thata.s., for small h, supJ0,h

(∆iX)2 ≤ r(h), since this implies that for all i such thatI∆iN=0 = 1, we have I(∆iX)2≤r(h) = 1, which means that

∀i ∈ 1, ..., n I∆iN=0 ≤ I(∆iX)2≤r(h).

To evaluate the supJ0,h(∆iX)2, remark that

supi∈J0,h

(∆iX)2

r(h)= sup

J0,h

|

∫ titi−1

asds +∫ titi−1

σsdWs|√2h log 1

h

2

· 2h log 1h

r(h); (5)

since the first factor is a.s. bounded and the second one approaches zero, then a.s.supi∈J0,h

(∆iX)2

r(h) → 0. In particular, for small h, the supi∈J0,h

(∆iX)2

r(h) is dominatedby 1, that is

supJ0,h

(∆iX)2 ≤ r(h).

2) Now we establish the other inequality. For any ω set J1,h = i ∈ 1, ..., n :∆iN 6= 0. We are going to show that, for small h, infi∈J1,h

(∆iX)2 > r(h), sinceby that we will have: for each i such that I∆iN 6=0 = 1 then (∆iX)2 > r(h), orI(∆iX)2>r(h) = 1, therefore

∀ i ∈ 1, ..., n I∆iN 6=0 ≤ I(∆iX)2>r(h),

that is I∆iN=0 = 1− I∆iN 6=0 ≥ 1− I(∆iX)2>r(h) = I(∆iX)2≤r(h), as we need.

In order to evaluate infi∈J1,h

(∆iX)2

r(h) , remark that

∀i ∈ J1,h,(∆iX)2

r(h)=

(∫ titi−1

asds + ∆iσ.W)2

2h log 1h

2h log 1h

r(h)+

+2

∫ titi−1

asds + ∆iσ.W√

r(h)

∑∆iN`=1 γ`√r(h)

+(∑∆iN

`=1 γ`)2

r(h)

the first term tends a.s. to zero uniformly with respect to i. As for the other terms,for small h we have that ∆iN ≤ 1 for each i, so that they become

γτ (i)√r(h)

∫ titi−1

asds + ∆iσ.W√

r(h)+

γτ (i)√r(h)

.

Again the contribution of the first term within brackets tends a.s. to zero uniformlyon i, and thus

limh

infi∈J1,h

(∆iX)2

r(h)= lim

h

γ2τ (i)

r(h)≥ lim

h

γ2

r(h)= +∞.


•

Remarks.

Theorem 3.1 is an extension of the analogous theorem in Mancini (2001) whichwas proved in the case where X has constant coefficients.

Assumption 1) simply requests that the sequence(supi |

∫ titi−1

asds|)

/(h log 1

h

)1/2

keeps bounded as h → 0. It is satisfied if e.g. pathwise (as(ω))s is bounded on[0, T ]. In particular assumption 1) is satisfied in a model with mean reverting driftas = kθ − kXs (since Xs has a finite number of jumps in [0, T ] and the realizationsof the sizes of jump are real numbers).

If we assume that in equation (1) a and σ are processes with right continuouspath with left limits (cadlag), then assumptions 1) and 2) are immediately satisfied,since a.s. such paths are bounded on [0,T].

The assumption that J is a compound Poisson process means in particular that∀t ∈ [0, T ] P∆Nt 6= 0, γt = 0 = 0. In fact the Levy measure of a Levy one dimen-sional process is defined on IR−0, and extended on 0 by defining ν0 = 0, whichis the case for instance of the compound Poisson process having Gaussian sizes ofjump. This is very natural, since if a jump occured with zero size then in fact thiswould have no effect at t and we could say that no jumps occurred.Anyway theorem 3.1 holds true even when J is any finite activity jump process∑Nt

j=1 cj where N is a non explosive counting process and cj satisfy that ∀t ∈ [0, T ]P∆Nt 6= 0, γt = 0 = 0.

Assumption 3) says us how to choose the threshold r(h). The key idea for whichsuch a choice works is that the (absolute value of) increments of the path of a Brow-nian motion (and thus of a stochastic integral with respect to a Brownian motion)tend a.s. to zero like as the deterministic function

√2h ln 1

h . Therefore, for small h,when we find that the squared increment (∆iX)2 is bigger than r(h) > 2h ln 1

h , wecan think that it could not be generated by the Brownian part of X (least of all bythe drift part), and thus some jumps have to be occurred.

