Mean Exit Time of Equity Assets

Mean Exit Time of Equity AssetsMean Exit Time of Equity AssetsSalvatore MiccichèSalvatore Miccichè

http://lagash.dft.unipa.ithttp://lagash.dft.unipa.it

Observatory of Complex Observatory of Complex SystemsSystems

Dipartimento di Fisica e Tecnologie RelativeDipartimento di Fisica e Tecnologie Relative Università degli Studi di Palermo Università degli Studi di Palermo

Progetto Strategico - Incontro di progetto III anno - Roma 20 Giugno Progetto Strategico - Incontro di progetto III anno - Roma 20 Giugno 20072007

Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets

Observatory of Complex SystemsObservatory of Complex Systems

S. MiccichèS. Miccichè

F. LilloF. LilloR. N. MantegnaR. N. Mantegna

F. TerzoM. TumminelloG. Vaglica

C. CoronnelloC. Coronnello

EconophysicsEconophysics BioinformaticsBioinformatics Stochastic ProcessesStochastic Processes

M. Spanò

J. MasoliverJ. MasoliverM. MonteroM. MonteroJ. PerellóJ. Perelló

BarcellonaBarcellona

S

Aim of the ResearchAim of the Research

The long-term aim is to use CTRW (Markovian The long-term aim is to use CTRW (Markovian process) as a process) as a stochastic processstochastic process able to able to

provide a sound description of extreme times provide a sound description of extreme times in financial datain financial data

ExplorativeExplorative analysis of the capability of CTRW analysis of the capability of CTRW to explain some empirical features of tick-by-tick to explain some empirical features of tick-by-tick data, data, role of tick-by-tick volatilityrole of tick-by-tick volatility..

METMETLL22 andand data collapsedata collapse

Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets S

The set of investigated stocksThe set of investigated stocksWe consider: Mean Exit Times - the 20 most capitalized stocks in 1995-1998 at NYSE

the 100 most capitalized stocks in 1995-2003 at NYSE

We hereafter consider high-frequency (intradayintraday) data: tick-by-tick datatick-by-tick data

TTrades AAnd QQuotes (TAQTAQ) database maintained by NYSE (1995-20031995-2003)


Mean Exit Time (MET)Mean Exit Time (MET)

• The “extreme events” we consider will be related with the first crossing of any of the two barriers.

• The Mean Exit Time (MET) is simply the expected value of the time interval

)( )(

)(

0],[0],[

0],[

xtExT

xt

baba

ba

Financial InterestFinancial Interest: the MET provides a timescale for

market movements.


2L2L

S

For the Wiener process:

An example: a Wiener stochastic processAn example: a Wiener stochastic process

DLLaT ba 4/)2/( 2],[

DD is the diffusion coefficient


xtxttx

ttDx

)()(

)(

the MET is:

(t)(t) is a -correlated gaussian distributed noise

S

The Continuous Time Random Walk (CTRW) is a natural extension of Random Walks (Ornstein-Uhlembeck, Wiener, ... ).

A (one dimensional) random walk is a random process in which, at every time step, you can move in a grid either up or down, with different probabilities.

The key point is that in a CTRW not only the size of the movements but also the time lags between themtime lags between them are random.

Stochastic Process: Stochastic Process: CTRWCTRW


• CTRW first CTRW first developed by developed by Montroll and Montroll and Weiss (1965)Weiss (1965)

• Microstructure of Microstructure of Random ProcessRandom Process

S

The relevant variables: I - price changesThe relevant variables: I - price changes

• Log-prices:

• Log-Returns:Log-Returns:

• Return changes conform a stationary random process with a (marginal) probability density function:

)](log[)( nn tStX

)()()( 1 nnn tXtXtX

))({)( dxxtXxPdxxh n


• The process only may change at “random” times

remaining constant between these jumps. • The waiting timeswaiting times

also are characterized by a (marginal) probability density function:

,,,,,,, 2101 nttttt

1 nnn tt

}{)( dPd n

The relevant variables: II- waiting timesThe relevant variables: II- waiting timesMean Exit Times of Equity AssetsMean Exit Times of Equity Assets S

The relevant variables: joint pdfThe relevant variables: joint pdf

The system is characterized by the following JOINT probability density function

};)({),( ddxxtXxPdxdx nnn

)( and )( xh are just two marginal density functions:

dxxdxxh ),()( , ),()(0


P(X,t) probability that a particle is at position X at time t(X,t) probability of making a step of length X in the interval [t,t+dt]

)','()','(''),(0

tXPttXXdxdttXPxt

S

A simple modelA simple model

The The uncoupleduncoupled i.i.d. case of CTRW: MET i.i.d. case of CTRW: METMean Exit Times of Equity AssetsMean Exit Times of Equity Assets S

The The uncoupleduncoupled i.i.d. case of CTRW: setup i.i.d. case of CTRW: setup

• If we assume that the system has no memory at all, all the pairs

will be independent and identically distributed (Separability Ansatz).

