Observatory of Complex Systems http lagash dft unipa
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
S Mean Exit Times of Equity Assets 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 Econophysics Bioinformatics Stochastic Processes
Mean Exit Times of Equity Assets 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-bytick volatility. MET L 2 and data collapse S
Mean Exit Times of Equity Assets 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) TAQ database maintained by NYSE (1995 -2003) -2003 We hereafter consider high-frequency (intraday) intraday data: tick-by-tick data S
S Mean Exit Times of Equity Assets 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 Financial Interest: Interest the MET provides a timescale for market movements. 2 L
Mean Exit Times of Equity Assets 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 S
Mean Exit Times of Equity Assets 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 S
Mean Exit Times of Equity Assets The relevant variables: I - price changes • Log-prices: • Log-Returns: • Return changes conform a stationary random process with a (marginal) probability density function: S
Mean Exit Times of Equity Assets 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: S
Mean Exit Times of Equity Assets The relevant variables: joint pdf The system is characterized by the following JOINT probability density function are just two marginal density functions: P(X, t) probability that a particle is at position X at time t probability of making a step of length X in the interval [t, t+dt] S
Mean Exit Times of Equity Assets The uncoupled i. i. d. case of CTRW: MET A simple model S
Mean Exit Times of Equity Assets S The uncoupled 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
S Mean Exit Times of Equity Assets 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. E 71, 056130 (2005) • If one now assumes that tick-by-tick volatility then one would observe that vs is a universal curve
S Mean Exit Times of Equity Assets The uncoupled i. i. d. case of CTRW: MET In particular, if one assumes that c is the basic jump size Q is the probability that the price is unchanged (three state i. i. d. discrete model) then one can prove that The quadratic dependance of MET is recovered
S Mean Exit Times of Equity Assets The uncoupled i. i. d. case of CTRW: MET for the 20 stocks rescaled variables 20 stocks 1995 -1998 No data collapse is observable
Mean Exit Times of Equity Assets 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 ! S
S Mean Exit Times of Equity Assets 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. A good data collapse is observable: then h(x) is “the same” for all stocks 20 stocks 1995 -1998
S Mean Exit Times of Equity Assets 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 H 2) H 2 returns and waiting times and preserving the crosscorrelations, i. e. the pairs (green) green H 3) H 3 waiting times only (blue) blue H 4) H 4 returns only (magenta) magenta dashed black=original data red=H 1 GE stock
S Mean Exit Times of Equity Assets Fourier Shuffling Experiments black=blue neglecting the autocorrelation of waiting times is not important green=red neglecting the cross-correlations is not important magenta black: 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)
S Mean Exit Times of Equity Assets 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
S Mean Exit Times of Equity Assets 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: 01/01/1995 24/06/1997 25/06/1997 28/01/2001 29/01/2001 31/12/2003 Therefore 3 time periods are also different for the trading activity !!
S Mean Exit Times of Equity Assets 2) Jump size & Trading Activity Each point is the mean over 100 stocks The error bar is the standard deviation 100 stocks Nothing changes for the shufflings ! T/E[ ] BUT The standard deviation is smaller in 01 -03 than in 9597. i. e. 100 stocks The collapse on a single curve is better in 01 -03 than in 95 -97. L/2 k GE: E[ ] 5. 3 s =3. 3 10 -3
Mean Exit Times of Equity Assets The uncoupled i. i. d. case of CTRW: MET A more sophisticated model S
Mean Exit Times of Equity Assets The uncoupled not-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. process to a one step markovian chain S
Mean Exit Times of Equity Assets The uncoupled not-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. E 72, 056101 (2005) S
S Mean Exit Times of Equity Assets The uncoupled not-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: By inspection: 2=c 2 NEW extra factor !
S Mean Exit Times of Equity Assets The uncoupled not-i. i. d. case of CTRW: MET rescaled T MET for the 100 stocks rescaled variables in the 3 time periods considered L/2 k The observed data collapse is improved, although it is still not completely satisfactory jump size or trading activity? activity
S Mean Exit Times of Equity Assets The uncoupled not-i. i. d. case of CTRW: MET CTRW WIENER D is the diffusion coefficient In a sense, our results are not worth all the efforts done by introducing this more complicated model !!!! However, the model gives an HINT about the “INGREDIENTS” of the diffusion coefficient !!!
Mean Exit Times of Equity Assets 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. S
Mean Exit Times and Survival Probability of Equity Assets The End micciche@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 powerlaw function: MET = (C+A L) The dependance from the capitalization is not so dramatic !!!
S Mean Exit Times of Equity Assets The uncoupled not-i. i. d. case of CTRW: MET dispersion Again, the data collapse is better in 01 -03 than in 95 -97 L/2 k The observed data collapse is improved, although it is still not completely satisfactory jump size or trading activity? activity
Mean Exit Times of Equity Assets The uncoupled not-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 T/E[ ] Nothing changes for the shufflings ! London Stock Exchange (SET 1 - electronic transactions only) L/2 k
Mean Exit Times of Equity Assets 2) Jump size & Trading Activity T/E[ ] Nothing changes for the shufflings ! Milan Stock Exchange L/2 k
Mean Exit Times of Equity Assets 2) Jump size & Trading Activity T/E[ ] Nothing changes for the shufflings ! NYSE LSE MIB 30 L/2 k
Mean Exit Times of Equity Assets 2) Jump size & Trading Activity T/E[ ] If the higher moments exist. . . III moment L 4 L/2 k It depends on the tails of the Survival Probability distribution. . . T/E[ ] II moment L 6 L/2 k
Mean Exit Times of Equity Assets The uncoupled not-i. i. d. case of CTRW: MET rescaled T MET for the 100 stocks rescaled variables in the 3 time periods considered L/2 k The observed data collapse is improved, although it is still not completely satisfactory jump size or trading activity? activity
Mean Exit Times of Equity Assets The uncoupled not-i. i. d. case of CTRW: MET dispersion Again, the data collapse is better in 01 -03 than in 95 -97 L/2 k The observed data collapse is improved, although it is still not completely satisfactory jump size or trading activity? activity
Mean Exit Times of Equity Assets The uncoupled not-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
Mean Exit Times of Equity Assets CTRW: origin and applications • CTRW first developed by Montroll and Weiss (1965) Applications: • Transport in random • • • Microstructure of Random Process • • media Random networks Self-organized criticality Earthquake modeling Finance!
Mean Exit Times and Survival Probability of Equity Assets 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 interest: 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 Va. R, 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: 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:
Mean Exit Times and Survival Probability of Equity Assets The uncoupled not-i. i. d. case of CTRW : SP
Mean Exit Times and Survival Probability of Equity Assets The uncoupled not-i. i. d. case of CTRW : SP
Mean Exit Times and Survival Probability of Equity Assets The uncoupled not-i. i. d. case of CTRW : SP L=0 1995 -2003 L=0 2001 -2003
Mean Exit Times of Equity Assets The uncoupled not-i. i. d. case of CTRW: MET for the 20 stocks rescaled variables 20 stocks 1995 -1998 The observed data collapse is improved, although it is still not completely satisfactory
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