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## Mean Exit Time of Equity Assets

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**Observatory of Complex Systems**http://lagash.dft.unipa.it Mean Exit Time of Equity Assets Salvatore Miccichè Dipartimento di Fisica e Tecnologie Relative Università degli Studi di Palermo Progetto Strategico - Incontro di progetto III anno - Roma 20 Giugno 2007**Mean Exit Times of Equity Assets**S Observatory of Complex Systems C. Coronnello F. Lillo R. N. Mantegna S. Miccichè M. Spanò F. Terzo M. Tumminello G. Vaglica J. Masoliver M. Montero J. Perelló Barcellona EconophysicsBioinformatics Stochastic Processes**Mean Exit Times of Equity Assets**S Aim of the Research The long-term aim is to use CTRW (Markovian process) as a stochastic process able to provide a sound description of extreme times in financial data Explorative analysis of the capability of CTRW to explain some empirical features of tick-by-tick data, role of tick-by-tick volatility. METL2anddata collapse**Mean Exit Times of Equity Assets**S The set of investigated stocks We consider: Mean Exit Times - the 20 most capitalized stocks in 1995-1998 at NYSE the 100 most capitalized stocks in 1995-2003 at NYSE Trades And Quotes (TAQ) database maintained by NYSE (1995-2003) We hereafter consider high-frequency (intraday) data: tick-by-tick data**Mean Exit Time (MET)**Mean Exit Times of Equity Assets S • 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 Financial Interest: the MET provides a timescale for market movements. 2L**Mean Exit Times of Equity Assets**S An example: a Wiener stochastic process For the Wiener process: the MET is: D is the diffusion coefficient (t) is a -correlated gaussian distributed noise**Mean Exit Times of Equity Assets**S Stochastic Process: CTRW 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 them are random. • CTRW first developed by Montroll and Weiss (1965) • Microstructure of Random Process**The relevant variables: I - price changes**Mean Exit Times of Equity Assets S • Log-prices: • Log-Returns: • Return changes conform a stationary random process with a (marginal) probability density function:**Mean Exit Times of Equity Assets**S The relevant variables: II- waiting times • The process only may change at “random” times remaining constant between these jumps. • The waiting times also are characterized by a (marginal) probability density function:**Mean Exit Times of Equity Assets**S The relevant variables: joint pdf The system is characterized by the following JOINT probability density function 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] are just two marginal density functions:**Mean Exit Times of Equity Assets**S The uncoupled i.i.d. case of CTRW: MET A simple model**The uncoupled i.i.d. case of CTRW: setup**Mean Exit Times of Equity Assets S • 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**Mean Exit Times of Equity Assets**S The uncoupled i.i.d. case of CTRW: MET • The MET for i.i.d. CTRW process fulfils a renewal equation: J. Masoliver, M. Montero, J. Perelló, Phys. Rev. E71, 056130 (2005) • If one now assumes that then one would observe that tick-by-tick volatility vs is a universal curve**Mean Exit Times of Equity Assets**S The uncoupled i.i.d. case of CTRW: MET In particular, if one assumes that (three state i.i.d. discrete model) then one can prove that c is the basic jump size Q is the probability that the price is unchanged The quadratic dependance of MET is recovered**Mean Exit Times of Equity Assets**S The uncoupled i.i.d. case of CTRW: MET MET for the 20 stocks rescaled variables 20 stocks 1995-1998 No data collapse is observable**Mean Exit Times of Equity Assets**S The uncoupled i.i.d. case of CTRW: summary No data collapse is observable The quadratic dependance of MET is recovered • What is the reason why we do not observe data collapse? • Is H(u) not universal? • Is the uncoupled case too simple? • Is there any role of capitalization ? • Is there any role of tick size ? • Is there any role of trading activity ? Let us go back to the empirical data !**Mean Exit Times of Equity Assets**S 1) Shuffling Experiments Hypothesis 1: h(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. 20 stocks 1995-1998 A good data collapse is observable: then h(x) is “the same” for all stocks**Mean Exit Times of Equity Assets**S 1) Shuffling Experiments Hypothesis 2: There is a role of the cross-correlations between returns and jumps Hypothesis 3: There is a role of the autocorrelation of waiting times Hypothesis 4: There is a role of the autocorrelation of returns 1995-1998 We can test these hypothesis by shuffling H2) returns and waiting times and preserving the crosscorrelations, i.e. the pairs (green) H3) waiting times only (blue) H4) returns only (magenta) dashed black=original data red=H1 GE stock**Mean Exit Times of Equity Assets**S Fourier Shuffling Experiments black=blue neglecting the autocorrelation of waiting times is not important green=red neglecting the cross-correlations is not important magentablack: There is a role of the autocorrelation of returns GE stock Two possible sources of (auto)-correlation in returns: linear (bid-ask bounce) nonlinear (volatility)**Mean Exit Times of Equity Assets**S Fourier Shuffling Experiments Shuffling that destroys only the nonlinear (auto)-correlation properties of a time-series red =phase randomized data of Xn red=black neglecting the volatility (nonlinear) correlation is not important GE stock dashed black=original data**Mean Exit Times of Equity Assets**S 2) Jump size & Trading Activity On 24/06/1997 the tick size changed from 1/8$ to 1/16$ On 29/01/2001 the tick size changed from 1/16$ to 1/100$ Therefore we decided to consider a larger set of 100 stocks continuously traded from 1995 to 2003 and considered 3 time periods: 29/01/200131/12/2003 01/01/199524/06/1997 25/06/199728/01/2001 Therefore 3 time periods are also different for the trading activity !!