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 Times of Equity AssetsMean Exit Times of Equity Assets S
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.
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
• 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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets S
• 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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets S
• 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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets S
The The uncoupleduncoupled i.i.d. case of CTRW: MET i.i.d. case of CTRW: MET
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
20],[
2][
)(
L
E
xT ba
S
The The uncoupleduncoupled i.i.d. case of CTRW: MET i.i.d. case of CTRW: MET
MET for the 20 stocksMET for the 20 stocksrescaled variablesrescaled variables
No data collapse is observableNo data collapse is observable
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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 !
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets S
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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)
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets S
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets S
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets S
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
The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: METMET
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: METMET
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
],[
DD is the diffusion coefficient
][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 !!!!
The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: METMET
WIENERWIENERCTRWCTRW
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets
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 !!!
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: METMET
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
jump sizejump sizeor
trading trading activityactivity?
S
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: METMET
Additional: other marketsAdditional: other markets
Mean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets
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 !
L/2k
T/E
[]
London Stock London Stock ExchangeExchange
(SET1 - (SET1 -
electronic electronic transactions transactions
only)only)
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 !
L/2k
T/E
[]
Milan Stock Milan Stock ExchangeExchange
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 !
L/2k
T/E
[]
NYSENYSELSELSE
MIB30MIB30
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: METMET
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?
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: METMET
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
jump sizejump sizeor
trading trading activityactivity?
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
The The uncoupleduncoupled notnot-i.i.d. case of CTRW: -i.i.d. case of CTRW: METMET
Additional: old slidesAdditional: old slides
Mean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets
CTRW: The ideaCTRW: The idea
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
• 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
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
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
Mean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets
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
DD is the diffusion coefficient
One barrier to infinity
Mean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets
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 : SPMean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets
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
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
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
Mean Exit Times and Survival Probability of Equity AssetsMean Exit Times and Survival Probability of Equity Assets
The The uncoupleduncoupled not-i.i.d. case of CTRW: not-i.i.d. case of CTRW: METMET
Mean Exit Times of Equity AssetsMean Exit Times of Equity Assets
MET for the 20 stocksMET for the 20 stocksrescaled variablesrescaled variables
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
20 stocks1995-
1998