Not equally spaced observations. In the case, in fact frequent, where the availablerecord X0, Xt1 , ..., Xtn−1 , Xtn has not constant step ∆ti

.= ti − ti−1 between theobservations, theorem 3.1, and thus also (6) below, is again valid. In fact if we seth

.= maxi ∆ti all the fundamental ingredients of the proof of theorem 3.1 will hold:

limh→0

supi∈1,...,n

|∆iW |√2h log 1

h

≤ limh→0

supi∈1,...,n

|∆iW |√2∆ti log 1

∆ti

≤ 1,


by (4) and the monotonicity of x ln 1x , and

a.s. ∀i ∆iY < ∆ti M(ω) ≤ hM(ω).

This implies again that

a.s. supi∈1,...,n


h

≤ M(ω).

Corollary 3.2 Under the assumptions of theorem 3.1 we have

P limh→0

n∑

i=1

(∆iX)2I(∆iX)2≤r(h) =∫ T

0σ2

t dt. (6)

Proof. Since a.s. for small h we have that∀i = 1, ..., n, I∆iN=0(ω) = I(∆iX)2≤r(h)(ω), then for such small h it holds thatP lim

h→0

∑i(∆iX)2I(∆iX)2≤r(h) = P lim

h→0

∑i(∆iX)2I∆iN=0 =

∫ t0 σ2

udu. •

3.2 Central limit theorem

As a corollary of results in Barndorff-Nielsen and al. (2005), in Barndorff-Nielsenand Shephard (2004a) and of our theorem 3.1 we obtain two threshold type es-timators of

∫ T0 σ4

t dt, which are alternative to the power and multipower variationestimators, given by them, when X has a finite activity jump component. An es-timate of

∫ T0 σ4

t dt is needed in order to give the asymptotic law of the error takenwhen approximating

∫ T0 σ2

t dt by∑n

i=1(∆iX)2I(∆iX)2≤r(h). Let us recall the resultsof Barndorff-Nielsen and al. (2005) and of Barndorff-Nielsen and Shephard (2004a).

In Barndorff-Nielsen and al. (2005), theorem 2.2 with r = 4, s = 0, we have (inparticular): if dX = asds + σsdWs, where a is predictable and locally bounded andσ is cadlag, then for h → 0

13

∑i(∆iX)4

h→P

∫ T

0σ4

t dt

(power variation estimator).

Moreover Barndorff-Nielsen and Shephard (2004a, p. 29) state that for dX =asds + σsdWs +

∑Ntj=1 cj we have

∑ni=4 Π3

k=0|∆i−kX|h

→P µ41

∫ T

0σ4

t dt,

(quadpower variation estimator), where µ1 = E[|u|] =√

2π , u ∼ N(0, 1).

In the light of these results we now state the following asymptotic properties ofthe threshold estimators.


Proposition 3.3 Under the same assumptions of theorem 3.1 and the assumptionsof Barndorff-Nielsen and al. (2005) (or of Barndorff-Nielsen and Shephard (2004a),respectively) we have that

13

∑i(∆iX)4I(∆iX)2≤r(h)

h→P

∫ T

0σ4

t dt,

(or ∑ni=4 Π3

k=0|∆i−kXI∆i−kX2≤r(h)|h

→P µ41

∫ T

0σ4

t dt,

respectively).

Proof.

P lim13


h= P lim

13

∑i(∆iX)4I∆iN=0

h=

P lim13

∑ni=1(

∫ titi−1

asds +∫ titi−1

σsdWs)4

h+

−P lim13

∑ni=1(

∫ titi−1

asds +∫ titi−1

σsdWs)4I∆iN 6=0h

:

the limit in probability of the first term is∫ T0 σ4

t dt as Barndorff-Nielsen and al.(2005) have shown, while the second term is dominated by

P lim13

supi

∫ titi−1

asds +∫ titi−1

σsdWs√h ln 1

h

4

NT(h ln 1

h)2

h≤

P lim (C(ω) + M(ω))4NT (ω)

3(h ln 1

h)2

h= 0.

As for the quadpower-kind estimator, we can write X = X(c) + X(d), whereX

(c)t =

∫ t0 asds +

∫ t0 σsdWs and X

(d)t =

∑Ntj=1 γj , and thus, by theorem 3.1, we have

P lim∑n

i=4 Π3k=0|∆i−kXI(∆i−kX)2≤r(h)|

h= P lim

∑ni=4 Π3

k=0|∆i−kX(c)I∆i−kN=0|

h

= P lim∑n

i=4 Π3k=0|∆i−kX

(c) −∆i−kX(c)I∆Ni−k 6=0|

h

≤ P lim∑n

i=4 Π3k=0|∆i−kX

(c)|h

+ P lim supj

|∆jX(c)|4

hc NT . (7)

Since a.s.

supj

|∆jX(c)|4

h= sup

j

|

∫ ti−kti−k−1

asds +∫ ti−kti−k−1

σsdWs|√h ln 1

h

4

h ln2 1h→ 0,


then the second term in (7) is in fact zero, and, thanks to the result in Barndorff-Nielsen and Shephard (2004a), the proof is complete. •

Finally, as a corollary of theorem 1 in Barndorff-Nielsen and Shephard (2004b)we have the following result of asymptotic normality for our estimator of the inte-grated volatility.

Theorem 1 in Barndorff-Nielsen and Shephard (2004b): if dX = asds + σsdWs,where a and σ are cadlag processes such that

∫ T0 σ2

sds < ∞ and∫ T0 a2

sds < ∞, then,as h → 0

[Xh]T − [X]T√h

→d

√2

∫ T

0σ2

udBu

where B is a Brownian motion independent from X (recall from the notations that[Xh]T =

∑ni=1(∆iX)2). •

Proposition 3.4 Under the assumptions of theorem 3.1 and under the assumptionsthat a and σ are cadlag processes FX-measurable, then we have

∑ni=1(∆iX)2I(∆iX)2≤r(h) −

∫ T0 σ2

t dt√23


→d N (0, 1) .

Proof.

d lim∑n

i=1(∆iX)2I(∆iX)2≤r(h) −∫ T0 σ2

t dt√23


=

d lim

∑n

i=1(∫ ti

ti−1asds+

∫ titi−1

σsdWs)2−∫ T

0σ2

t dt√

23


+

−d lim

∑n

i=1(∫ ti

ti−1asds+

∫ titi−1

σsdWs)2I∆iN 6=0√2h

∫ T

0σ4

t dt

:

(8)

the second term tends to zero in probability, since

supi

(∫ titi−1

asds +∫ titi−1

σsdWs)2√h

NT ≤ (C(ω) + M(ω))√

h ln1h

NT →a.s. 0.

The first term of (8) coincides with

d lim∑n

i=1(∆iX(c))2 − ∫ T

0 σ2t dt√

23

∑i(∆iX(c))4 − 2

3

∑i(∆iX(c))4I∆iN 6=0

that is it coincides with

d lim[X(c)

h ]T − [X(c)]T√

h

√23

∑i(∆iX(c))4

h

= d lim[X(c)

h ]T − [X(c)]T√h

1√2

1√∫ T0 σ4

t dt.


The first factor tends in law to√

2∫ T0 σ2

udBu, by Barndorff-Nielsen and Shephard re-sult (2004b, theorem 1). However the assumption (a, σ) ∈ FX ensures that a and σare independent of B and thus, conditionally on (a, σ), B is again a Brownian motionand

∫ T0 σ2

udBu is Gaussian with law N(0,∫ T0 σ4

udu). Thus, conditionally on (a, σ),

we have that [X(c)h ]T−[X(c)]T√2h

∫ T

0σ4

udu

→d N(0, 1), and since the limit law does not depend on

(a, σ) then in fact the convergence in distribution holds even without conditioning. •

It is clear, from theorem 3.1 and from the fact that for small h it is low theprobability of more than one jump over an interval ]ti−1, ti], that an estimator ofeach instant of jump is (see e.g. Mancini, 2004)

∆i,hN.= I(∆iX)2>r(h).

Moreover it is natural to think that an estimate of each realized size of jump is givenby

γ(i) .= ∆iXI(∆iX)2>r(h),

since when a jump occurs then the contribution of∫ titi−1

audu +∫ titi−1

σudWu to ∆iXis asymptotically negligible. In Mancini (2004) we have shown the consistency ofeach γ(i). However we only gave a lower bound for the speed of convergence whenthe coefficients σ and a are stochastic. Here we show that, at least in the specialcase of constant σ, the speed is

√n.

Theorem 3.5 If X is as in theorem 3.1 with moreover a.s. lim suph→0

supi |∫ ti

ti−1asds|

hµ ≤C(ω) < ∞ for some µ > 0.5 (which is the case if a is cadlag), and σ is constant,then

√n

∑

i

(γ(i) − γ(i)I∆iN≥1

)→d

NT∑

j=1

Zj

where Zj are i.i.d. N(0, σ2T ).

Proof. √n

∑

i

(γ(i) − γ(i)I∆iN≥1

)=

√n

∑

i

∆iXI(∆iX)2>r(h), ∆iN=0+√

n∑

i

(∆iXI(∆iX)2>r(h), ∆iN=1 − γ(i)I∆iN=1

)

+√

n∑

i

(∆iXI(∆iX)2>r(h), ∆iN≥2 − γ(i)I∆iN≥2

):

a.s. for small h we have that, for each i, I(∆iX)2>r(h), ∆iN=0 = 0, and thus the firstterm tends to zero in distribution. Also the third term tends to zero in distribution,since

P√n∑

i

(∆iXI(∆iX)2>r(h) − γ(i)

)I∆iN≥2 6= 0 ≤


P (∪i∆iN ≥ 2) ≤ nO(h2) = O(h).

Therefored lim

√n

∑

i

(γ(i) − γ(i)I∆iN≥1

)=

d lim√

n∑

i

(∆iXI(∆iX)2>r(h),∆iN=1 − γ(i)I∆iN=1

)=

= d lim√

n∑

i

((∫ ti

ti−1

asds +∫ ti

ti−1

σsdWs)I(∆iX)2>r(h),∆iN=1

−γ(i)I(∆iX)2≤r(h),∆iN=1)

and this coincides in fact with

d lim√

n∑

i

(∫ ti

ti−1

asds +∫ ti

ti−1

σsdWs)I(∆iX)2>r(h),∆iN=1 (9)

since a.s. for small h we have√

n∑

i γ(i)I(∆iX)2≤r(h),∆iN=1 = 0. Moreover (9)

coincides withd lim

√n

∑

i

∫ ti

ti−1

σsdWsI(∆iX)2>r(h),∆iN=1,

sinceP√n

∑

i

|∫ ti

ti−1

asds|I(∆iX)2>r(h),∆iN=1 > ε ≤

P√nC(ω)hµ∑

i

I∆iN=1 > ε ≤ Phµ−0.5√

TC(ω)NT > ε →h 0.

Finally, since a.s. for small h we have for all i that I(∆iX)2>r(h),∆iN=1 = I∆iN=1,we only need now to compute

d lim√

n∑

i

∫ ti

ti−1

σsdWsI∆iN=1 = d lim√

nσ∑

i

∆iWI∆iN=1,

for which we consider the characteristic function

E[eiθσ√

n∑

i∆iWI∆iN=1 ] = E[eiθσ

√T

∑i

∆iW√h

I∆iN=1 ] :

∆iW√h

are independent standard Gaussian random variables, thus we have

= En[eiθσ√

T∆1W√

h I∆iN=1 + I∆iN 6=1] =

=(

1 + (e−12θ2σ2T − 1)e−λhλ

T

n

)n

→h e(e−12 θ2σ2T−1)λT ,

which is the characteristic function of a compound Poisson process∑NT

i=1 Zi withZi ∼ N(0, σ2T ). •


4 The case of infinite activity jump component

Let us now consider the case of a log price process containing a jump part J withpossibly infinite activity. In fact our estimator again is able to extract the integratedvolatility form the observed data, as soon as J is a pure jump Levy process.

Theorem 4.1 Let the assumptions 1) (pathwise integrability condition on a) 2)(pathwise integrability condition on σ) and 3) (choice of the function r(h): thatchoice implies that also h

r(h) →h 0) of theorem 3.1 hold.

Let J be the process defined by Jt ≡∫ t0

∫x∈IR,|x|>1 xµ(ds, dx)+

∫ t0

∫x∈IR,|x|≤1 xµ(ds, dx),

where µ is the Poisson random jump measure of a Levy process, µ = µ − ν is thecompensated measure, and ν is the Levy measure of J . Let ν be independent of W .Then

P limh→0

n∑

i=1

(∆iX)2I(∆iX)2≤r(h) =∫ T

0σ2

t dt. (10)

To show the result we will decompose X into the sum of a stochastic volatilityand finite activity jump process X1 plus an infinite activity compensated process J2

of small jumps. We will use theorem 3.1 for the first term, and we will show thatthe contribution of the second term J2 to the sum of all the squared increments ofX is in fact negligible.

Proof. We decompose the Levy jump process J as the sum of the jumps bigger than1 and the sum of the compensated jumps smaller than 1. Let us define

X0s.=

∫ s0 atdt +

∫ s0 σtdWt,

J1s.=

∫ s0

∫|x|>1 xµ(dt, dx),

X1.= X0 + J1,

J2s.=

∫ s0

∫|x|≤1 x(µ(dt, dx)− dtν(dx)).

(11)

The jump part J1 is a compound Poisson process: J1s =∑N1

si=1 γ1

i . For each s,V ar(J2s) = s

∫|x|≤1 x2ν(dx) .= σ2(1). Note that, since J = J2 +J1 is a Levy process,

then∫|x|≤1 x2ν(dx) < +∞, and thus we have that

σ2(ε) .= s

∫

|x|≤εx2ν(dx) →ε→0 0.

Let us show (10). Since X = X1 + J2, we can write∣∣∣∣∣∑

i

(∆iX)2I(∆iX)2≤r(h) −∫ t

0σ2dt

∣∣∣∣∣ ≤

Estimating the integrated volatility 14∣∣∣∑i(∆iX1)2I(∆iX1)2≤4r(h) −

∫ t0 σ2dt

∣∣∣ +

+∣∣∣∑i(∆iX1)2(I(∆iX)2≤r(h) − I(∆iX1)2≤4r(h))

∣∣∣ +

+2∣∣∣∑i ∆iX1∆iJ2I(∆iX)2≤r(h)

∣∣∣ +∣∣∣∑i(∆iJ2)2I(∆iX)2≤r(h)

∣∣∣ .

(12)

By theorem 3.1 we know that the first term tends to zero in probability. We noware going to show that the P lim of each one of the other three terms is zero.

Let us deal with the second term:∣∣∣∣∣∑

i

(∆iX1)2(I(∆iX)2≤r(h) − I(∆iX1)2≤4r(h))

∣∣∣∣∣ =

∣∣∣∣∣∑

i

(∆iX1)2(I(∆iX)2≤r(h),(∆iX1)2>4r(h) − I(∆iX)2>r(h),(∆iX1)2≤4r(h)

)∣∣∣∣∣ (13)

If I(∆iX)2≤r(h),(∆iX1)2>4r(h) = 1, since

2√

r(h)− |∆iJ2| < |∆iX1| − |∆iJ2| ≤ |∆iX1 + ∆iJ2| ≤√

r(h),

then |∆iJ2| >√

r(h). Thus∑

i

(∆iX1)2I(∆iX)2≤r(h),(∆iX1)2>4r(h) ≤∑

i

(∆iX1)2I(∆iJ2)2>r(h) ≤

2∑

i

(∆iX0)2 I(∆iJ2)2>r(h) + 2

∑

i

(∆iN

1∑

j=1

γ1j )2I(∆iJ2)2>r(h). (14)

The first term is dominated by

2 supi

(∆iX0)2

h ln 1h

h ln1h

∑

i

I(∆iJ2)2>r(h), (15)

where the first factor is a.s. bounded and the second one tends to zero in probability,since

h ln1h

E[∑

i

I(∆iJ2)2>r(h)] = h ln1h

nP(∆iJ2)2 > r(h) ≤

h ln1h

nE[(∆iJ2)2]

r(h)= nhσ2(1)

h ln 1h

r(h)→ 0,

and thus the first term of (14) tends to zero in probability.Within the second term of (14) J1 is independent of J2, therefore

P

∑

i

(∆iN

1∑

j=1

γ1j )2I(∆iJ2)2>r(h) 6= 0

≤ P

(∪i∆iN

1 6= 0, (∆iJ2)2 > r(h))≤


nP∆1N1 6= 0E[(∆1J2)2]

r(h)= nO(h)

hσ2(1)r(h)

→ 0.

As for the second term of (13) we note that |∆iJ1| − |∆iX0| ≤ |∆iX1|, therefore if|∆iX1| ≤ 2

√r(h) then

∆iN1 < |∆iJ1| < |∆iX0|+2

√r(h) ≤ sup

i

| ∫ titi−1

asds +∫ titi−1

σsdWs|√h ln 1

h

√h ln

1h

+2√

r(h) :

the righthand term does not depend on i and tends to zero a.s. with order√

r(h),and thus, a.s. ∀i = 1..n on (∆iX1)2 ≤ 4r(h) we have ∆iN

1 → 0, and in particularfor small h we have that, ∀i = 1..n, ∆iN

1 = 0. This allows us to say that for smallh

(∆iX)2 > r(h), (∆iX1)2 ≤ 4r(h) ⊂ (∆iX0 + ∆iJ2)2 > r(h)

⊂ (∆iX0)2 >r(h)

4 ∪ (∆iJ2)2 >

r(h)4.

However for small h a.s. ∀i = 1..n I(∆iX0)2>r(h)

4 = 0 and thus for small h we have

∑

i

(∆iX1)2I(∆iX)2>r(h),(∆iX1)2≤4r(h) ≤

∑

i

(∆iX0)2I(∆iJ2)2>r(h)

4 ≤

supi

(∆iX0)2

h ln 1h

h ln1h

∑

i

I(∆iJ2)2>r(h)

4,

which tends to zero in probability like as before in (15).

Let us now deal with the third term of (12):∑

i

∆iX1∆iJ2I(∆iX)2≤r(h) =∑

i

∆iX1∆iJ2I|∆iX|≤√

r(h),|∆iJ2|≤2√

r(h)+

+∑

i


r(h),|∆iJ2|>2√

r(h) .

If |∆iX| ≤√

r(h) and |∆iJ2| ≤ 2√

r(h) then

|∆iJ1| − |∆iX0 + ∆iJ2| < |∆iX| ≤√

r(h),

and ∆iN1 < |∆iJ1| <

√r(h) +

√h ln 1

h supi

|∫ ti

ti−1asds+

∫ titi−1

σsdWs|√h ln 1

h

+ |∆iJ2|, which

tends to zero with order√

r(h), uniformly in i. Therefore on |∆iX| ≤√

r(h), |∆iJ2|≤ 2

√r(h) we have a.s. ∀i lim∆iN

1 = 0, and in particular a.s. for small h,∀i, ∆iN

1 = 0. However on |∆iX| ≤√

r(h), ∆iN1 = 0, |∆iJ2| ≤ 2

√r(h) =


|∆iX0 + ∆iJ2| ≤√

r(h),∆iN1 = 0, |∆iJ2| ≤ 2

√r(h) we have also |∆iX0| −

|∆iJ2| ≤ |∆iX0 + ∆iJ2| ≤√

r(h) and so |∆iX0| ≤ 3√

r(h). Thus

P lim∑

i


r(h),|∆iJ2|≤2√

r(h) =

P lim∑

i

∆iX0I|∆iX0|≤3√

r(h),∆iN1=0∆iJ2I|∆iJ2|≤2√

r(h) :

as before a.s. for small h, ∀i, I|∆iX0|≤3√

r(h) = 1, thus we have

P lim∑

i

∆iX0I∆iN1=0∆iJ2I|∆iJ2|≤2√

r(h) =

P lim∑

i

∆iX0∆iJ2I|∆iJ2|≤2√

r(h) − P lim∑

i

∆iX0I∆iN1 6=0∆iJ2I|∆iJ2|≤2√

r(h).

(16)The second sum contains only at most N1

T terms and∑

i

|∆iX0|I∆iN1 6=0|∆iJ2|I|∆iJ2|≤2√

r(h) ≤

supi|∆iX0|

∑

i

I∆iN1 6=0|∆iJ2|I|∆iJ2|≤2√

r(h) :

the first factor tends to zero a.s. since it is bounded by√

h ln 1h ; the second factor

tends to zero in probability since, by the independence of N1 on J2,

E[∑

i

I∆iN1 6=0|∆iJ2|I|∆iJ2|≤2√

r(h)] = E[N1T ]E[|∆iJ2|I|∆iJ2|≤2

√r(h)]

≤ E[N1T ]

√E[|∆iJ2|2I|∆iJ2|≤2

√r(h)] = O(

√h).

While, by remark 4.2, the first Plim in (16) coincides with

P lim[X0, Y(h)]T = 0, (17)

since (Ikeda and Watanabe, 1981, p.77) µ and W turn out to be stochastically in-dependent and by assumption also ν is independent on W , thus (Cont and Tankov,2004, p. 269) [W,Y (h)] ≡ 0, so that [X0, Y

(h)]T = [σ.W, Y (h)] =∫ T0 σsd[W,Y (h)]s ≡

0.

In order to deal with∑

i


r(h),|∆iJ2|>2√

r(h) ,

let us remark that on |∆iX| ≤√

r(h), |∆iJ2| > 2√

r(h) we have 2√

r(h) −|∆iX1| < |∆iJ2| − |∆iX1| ≤ |∆iX| ≤

√r(h) and then |∆iX1| >

√r(h), and so

|∆iJ1|+ |∆iX0| > |∆iJ1 + ∆iX0| >√

r(h) ⇒


or |∆iJ1| >√

r(h)2

either |∆iX0| >√

r(h)2

. (18)

However for small h uniformly in i we have I|∆iX0|>

√r(h)

2

= 0, while on |∆iJ1| >√

r(h)

2 we have that ∆iN1 ≥ 1, and thus

P lim∑

i

|∆iX1∆iJ2|I|∆iX|≤√

r(h), |∆iJ2|>2√

r(h) ≤

P lim∑

i

|∆iX1∆iJ2|I|∆iJ2|>2√

r(h), ∆iN1 6=0 = 0,

sinceP (

∑

i

|∆iX1∆iJ2|I|∆iJ2|>2√

r(h), ∆iN1 6=0 6= 0) ≤

nP|∆iJ2| > 2√

r(h)P∆iN1 6= 0 ≤ n

E[|∆iJ2|2]4r(h)

O(h) = nhσ2(1)4r(h)

O(h) → 0.

Finally let us show that the fourth term of (12) tends to zero in probability.First we show that on (∆iX)2 ≤ r(h) it is negligible the weight of (∆iJ2)2 >4r(h).In fact, like as after (18), on (∆iX)2 ≤ r(h), (∆iJ2)2 > 4r(h) we have, for smallh, uniformly in i, that ∆iN

1 > 0 and thus again

P∑

i

(∆iJ2)2I|∆iX|≤√

r(h), |∆iJ2|>2√

r(h) 6= 0 ≤

P

(∪i|∆iJ2| > 2

√r(h), ∆iN

1 6= 0)→ 0.

We then can conclude that

P lim∑

i

(∆iJ2)2I|∆iX|≤√

r(h) =

P lim∑

i

(∆iJ2)2I|∆iX|≤√

r(h), |∆iJ2|≤2√

r(h) ≤

P lim∑

i

(∆iJ2)2I∑

s∈]ti−1,ti](∆J2,s)2≤9r(h) = P lim

∑

i

(∆iY(h))2, (19)

having used remark 4.2. And thus we have

P lim [Y (h)]T = P lim∫ T

0

∫

|x|<1∧3√

r(h)x2µ(ds, dx) = 0,

since last term has expectation σ2(1 ∧ 3√

r(h) ) →h 0. •


Remark 4.2 A.s., for small h, uniformly in i, on (∆iJ2)2 ≤ 4r(h) we have that

(∆J2,s)2 ≤ 9r(h) ∀s ∈]ti−1, ti] and ν ]3√

r(h), 1] = 0.

Thus ∆iJ2I(∆iJ2)2≤4r(h) are in fact the increments of a ν-compensated pure jump

Levy process having jumps |∆J2,s| bounded both by 1 and by 3√

r(h). That is∆iJ2I(∆iJ2)2≤4r(h) are the increments of the process Y (h) given by

Y(h)t

.=∫ t

0

∫

|x|<1∧3√

r(h)x[µ(ds, dx)− dsν(dx)].

In fact (Metivier, 1982, theorem 25.1) for any semimartingale Z we have notonly that

n∑

i=1

(∆iZ)2 →P [Z]T

but, defined Πn(t) the partition of [0, t] induced by 0, t1, ...tn = T and Rn(t, Z) .=∑ti∈Πn(t)(∆iZ)2, we also have that there exists a subsequence nk for which

a.s. Rnk(·, Z) → [Z]. uniformly in t ∈ [0, T ]. (20)

For Z ≡ J2 we have that the quadratic variation [Z]t is precisely the sum ofthe squared sizes of the jumps occurred until t (Cont and Tankov, 2004, p.266):[Z]t =

∑s≤t(∆J2,s)2. This allows us to say that a.s., defined fk(t) = Rnk(t, J2) −∑

s≤t(∆J2,s)2,

supi|(∆iJ2)2 −

∑

s∈]ti−1,ti]

(∆J2,s)2| = supi|fk(ti)− fk(ti−1)| ≤

2 supt∈[0,T ]

|fk(t)| →k 0.

Therefore, given arbitrary δ > 0, for small h we have

supi|(∆iJ2)2 −

∑

s∈]ti−1,ti]

(∆J2,s)2| < δ,

and thus

∀i |∑

s∈]ti−1,ti]

(∆J2,s)2| − |(∆iJ2)2| < |(∆iJ2)2 −∑

s∈]ti−1,ti]

(∆J2,s)2| < δ

so that for all i on (∆iJ2)2 ≤ 4r(h)∑

s∈]ti−1,ti]

(∆J2,s)2 < δ + (∆iJ2)2 ≤ δ + 4r(h).


In particular we cannot have, on (∆iJ2)2 ≤ 4r(h), that e.g.∑

s∈]ti−1,ti](∆J2,s)2

> 9r(h). Thus, for small h, for all i we have that on (∆iJ2)2 ≤ 4r(h) it holds∑

s∈]ti−1,ti]

(∆J2,s)2 ≤ 9r(h).

However if∑

s∈]ti−1,ti](∆J2,s)2 ≤ 9r(h) then each term (∆J2,s)2 with s ∈]ti−1, ti] willbe dominated by 9r(h).

Moreover, since E[∫ T0

∫IR |x|2I3√r(h)<|x|≤1dtν(dx)] < ∞, then (Cont and Tankov,

p.262) the process(∫ t

0

∫3√

r(h)<|x|≤1|x|dJ2

)

tis a square integrable martingale, so in

particular

E

[∫ T

0

∫

3√

r(h)<|x|≤1|x|(µ(dt, dx)− dtν(dx))

]= 0.

We can separate the two terms since µ has a.s. a finite number of jumps of sizebigger than 3

√r(h), thus we have

E

[∫ T

0

∫

3√

r(h)<|x|≤1|x|µ(dt, dx)−

∫ T

0

∫

3√

r(h)<|x|≤1|x|dtν(dx)

]= 0.

However the first term is zero since µ has no jumps of size bigger than 3√

r(h), thus

3T√

r(h) ν ]3√

r(h), 1] ≤ TE

[∫

3√

r(h)<|x|≤1|x|ν(dx)

]= 0,

so ν ]3√

r(h), 1] = 0. •

Remarks.

Not equally spaced observations. Everything is still valid if we have not con-stant steps ∆ti. In fact setting, like as in the remark of the previous section,h

.= maxi ∆ti, for the term E[I(∆iJ2)2>r(h)], we often meet from (15) on, we still

have P(∆iJ2)2 > r(h) ≤ E[(∆iJ2)2]r(h) = ∆tiσ

2(1)r(h) ≤ c h

r(h) ; and (17) and (20) still hold.

Since (20) holds true for any semimartingale, we suspect that this theorem canhold for any semimartingale X, since even for it we can use a pathwise decomposi-tion analogous to (11).

We are not able, for the moment, to verify the asymptotic normality of our esti-mator of

∫ T0 σ2

sds in the case of a model X with an infinite jump activity component.Anyway, based on some simulations and some precious suggestions by Jean Jacod,we suspect that: the asymptotic normality holds when the jump component has amoderate jump activity, while it does not hold if such activity is too much wild.


5 Simulations

We present the following results showing the performance of our integrated volatilityestimator. We implement the threshold estimator within three different simulatedmodels: a jump diffusion process with jump part given by a compound Poissonprocess with Gaussian sizes of jump; an analogous model having however stochasticvolatility correlated with the Brownian motion leading the dynamics of X; and ajump diffusion model having an infinite activity (finite variation) Variance Gammajump part.

MODEL 1. Let us begin with the case of a jump diffusion process with finiteactivity compound Poisson jump part. We generated N = 5000 trajectories of aprocess of kind

dXt = σdWt +NT∑

i=1

Zi

with Zi i.i.d. with law N(0, η2), where η = 0.6, λ = 5 and σ = 0.3, like as in Aıt-Sahalia (2003). To generate each path we discretized EDS (1) and took n = 1000equally spaced observations Xti by step h = 1

n so that T = 1. We have chosenr(h) = h0.9. Figure 1 shows the histogram of the 5000 values assumed by theestimator YT =

∑ni=1(∆iX)2I(∆iX)2≤r(h): one value for each simulated path.

[FIG.1]

Given L = 22 and step = 0.001, each point in figure 1 represents the number oftimes, among the 5000, the integrated volatility estimator YT assumes values within[m + (K − L)× step, m + (K − L + 1)× step[, divided by the step. m is the empi-rical mean realized by the 5000 random variables YT , K ranges from 1 to 2L. Thepoints give the empirical density of the distribution of YT . The continuous line isthe theoretical density of the Gaussian law N(σ2T, 2hσ4T ).

MODEL 2. Let us now consider a process with jump part given by a finite activ-ity compound Poisson process and stochastic volatility correlated to the Brownianmotion leading X. We generated N = 5000 trajectories of a process of kind

Xt = ln(St)

wheredSt

St−= µdt + σdW

(1)t + dJt Jt =

Nt∑

i=1

Zi, Zi ∼ N(mG, ν2),

so thatdXt = (µ− σ2/2)dt + σdW

(1)t + ln(1 + ∆Jt),

and where

σt = eHt , dHt = −k(Ht − H)dt + ηdW(2)t , d < W (1), W (2) >t= ρdt.


We chose µ = 0 and a negative correlation coefficient ρ = −0.01; then we havetaken H0 ≡ ln(0.3), k = 1, H = ln(0.25), η = 0.01 so to ensure that a path of σwithin [0, T ] varies most between 0.4 and 0.2. Moreover mG = 0.001, ν =

√0.02 give

relative amplitudes of the jumps of S most between 0.01 and 0.20. Finally we againhave taken n = 1000 equally spaced observations Xti by step h = 1

n and r(h) = h0.9.Figure 2 shows empirical density (histogram) of the distribution of

ˆIV T −∫ T0 σ2

sds√23


,

L = 100, step = 0.1; the continuous line being the theoretical density of the lawN(0, 1).

[FIG.2]

MODEL 3. Figure 3 shows the histogram obtained in the case of a VarianceGamma jump component. The VG process is a purely jumping process with infiniteactivity and finite variation. We add to it a diffusion component:

Xt = σBt + cGt + ηWGt ,

where the VG component is cGt + ηWGt . The subordinator G is a Gamma processhaving V ar(G1) = b, B and W are independent Brownian motions; we have chosenn = 6000 and h = 1/n. n = 6000 corresponds to consider, when we apply ourestimators to market data, 24 observations each day the market is open, which isstill plausible. b = 0.23, c = −0.2 and η = 0.2 are chosen like as in Madan (2001);σ = 0.2 is chosen so that V ar(X1) = η2 + c2b + σ2 = .0892 is comparable with theV ar(X1) we obtained for model 1. Finally r(h) = h0.99, N = 1000, L = 22 andstep = 0.0002.

[FIG.3]

This figure seems to show that at least for some kind of infinite activity jumpcomponents our estimator of the integrated volatility is asymptotically Gaussian.

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