• The relevant probability density function are simply

),( nnX

)( and )( xh


• The MET for i.i.d. CTRW process fulfils a renewal equation:

ba baba xTxxhExT )()(][)( ],[00],[

The The uncoupleduncoupled i.i.d. case of CTRW: MET i.i.d. case of CTRW: MET

0

)(][ dttE

• If one now assumesassumes that

then one wouldwould observe that

x

Hxh1

)(

J. Masoliver, M. Montero, J. Perelló, Phys. Rev. E 71, 056130 (2005)

]][[ 222nn XEXE

tick-by-tick volatilitytick-by-tick volatility

][

)( 0],[

E

xT bavs

2

Lis a universal curveis a universal curve



In particular, if one assumesassumes that

(three state i.i.d. discrete model)

then one can prove can prove that that

)()(2

1)()( cxcx

QxQxh

c is the basic jump sizeQ is the probability that the price is unchanged

20],[

2

11

1

1

][

)(

QL

QE

xT ba

The quadratic dependance of MET is recoveredThe quadratic dependance of MET is recovered


20],[

2][

)(

L

E

xT ba

S


MET for the 20 stocksMET for the 20 stocksrescaled variablesrescaled variables

No data collapse is observableNo data collapse is observable


20 stocks1995-

1998

S

The The uncoupleduncoupled i.i.d. case of CTRW: i.i.d. case of CTRW: summarysummary

The quadratic dependance of MET is recoveredThe quadratic dependance of MET is recovered

No data collapse is observableNo data collapse is observable

What is the reason why we do not observe data collapse?What is the reason why we do not observe data collapse?

•Is H(u) not universal?Is H(u) not universal?•Is the uncoupled case too simple?Is the uncoupled case too simple?•Is there any role of capitalization ?Is there any role of capitalization ?•Is there any role of tick size ? Is there any role of tick size ? •Is there any role of trading activity ? Is there any role of trading activity ?

Let us go back to the empirical data !Let us go back to the empirical data !


1) Shuffling Experiments1) Shuffling Experiments

Hypothesis 1Hypothesis 1: : h(x) is functionally different for different stocksh(x) is functionally different for different stocks

•We can test this hypothesis by shuffling independently Xn and n.

•This destroys the autocorrelation in both variables and the cross-correlation between them.

•However the distributions h(x) and () are preserved.

A good data collapse is observable:A good data collapse is observable:then h(x) is “the same” for all stocks then h(x) is “the same” for all stocks


20 stocks1995-

1998

S

1) Shuffling Experiments1) Shuffling ExperimentsHypothesis 2Hypothesis 2: : There is a role of the cross-correlations between There is a role of the cross-correlations between returns and jumpsreturns and jumps

Hypothesis 3Hypothesis 3: : There is a role of the autocorrelation of waiting timesThere is a role of the autocorrelation of waiting times

Hypothesis 4Hypothesis 4: : There is a role of the autocorrelation of returnsThere is a role of the autocorrelation of returns

We can test these hypothesis by shuffling

H2H2) returns and waiting times and preserving the crosscorrelations, i.e. the pairs (greengreen)

H3H3) waiting times only (blueblue)H4H4) returns only (magentamagenta)

dashed black=original data

red=H1GE stock


1995-1998

S

Fourier Shuffling ExperimentsFourier Shuffling Experiments

black=black=blue blue neglecting the autocorrelation of waiting times is not importantneglecting the autocorrelation of waiting times is not important

magentamagentablack: black: There is a role of the autocorrelation of returnsThere is a role of the autocorrelation of returns

greengreen==red red neglecting the cross-correlations is not importantneglecting the cross-correlations is not important

Two possible sources of (auto)-Two possible sources of (auto)-correlation in returns:correlation in returns:linearlinear (bid-ask bounce) (bid-ask bounce)

nonlinearnonlinear (volatility) (volatility)


GE stock

S

Fourier Shuffling ExperimentsFourier Shuffling Experiments

dashed black=original data

GE stock

red =phase randomized data of Xn

red=black neglecting the volatility

(nonlinear) correlation is not important

Shuffling that destroys only the nonlinearnonlinear (auto)-correlation properties of a time-series



2) Jump size & Trading Activity2) Jump size & Trading ActivityOn 24/06/1997 the tick size changed from 1/8$1/8$ to 1/16$1/16$ On 29/01/2001 the tick size changed from 1/16$1/16$ to 1/100$1/100$ Therefore we decided to consider a larger set of 100 stocks continuously traded from 1995 to 2003 and considered 3 time periods:

01/01/199524/06/1997

25/06/199728/01/2001

29/01/200131/12/2003

Therefore 3 time periods are also differentdifferent for the trading trading activity !!activity !!

S

2) Jump size & Trading Activity2) Jump size & Trading ActivityMean Exit Times of Equity AssetsMean Exit Times of Equity Assets

Nothing Nothing changes for changes for

the the shufflings !shufflings !

Each point is the mean over 100 stocks

The error bar is the standard deviation

The standard deviation is

smaller in 01-01-0303 than in 95-95-

9797..

BUT

The collapse on collapse on a single curvea single curve is betterbetter in 01-01-0303 than in 95-95-

9797..

i.e.

GE: E[GE: E[]]5.3 s5.3 s =3.3 =3.3

1010-3-3

100 stocks

100 stocks

100 stocks

L/2k

T/E

[]

S

A more A more sophisticated sophisticated

modelmodel

The The uncoupleduncoupled i.i.d. case of CTRW: MET i.i.d. case of CTRW: METMean Exit Times of Equity AssetsMean Exit Times of Equity Assets S

The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: setupsetup

The only important thing is the bid-ask The only important thing is the bid-ask bounce !!!!bounce !!!!

Since this is a short range effect, it is reasonable to assume that we can modify the previous CTRW by

changing from an i.i.d. processfrom an i.i.d. process to ato a one step one step markovian chainmarkovian chain.

}','|;{),( 11 nnnn xXddxxXxPdxdx

};)({),( ddxxtXxPdxdx nnn


We can modify the previous expression for the MET equation in order to include the last-change memory (which is the most relevant information in this case):

M. Montero, J. Perelló, J. Masoliver, F. Lillo, S. Miccichè, and R.N. Mantegna, Phys. Rev. E 72, 056101 (2005)

dxXxTXxxhXEXxTb

a baba )|()|(]|[)|( ],[00000],[

The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: METMET


If we consider a two-state Markov chaintwo-state Markov chain model:

we can obtain a scale-free expression for the symmetrical METscale-free expression for the symmetrical MET in terms of the width L of the interval:

)()()|( cxc

ryccx

c

rycyxh

2],[

],[ 21

1

][

)2/()2/(

c

L

r

r

E

LaTLaT ba

ba



r is the correlation between two consecutive jumps:

1

1

varvar

,cov

nn

nn

XX

XXr

By inspection: 2=c2

)()(2

1)()( cxcx

QxQxh

NEWNEW

extra factor !extra factor !

S



rescale

d T

L/2k

The observed data collapse is improved, The observed data collapse is improved, although it is still not completely satisfactoryalthough it is still not completely satisfactory

MET for the 100 MET for the 100 stocksstocks

rescaled variablesrescaled variablesin the 3 time in the 3 time

periods consideredperiods considered

jump sizejump sizeor

trading trading activityactivity?

S

D

LLaT ba 4

)2/(2

],[


][1

11

1

4)2/(

21

1

][

)2/(

)2/(

2

2

],[

2],[

],[

Err

c

LLaT

c

L

r

r

E

LaT

LaT

ba

ba

ba

In a sense, our results are In a sense, our results are notnot worth all the efforts worth all the efforts done by introducing this more complicated model !!!!done by introducing this more complicated model !!!!


WIENERWIENERCTRWCTRW


However, the model gives an HINT about the However, the model gives an HINT about the “INGREDIENTS” of the “INGREDIENTS” of the diffusion coefficientdiffusion coefficient !!! !!!

S

• The CTRW is a well suited tool for modeling market changes at very low scales (high frequency data) and allows a sound description of extreme times under a very general setting (Markovian process)

• MET properties: • It grows quadratically with the barrier LIt grows quadratically with the barrier L• depends only from the bid-ask bounce rdepends only from the bid-ask bounce r• seems to scale in a similar way for different assets, seems to scale in a similar way for different assets, better when the better when the

thick size is smaller.thick size is smaller.

• The CTRW describes the quadratic dependence and seems to give indications about the data collapse.

• As far as the data collapse in concerned, the CTRW models seem to give the best contribution when the thick sie is larger.

ConclusionsConclusionsMean Exit Times of Equity AssetsMean Exit Times of Equity Assets S

The EndThe End

[email protected]@lagash.dft.unipa.it

Mean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets

Additional: other marketsAdditional: other markets


3) Capitalization3) CapitalizationMean Exit Times of Equity AssetsMean Exit Times of Equity Assets

Fit with a Fit with a power-law power-law function:function:

MET = (C+A MET = (C+A L)L)

The The dependance dependance

from the from the capitalization capitalization

is not so is not so

dramatic !!!dramatic !!!



dis

pers

ion

L/2kThe observed data collapse is improved, The observed data collapse is improved,

although it is still not completely satisfactoryalthough it is still not completely satisfactory

Again, theAgain, the data data collapsecollapse is is betterbetter in in 01-0301-03 than inthan in

95-9795-97



S



Additional: other marketsAdditional: other markets





L/2k

T/E

[]

London Stock London Stock ExchangeExchange

(SET1 - (SET1 -

electronic electronic transactions transactions

only)only)




L/2k

T/E

[]

Milan Stock Milan Stock ExchangeExchange




L/2k

T/E

[]

NYSENYSELSELSE

MIB30MIB30


III momentIII moment

II momentII moment

If the higher If the higher moments exist moments exist

......

It depends on the It depends on the tails of the Survival tails of the Survival

Probability Probability distribution ...distribution ...

LL44

LL66

T/E

[]

T/E

[]

L/2k

L/2k

2) Jump size & Trading Activity2) Jump size & Trading Activity



rescale

d T

L/2k


MET for the 100 MET for the 100 stocksstocks

rescaled variablesrescaled variablesin the 3 time in the 3 time

periods consideredperiods considered





dis

pers

ion

L/2kThe observed data collapse is improved, The observed data collapse is improved,

although it is still not completely satisfactoryalthough it is still not completely satisfactory

Again, theAgain, the data data collapsecollapse is is betterbetter in in 01-0301-03 than inthan in

95-9795-97





Additional: old slidesAdditional: old slides


CTRW: The ideaCTRW: The idea


• CTRW first developed by Montroll and Weiss (1965)

• Microstructure of Random Process

Applications:• Transport in random

media• Random networks• Self-organized

criticality• Earthquake

modeling• Finance!

CTRW: origin and applicationsCTRW: origin and applications


Instrument II: Survival Probability (SP)Instrument II: Survival Probability (SP)

• The Survival Probability (SP) measures the likelihood that, up to time t the process has been never outside the interval [a,b]:

• Financial interestFinancial interest: It may be very useful in risk control. Note, for instance, the case .The SP measures, not only the probability that you do not loose more than a at the end of your investment horizon, like VaR, but in any previous instant.

)( min)( , )( max)(

)(|)(,)(,)(

);(

00

00

00],[

tXtmtXtM

xtXatmbtMbtXaP

xttS

tttttt

ba

b


0 0],[0 0],[0 0],[

0],[0 0 00],[0],[

);(}|{}|{

}|{ }|{)(

duxuSduxutPduxvtdP

xvtdPduxvtvdPxT

babav ba

ba

v

baba

);0(ˆ)( 0],[0],[ xsSxT baba

We can recover the Mean Exit TimeMean Exit Time from the Laplace TransformLaplace Transform of the Survival ProbabilitySurvival Probability:

Therefore:

Because: 00

00],[

)(|)(,)(,)(

|

xtXatmbtMbtXaP

xtttP ba

Instrument III: relation between SP and METInstrument III: relation between SP and METMean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets

The MET and SP for the Wiener process are:

||2

exp11

);(ˆ);(ˆ00],[0],[ xc

D

s

sxsSxsS cc

SP and MET for a Wiener processSP and MET for a Wiener process

DLLaT ba 4/)2/( 2],[

2)12(

)1(

)12(

8

)2/;(ˆ

2220

2

],[

sLkDk

L

LasS

k

k

ba


One barrier to infinity


The renewal equations for the SP, if the process is only depending on the size of last the jump, are:

)|,( )|(

)|,()|,(

)|()|;( :domain Time

00000

00],[000

00000],[

0

XttxxdxtdXtt

XxxttSXttxxdxtd

XttXxttS

t

ba ba

tt

ba

b

a ba

ba

XxxsSXsxxdx

XsXxsS

)|,(ˆ)|,(ˆ

)|(ˆ)|;(ˆ :domain Laplace

0],[00

000],[

The The uncoupleduncoupled not-i.i.d. case of CTRW : SP not-i.i.d. case of CTRW : SPMean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets

Some examples:

||)(ˆ1exp)(ˆ11

)(ˆ1

1

);(ˆ);(ˆ

2/)(ˆ1cosh2/)(ˆ1sinh)(ˆ1

)(ˆ1

1

)2/;(ˆ

)(ˆ2

)|,(ˆ :Assumption

0

0],[0],[

],[

||00

0

xcss

s

s

xsSxsS

LsLss

s

s

LasS

seXsxx

cc

ba

xx




The The uncoupleduncoupled not-i.i.d. case of CTRW : SP not-i.i.d. case of CTRW : SP

L=0 2001-2003

L=0 1995-2003


The The uncoupleduncoupled not-i.i.d. case of CTRW: not-i.i.d. case of CTRW: METMET


MET for the 20 stocksMET for the 20 stocksrescaled variablesrescaled variables


20 stocks1995-

1998

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Mean Exit Time of Equity Assets