testing for jumps and estimating their degree of activity...

Testing for Jumps and Estimating their Degree ofActivity in High Frequency Financial Data

Yacine A��t-Sahalia Jean JacodPrinceton University Universit�e de Paris VI

1

1 INTRODUCTION

1. Introduction

� Di�erent types of jumps

{ Large jumps, which are rather infrequent, are easy to pick out.

{ But visual inspection of most time series in �nance does not provide

clear evidence for either the presence or the absence of smaller,

more frequent, jumps.

2

1 INTRODUCTION

� Models with and without jumps do have quite di�erent properties,both mathematical and �nancial:

{ Model calibration

{ Volatility estimation

{ Market (in)completeness

{ Option pricing and hedging

{ Risk management

{ Portfolio choice

3

1 INTRODUCTION

� Detecting jumps: other approaches

{ Tests: A��t-Sahalia (2002), Carr and Wu (2003), BNS (2004), ABD

(2004), Huang and Tauchen (2006), Lee and Mykland (2005),

Jiang and Oomen (2006)

{ Separating jumps from volatility: Mancini (2001, 2004), A��t-Sahalia

(2004), A��t-Sahalia and Jacod (2005), Woerner (2006)

4

1 INTRODUCTION

� This paper: we propose a very simple family of test statistics for jumpswhich converge as �n ! 0 :

{ To 1 if there are jumps

{ To 2 if there are no jumps.

� We provide a distribution theory (hence a test) for the null hypothesiswhere no jumps are present, but also one for the null where jumps are

present.

5

1 INTRODUCTION

� This works as soon as the process X is an Ito semimartingale

{ This is a much weaker condition than what is usually assumed

(compound Poisson processes, or jump-di�usions)

{ The limit depends neither on the law of the process nor on the

coe�cients of the (possibly very complicated) SDE

{ So the test does not require any estimation of these coe�cients.

6

2 THE SETUP

2. The Setup

� The structural assumption is that X is an Ito semimartingale on some

�ltered space (;F ; (Ft)t�0;P):

Xt = X0 +At +Mt

= X0 +ACt +A

Jt +M

Ct +M

Jt

where At is a �nite variation and predictable mean component and

Mt is a local martingale

� Each decomposable into a continuous and a pure jump part.

7

2 THE SETUP

� The drift, volatility and jump measure are themselves possibly sto-chastic and can possibly jump.

� We assume that the continuous part of X is never degenerate, i.e., we

haveR t0 j�sjds > 0 a.s. for all t > 0.

8

3 THE TESTING PROBLEM

3. The Testing Problem

� X is discretely observed at times i�n for all i = 0; 1; � � � ; n with

n�n = T:

� When the jump measure is �nite, there is a positive probability thatthe path X(!) has no jump on [0; T ], although the model itself may

allow for jumps: this is the peso problem.

9

3.1 Various Measures of the Variability of X 3 THE TESTING PROBLEM

3.1. Various Measures of the Variability of X

� Here are processes which measure some kind of variability of X and

depend on the whole (unobserved) path of X:

A(p)t =Z t0j�sjpds; B(p)t =

Xs�t

j�Xsjp

where p > 0 and �Xs = Xs �Xs� are the jumps of X.

� The quadratic variation of the process is [X;X] = A(2) +B(2).

10


� The problem boils down to deciding whether whether B(p)T > 0 for

our particular observed path with any given p:

� Let the observed discrete increments of X (not necessarily due to

jumps) be

�niX = Xi�n �X(i�1)�nand for p > 0 de�ne the estimator

bB(p;�n)t = [t=�n]Xi=1

j�niXjp

11


� For r > 0; let

mr = E(jU jr) = ��1=22r=2 ��r + 1

2

�denote the rth absolute moment of a variable U � N(0; 1).

� We have the following convergences in probability, locally uniform in

t: 8>>>>>>>>><>>>>>>>>>:

p > 2; all X ) bB(p;�n)t P�! B(p)t

p = 2; all X ) bB(p;�n)t P�! A(2)t +B(2)t

p < 2; all X ) �1�p=2nmp

bB(p;�n)t P�! A(p)t

all p; X continuous ) �1�p=2nmp

bB(p;�n)t P�! A(p)t:

12

3.2 The Basic Idea 3 THE TESTING PROBLEM

3.2. The Basic Idea

�

8>><>>:p > 2; all X ) bB(p;�n)t P�! B(p)t

all p; X continuous ) �1�p=2nmp

bB(p;�n)t P�! A(p)t:

� We see that when p > 2 the limit B(p)t of bB(p;�n)t does not dependon �n, and B(p)t > 0 is strictly positive if X has jumps between 0

and t.

� On the other hand when X is continuous on [0; t]; then the limit

is B(p)t = 0 but, after a normalization which does depend on �n,bB(p;�n)t converges again to a limit A(p)t not depending on �n.13

3.2 The Basic Idea 3 THE TESTING PROBLEM

� These considerations lead us to compare bB(p;�n) on two di�erent�n�scales.

� Speci�cally, for an integer k, consider:

bS(p; k;�n)t = bB(p; k�n)tbB(p;�n)t :

� Theorem: For any t > 0 the variables bS(p; k;�n)t converge in proba-bility to (

1 if X jumps

kp=2�1 if X is continuous

14

4 TESTING FOR JUMPS

4. Testing for Jumps

� The previous theorem provides the �rst step towards constructing a

test for the presence or absence of jumps.

� But to construct actual tests, we need: rates of convergence andasymptotic variances.

� That are applicable under both nulls of jumps and no jumps.

� Consistent estimators of the variances.

15

4.1 CLT for Standardized Statistics 4 TESTING FOR JUMPS

4.1. CLT for Standardized Statistics

Theorem:

1. Let p > 3. The variables ( bV jn )�1=2 � bS(p; k;�n)t � 1� converge stablyin law, in restriction to the set

jt to a variable which, conditionally

on F , is centered with variance 1, and which is N(0; 1) if in additionthe processes � and X have no common jumps.

2. If X is continuous, then for p � 2

( bV cn)�1=2 � bS(p; k;�n)t � kp=2�1�! N(0; 1)

stably in law, conditionally on F .16

4.2 Practical Considerations 4 TESTING FOR JUMPS

4.2. Practical Considerations

� Since we must have p > 3, a rather natural choice seems to be p = 4.

� We see that the variances are increasing with k, so it is probably wiseto take k = 2 (although when k > 2 we have to separate the two

points 1 and k, which are further apart than 1 and 2).

17

5 SIMULATION RESULTS

5. Simulation Results

� We calibrate the values to be realistic for a liquid stock trading on theNYSE.

� We use an observation length of T = 1 day, consisting of 6:5 hours oftrading, that is 23; 400 seconds.

18


Simulations: Null of No Jumps, k = 2 and 3

No Jumps: Distribution of the Statistic

-4 -2 -1 0 1 2 4

0.1

0.2

0.3

0.4

No Jumps: k = 2, Standardized

-4 -2 -1 0 1 2 4

0.1

0.2

0.3

0.4

No Jumps: k = 3, Standardized

1.9 2 2.1

2

4

6

8

No Jumps: k = 2, Non−Standardized

2.8 3 3.2 3.4

1

2

3

4

No Jumps: k = 3, Non−Standardized

19


Simulations: Poisson Jumps

Poisson Jumps: Distribution of the Statistic

-4 -2 0 2 4

0.1

0.2

0.3

0.4

Poisson: 1 Jump per Day, Standardized

-4 -2 0 2 4

0.1

0.2

0.3

0.4

Poisson: 10 Jumps per Day, Standardized

0.95 1 1.05

5

10

15

20

25

30

35Poisson: 1 Jump per Day, Non−Standardized

0.95 1 1.05

5

10

15

20

25

30Poisson: 10 Jumps per Day, Non−Standardized

20


Simulations: Cauchy Jumps

Cauchy Jumps: Distribution of the Statistic

-4 -2 0 2 4

0.1

0.2

0.3

0.4

Cauchy Jumps θ = 10, Standardized

-4 -2 0 2 4

0.1

0.2

0.3

0.4

Cauchy Jumps θ = 50, Standardized

0.9 0.95 1 1.05 1.1

10

20

30

40

50

Cauchy Jumps θ = 10, Non−Standardized

0.975 1 1.025

50

100

150

200

Cauchy Jumps θ = 50, Non−Standardized

21


Simulations: Tiny or No Jumps

Tiny Jumps or No Jumps: Distribution of the Statistic

0.8 1 1.2 1.5 1.8 2 2.2

2

4

6

8

10

12

Poisson Jumps: 1 Jump per Day

Non−Standardized

0.8 1 1.2 1.5 1.8 2 2.2

1

2

3

4

5

6

Cauchy Jumps: θ = 1

Non−Standardized

22

6 REAL DATA ANALYSIS

6. Real Data Analysis

� In real data, observations of the process X are blurred by market

microstructure noise, which messes things up at very high frequency.

� Assume that each observation is a�ected by an additive noise, that isinstead of Xi�n we really observe Yi�n = Xi�n + "i, and the "i are

i.i.d. with E("2i ) and E("4i ) �nite.

� We show that, in the presence of noise, the limit of our test statisticsbS(4; k;�n)t becomes as �n ! 0:

bS(4; k;�n)t P�! 1

k

23

6 REAL DATA ANALYSIS

Real Data Analysis: 30 DJIA Stocks, All 2005 Trading Days

Empirical Distribution of the Test Statistic: DJIA30 All 2005 Trading Days

0.5 1 1.5 2 2.5 3

100

200

300

∆ = 15 seconds

0.5 1 1.5 2 2.5 3

50

100

150

200

250

∆ = 30 seconds

0.5 1 1.5 2 2.5 3

200

400

600

800∆ = 5 seconds

0.5 1 1.5 2 2.5 3

100

200

300

400

500∆ = 10 seconds

24

7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

7. Estimating the Degree of Jump Activity

� For modelling purposes one would like to infer the characteristics ofX; that is, its drift, its volatility and its L�evy jump measure, from the

observations.

{ When the time interval �n goes to 0; it is well known that one

can infer consistently the volatility, under very weak assumptions.

{ But such consistent inference is impossible for the drift or the L�evy

measure, if the overall time of observation [0; T ] is kept �xed.

{ In fact, even in the unrealistic case where the whole path of X is

observed over a �xed [0; T ], one can infer neither the drift nor the

L�evy measure.

25

7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

� One can however hope to be able to characterize the behavior of theL�evy measure near 0:

{ First whether it does not explode near 0, meaning that the number

of jumps is �nite;

{ Second, when the number of jumps is in�nite, we would like to be

able to say something about the concentration of small jumps.

26

7.1 De�ning an Index of Jump Activity 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

7.1. De�ning an Index of Jump Activity

� Recall our de�nition of the process B(p)t corresponding to the semi-martingale X :

B(p)t =Xs�t

j�Xsjp

where �Xs = Xs �Xs� is the size of the jump at time s, if any.

� De�ne

It = fp � 0 : B(p)t <1g:

� Necessarily, the (random) set It is of the form [�t;1) or (�t;1) forsome �t � 2, and 2 2 It always, and t 7! �t is non-decreasing.

27


� We call �T (!) the jump activity index for the path t 7! Xt(!) at time

T .

� We de�ne this index in analogy with the special case where X is a

L�evy process:

{ Then �T (!) = � does not depend on (!; T ), and it is also the

in�mum of all r � 0 such thatRfjxj�1g jxjrF (dx) < 1, where F

is the L�evy measure

{ This property shows that, for a L�evy process, the jump activity

index coincides with the Blumenthal-Getoor index of the process.

{ In the further special case where X is a stable process, then � is

also the stable index of the process.

28


� When X is a L�evy process, the index � is only a partial element of the

whole L�evy measure F

� But this is the most informative knowledge one can draw about F

from the observation of the path t 7! Xt for all t � T; T �nite.

� Things are very di�erent when T ! 1, though, since observing Xover [0;1) completely speci�es F .

� However, � captures an essential qualitative feature of F , which is itslevel of activity: when � increases, the (small) jumps tend to become

more and more frequent.

29

7.2 The Brownian Motion... 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

7.2. The Brownian Motion...

� Recall that the semimartingale X is only observed at times i�n, over

[0; T ].

� The problem is made more challenging by the presence in X of a

continuous, or Brownian, martingale part:

{ � characterizes the behavior of Fnear 0:

{ Hence it is natural to expect that the small increments of the

process are going to be the ones that are most informative about

�:

30

7.2 The Brownian Motion... 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

{ But that is where the contribution from the continuous martingale

part of the process is inexorably mixed with the contribution from

the small jumps.

{ We need to see through the continuous part of the semimartingale

in order to say something about the number and concentration of

small jumps.

31

7.3 Counting Increments 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

7.3. Counting Increments

� For �xed $ > 0 and � > 0, we consider the functionals

U($;�;�n)t =[t=�n]Xi=1

1fj�ni Xj>��$n g:

� U($;�;�n)t simply counts the number of increments whose magni-tude is greater than ��$n .

� In all cases below, we will set $ < 1=2:

� This way, we are retaining only those increments of X that are notpredominantly made of contributions from the continuous part, whichare Op(�

1=2n ):

32

7.3 Counting Increments 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

� A more general class of estimators can be constructed from the trun-

cated power variation functionals

Ur($;�;�n)t =[t=�n]Xi=1

j�niXjr1fj�ni Xj>��$n g:

� Here we focus on U = U0.

� While one could imagine looking at other (small) values of r; theredoes not appear to be immediate bene�ts from doing so in the present

problem.

33

7.4 Behavior of the L�evy Measure 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

7.4. Behavior of the L�evy Measure

� Our regularity assumption is that for some � 2 (0; 2) and �0 2 [0; �=2),we have for all (!; t):

Ft = F0t + F

00t + F

000t ;

where F 0t is locally of the ��stable form

F 0t(dx) =1

jxj1+�

a(+)t 1

f0<x�z(+)t g+ a

(�)t 1

f�z(�)t �x<0g

!dx;

for some predictable non-negative processes a(+)t ; a

(�)t ; z

(+)t and z

(�)t .

� Any additional components F 00 and F 000 in the L�evy measure beyondthe most active part F 0 must have jump activity indices (which areat most �0 and �=2; respectively) that are su�ciently apart from theleading jump activity index �:

34

7.4 Behavior of the L�evy Measure 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

� For example, any process of the following formwill satisfy the assump-tion

dXt = btdt+ �tdWt + �t�dYt + �0t�dY0t

where:

{ � and �0 are cadlag adapted processes

{ Y is ��stable

{ Y 0 is any L�evy process with jump activity index less that �.

35

7.5 Estimators of the Jump Activity Index 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

7.5. Estimators of the Jump Activity Index

� The key property of the functionals U($;�;�n) is

�$�n U($;�;�n)tP�!

�At

�

where �At =1�

R t0

�a(+)s + a

(�)s

�ds:

� This leads us to propose two di�erent estimators, at each stage n.

36


� For the �rst one, �x 0 < � < �0 and de�ne

b�n(t;$; �; �0) = log(U($;�;�n)t=U($;�0;�n)t)

log(�0=�);

� b�n is constructed from a suitably scaled ratio of two Us evaluated

on the same time scale �n but at two levels of truncation of the

increments, � and �0.

� Based on �$�n U($;�;�n)tP�! �At

��, this will be consistent.

37


� Our second estimator is

b�0n(t;$; �) = log(U($;�;�n)t=U($;�; 2�n)t)

$ log 2:

� b�0n is constructed from a suitably scaled ratio of two Us evaluated at

the same level of truncation �; but on two time scales, �n and 2�n.

� Based on �$�n U($;�;�n)tP�! �At

��, this will be consistent.

� One could look at a third estimator obtained from two Us evaluated attwo di�erent rates of truncation $ and $0; but there does not appearto be immediate bene�ts from doing so.

38

7.6 Asymptotic Distribution of the Estimators 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

7.6. Asymptotic Distribution of the Estimators

Theorem: Under regularity assumptions, both variables

log(�0=�)�1

U($;�0;�n)t� 1U($;�;�n)t

�1=2 � b�n(t;$; �; �0)� ��

$ log 2�1

U($;�;2�n)t� 1U($;�;�n)t

�1=2 � b�0n(t;$; �)� ��

converge stably in law, in restriction to the set f �At > 0g, to a standardnormal variable N (0; 1) independent of X.

39

7.6 Asymptotic Distribution of the Estimators 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

� The quali�er \in restriction to the set f �At > 0g" is essential in thisstatement.

{ On the (random) set f �At > 0g, the jump activity index is �.

{ On the complement set f �At = 0g; anything can happen: on thatset, the number � has no meaning as a jump activity index for X

on [0; T ]:

� These results are model-free, because the drift and the volatility processesare totally unspeci�ed apart from the regularity assumption on the L�evy

measures Ft.

40

7.7 Simulation Results 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

7.7. Simulation Results

� The data generating process is dXt=X0 = �tdWt + dYt

� Y is a pure jump process, ��stable or Compound Poisson (� = 0).

� Stochastic volatility �t = v1=2t

dvt = �(� � vt)dt+ v1=2t dBt + dJt;

� Leverage e�ect: E[dWtdBt] = �dt; � < 0

� With jumps in volatility: J is a compound Poisson process with uniformjumps.

41


Simulations: � = 1:25 and � = 1

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

b β = 1.25

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

b β = 1

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

200

bβ = 0.75

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

200

250

β = 0.5

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

bβ = 0.25

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

200

400

600

800

1000

b β = 0

Estimator Based on Two Truncation Levels

42


Simulations: � = 0:75 and � = 0:5

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

b β = 1.25

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

b β = 1

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

200

bβ = 0.75

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

200

250

β = 0.5

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

bβ = 0.25

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

200

400

600

800

1000

b β = 0


43


Simulations: � = 0:25 and � = 0

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

b β = 1.25

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

b β = 1

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

200

bβ = 0.75

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

200

250

β = 0.5

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

50

100

150

bβ = 0.25

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

200

400

600

800

1000

b β = 0


44

7.8 Empirical Results: Intel & Microsoft 2005 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

7.8. Empirical Results: Intel & Microsoft 2005

INTC�n 2 sec 5 sec 15 sec� 4 5 6 4 5 6 4 5 6

Qtr 1 1:70 1:69 1:69 1:86 1:87 1:76 1:61 1:36 1:46Qtr 2 1:06 1:06 1:05 1:23 1:13 1:09 1:09 1:13 1:14Qtr 3 1:15 1:20 1:40 1:20 1:21 1:18 1:27 1:34 1:45Qtr 4 1:32 1:51 1:59 1:54 1:35 1:42 1:77 1:72 1:42All Year 1:30 1:35 1:40 1:44 1:36 1:32 1:40 1:36 1:32

45

7.8 Empirical Results: Intel & Microsoft 2005 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY

MSFT�n 2 sec 5 sec 15 sec� 4 5 6 4 5 6 4 5 6

Qtr 1 1:72 1:92 1:94 1:74 1:86 1:86 1:75 1:89 2:00Qtr 2 1:59 1:60 1:43 1:60 1:48 1:56 1:47 1:17 1:27Qtr 3 1:50 1:60 1:63 1:52 1:54 1:63 1:66 1:81 1:97Qtr 4 1:64 1:79 1:72 1:82 1:66 1:65 1:71 1:37 1:24All Year 1:60 1:71 1:66 1:66 1:62 1:66 1:65 1:54 1:68

46

8 CONCLUSIONS

8. Conclusions

� Jumps are prevalent in these data

� Especially if one accounts for small, in�nite activity, jumps.

47

testing for jumps and estimating their degree of activity...

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