**Mean Exit Times of Equity Assets**S 2) Jump size & Trading Activity Each point is the mean over 100 stocks The error bar is the standard deviation Nothing changes for the shufflings ! 100 stocks 100 stocks BUT The standard deviation is smaller in 01-03 than in 95-97. T/E[] i.e. 100 stocks The collapse on a single curve is better in 01-03 than in 95-97. GE: E[]5.3 s =3.3 10-3 L/2k**Mean Exit Times of Equity Assets**S The uncoupled i.i.d. case of CTRW: MET A more sophisticated model**Mean Exit Times of Equity Assets**S The uncouplednot-i.i.d. case of CTRW: setup The only important thing is the bid-ask 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. processto aone step markovian chain.**Mean Exit Times of Equity Assets**S The uncouplednot-i.i.d. case of CTRW: MET 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. E72,056101 (2005)**Mean Exit Times of Equity Assets**S The uncouplednot-i.i.d. case of CTRW: MET If we consider a two-state Markov chain model: we can obtain a scale-free expression for the symmetrical MET in terms of the width L of the interval: r is the correlation between two consecutive jumps: NEW extra factor ! By inspection: 2=c2**Mean Exit Times of Equity Assets**S The uncouplednot-i.i.d. case of CTRW: MET MET for the 100 stocks rescaled variables in the 3 time periods considered rescaled T L/2k jump size or trading activity? The observed data collapse is improved, although it is still not completely satisfactory**Mean Exit Times of Equity Assets**S The uncouplednot-i.i.d. case of CTRW: MET CTRW WIENER In a sense, our results are not worth all the efforts done by introducing this more complicated model !!!! D is the diffusion coefficient However, the model gives an HINT about the “INGREDIENTS” of the diffusion coefficient !!!**Mean Exit Times of Equity Assets**S Conclusions • 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 L • depends only from the bid-ask bounce r • seems to scale in a similar way for different assets, better when the 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.**Mean Exit Times and Survival Probability of Equity Assets**The Endmicciche@lagash.dft.unipa.it**Mean Exit Times and Survival Probability of Equity Assets**Additional: other markets**Mean Exit Times of Equity Assets**3) Capitalization Fit with a power-law function:MET = (C+A L) The dependance from the capitalization is not so dramatic !!!**Mean Exit Times of Equity Assets**S The uncouplednot-i.i.d. case of CTRW: MET Again, thedata collapse is betterin 01-03than in 95-97 dispersion jump size or trading activity? L/2k The observed data collapse is improved, although it is still not completely satisfactory**Mean Exit Times of Equity Assets**The uncouplednot-i.i.d. case of CTRW: MET**Mean Exit Times and Survival Probability of Equity Assets**Additional: other markets**Mean Exit Times of Equity Assets**2) Jump size & Trading Activity Nothing changes for the shufflings ! T/E[] London Stock Exchange (SET1 - electronic transactions only) L/2k**Mean Exit Times of Equity Assets**2) Jump size & Trading Activity Nothing changes for the shufflings ! T/E[] Milan Stock Exchange L/2k**Mean Exit Times of Equity Assets**2) Jump size & Trading Activity Nothing changes for the shufflings ! T/E[] NYSE LSE MIB30 L/2k**Mean Exit Times of Equity Assets**2) Jump size & Trading Activity If the higher moments exist ... T/E[] III moment L4 L/2k II moment T/E[] It depends on the tails of the Survival Probability distribution ... L6 L/2k**Mean Exit Times of Equity Assets**The uncouplednot-i.i.d. case of CTRW: MET MET for the 100 stocks rescaled variables in the 3 time periods considered rescaled T L/2k jump size or trading activity? The observed data collapse is improved, although it is still not completely satisfactory**Mean Exit Times of Equity Assets**The uncouplednot-i.i.d. case of CTRW: MET Again, thedata collapse is betterin 01-03than in 95-97 dispersion jump size or trading activity? L/2k The observed data collapse is improved, although it is still not completely satisfactory**Mean Exit Times of Equity Assets**The uncouplednot-i.i.d. case of CTRW: MET**Mean Exit Times and Survival Probability of Equity Assets**Additional: old slides**Mean Exit Times of Equity Assets**CTRW: 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! Mean Exit Times of Equity Assets CTRW: origin and applications**Instrument II: Survival Probability (SP)**Mean Exit Times and Survival Probability of Equity Assets • The Survival Probability (SP)measures the likelihood that, up to time t the process has been never outside the interval [a,b]: • Financial 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.**Mean Exit Times and Survival Probability of Equity Assets**Instrument III: relation between SP and MET We can recover the Mean Exit Time from the Laplace Transform of the Survival Probability: Because: Therefore:**Mean Exit Times and Survival Probability of Equity Assets**SP and MET for a Wiener process The MET and SP for the Wiener process are: D is the diffusion coefficient One barrier to infinity**Mean Exit Times and Survival Probability of Equity Assets**The uncoupled not-i.i.d. case of CTRW : SP The renewal equations for the SP, if the process is only depending on the size of last the jump, are:**Mean Exit Times and Survival Probability of Equity Assets**The uncoupled not-i.i.d. case of CTRW : SP Some examples: