Mean Exit Time of Equity Assets Salvatore Miccichè Observatory of Complex Systems 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 Observatory of Complex Systems S. Miccichè F. Lillo R. N. Mantegna F. Terzo M. Tumminello G. Vaglica C. Coronnello Econophysics Bioinformatics Stochastic Processes Econophysics Bioinformatics Stochastic Processes M. Spanò J. Masoliver M. Montero J. Perelló Barcellona 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. MET  L 2 and data collapse Mean Exit Times of Equity Assets S

The set of investigated stocks We consider: Mean Exit Times - the 20 most capitalized stocks in at NYSE the 100 most capitalized stocks in at NYSE intradaytick-by-tick data We hereafter consider high-frequency (intraday) data: tick-by-tick data TAQTAQ Trades And Quotes (TAQ) database maintained by NYSE ( ) Mean Exit Times of Equity Assets S

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 Financial Interest: the MET provides a timescale for market movements. Mean Exit Times of Equity Assets 2L S

For the Wiener process: An example: a Wiener stochastic process D D is the diffusion coefficient Mean Exit Times of Equity Assets 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. time lags between them The key point is that in a CTRW not only the size of the movements but also the time lags between them are random. Stochastic Process: CTRW Mean Exit Times of Equity Assets CTRW first developed by Montroll and Weiss (1965) CTRW first developed by Montroll and Weiss (1965) Microstructure of Random Process Microstructure of Random Process S

The relevant variables: I - price changes Log-prices: Log-Returns:Log-Returns: Return changes conform a stationary random process with a (marginal) probability density function: Mean Exit Times of Equity Assets S

The process only may change at “random” times remaining constant between these jumps. waiting timesThe waiting times also are characterized by a (marginal) probability density function: The relevant variables: II- waiting times Mean Exit Times of Equity Assets S

The relevant variables: joint pdf The system is characterized by the following JOINT probability density function are just two marginal density functions: Mean 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] S

A simple model The uncoupled i.i.d. case of CTRW: MET 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 Mean Exit Times of Equity Assets S

The MET for i.i.d. CTRW process fulfils a renewal equation: The uncoupled i.i.d. case of CTRW: MET assumesIf one now assumes that would then one would observe that J. Masoliver, M. Montero, J. Perelló, Phys. Rev. E 71, (2005) 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 assumes In particular, if one assumes that (three state i.i.d. discrete model) can prove that 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 No data collapse is observable Mean Exit Times of Equity Assets 20 stocks S

The uncoupled i.i.d. case of CTRW: summary The quadratic dependance of MET is recovered No data collapse is observable 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 ! 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  X n 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 Mean Exit Times of Equity Assets 20 stocks 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 We can test these hypothesis by shuffling H2 green H2) returns and waiting times and preserving the crosscorrelations, i.e. the pairs (green) H3blue H3) waiting times only (blue) H4magenta 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 magenta  black: There is a role of the autocorrelation of returns green=red  neglecting the cross-correlations is not important Two possible sources of (auto)- correlation in returns: linear (bid-ask bounce) nonlinear (volatility) Mean Exit Times of Equity Assets GE stock S

Fourier Shuffling Experiments dashed black=original data GE stock red =phase randomized data of  X n  red=black  neglecting the volatility (nonlinear) correlation is not important nonlinear Shuffling that destroys only the nonlinear (auto)-correlation properties of a time-series Mean Exit Times of Equity Assets S

2) Jump size & Trading Activity 1/8$1/16$ On 24/06/1997 the tick size changed from 1/8$ to 1/16$ 1/16$1/100$ 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/ /06/ /06/ /01/ /01/ /12/2003 different trading activity !! Therefore 3 time periods are also different for the trading activity !! S

2) Jump size & Trading Activity Mean Exit Times of Equity Assets Nothing changes for the shufflings ! Each point is the mean over 100 stocks The error bar is the standard deviation The standard deviation is smaller in than in BUT collapse on a single curve better The collapse on a single curve is better in than in i.e. GE: E[  ]  5.3 s  =  = stocks L/2k T/E[  ] S

A more sophisticated model The uncoupled i.i.d. case of CTRW: MET Mean Exit Times of Equity Assets S

The uncoupled not-i.i.d. case of CTRW: setup The only important thing is the bid-ask bounce !!!! from an i.i.d. processto aone step markovian chain 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. Mean 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, (2005) The uncoupled not-i.i.d. case of CTRW: MET Mean Exit Times of Equity Assets S

two-state Markov chain If we consider a two-state Markov chain model: scale-free expression for the symmetrical MET we can obtain a scale-free expression for the symmetrical MET in terms of the width L of the interval: The uncoupled not-i.i.d. case of CTRW: MET Mean Exit Times of Equity Assets 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 L/2k The observed data collapse is improved, although it is still not completely satisfactory MET for the 100 stocks rescaled variables in the 3 time periods considered jump size or trading activity trading activity ? S

D D is the diffusion coefficient In a sense, our results are not worth all the efforts done by introducing this more complicated model !!!! The uncoupled not-i.i.d. case of CTRW: MET WIENERCTRW Mean Exit Times of Equity Assets However, the model gives an HINT about the “INGREDIENTS” of the diffusion 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, better when the thick size is smaller.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. Conclusions Mean Exit Times of Equity Assets S

The End Mean Exit Times and Survival Probability of Equity Assets

Additional: other markets Mean Exit Times and Survival Probability of Equity Assets

3) Capitalization Mean Exit Times of Equity Assets 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 The uncoupled not-i.i.d. case of CTRW: MET dispersion L/2k The observed data collapse is improved, although it is still not completely satisfactory Again, the data collapse is better in than in jump size or trading activity trading activity ? S

Mean Exit Times of Equity Assets The uncoupled not-i.i.d. case of CTRW: MET

Additional: other markets Mean Exit Times and Survival Probability of Equity Assets

2) Jump size & Trading Activity Mean Exit Times of Equity Assets Nothing changes for the shufflings ! L/2k T/E[  ] London Stock Exchange (SET1 - electronic transactions only)

2) Jump size & Trading Activity Mean Exit Times of Equity Assets Nothing changes for the shufflings ! L/2k T/E[  ] Milan Stock Exchange

2) Jump size & Trading Activity Mean Exit Times of Equity Assets Nothing changes for the shufflings ! L/2k T/E[  ] NYSELSEMIB30

Mean Exit Times of Equity Assets III moment II moment If the higher moments exist... It depends on the tails of the Survival Probability distribution... L4L4L4L4 L6L6L6L6 T/E[  ] L/2k 2) Jump size & Trading Activity

Mean Exit Times of Equity Assets The uncoupled not-i.i.d. case of CTRW: MET rescaled T L/2k The observed data collapse is improved, although it is still not completely satisfactory MET for the 100 stocks rescaled variables in the 3 time periods considered jump size or trading activity trading activity ?

Mean Exit Times of Equity Assets The uncoupled not-i.i.d. case of CTRW: MET dispersion L/2k The observed data collapse is improved, although it is still not completely satisfactory Again, the data collapse is better in than in jump size or trading activity trading activity ?

Mean Exit Times of Equity Assets The uncoupled not-i.i.d. case of CTRW: MET

Additional: old slides Mean Exit Times and Survival Probability of Equity Assets

CTRW: The idea Mean 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 applications Mean Exit Times 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 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. Mean Exit Times and Survival Probability of Equity Assets

Mean Exit TimeLaplace Transform Survival Probability We can recover the Mean Exit Time from the Laplace Transform of the Survival Probability: Therefore: Because: Instrument III: relation between SP and MET Mean Exit Times and Survival Probability of Equity Assets

The MET and SP for the Wiener process are: SP and MET for a Wiener process D D is the diffusion coefficient One barrier to infinity Mean 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: The uncoupled not-i.i.d. case of CTRW : SP Mean Exit Times and Survival Probability of Equity Assets

Some examples: 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 Mean Exit Times and Survival Probability of Equity Assets

The uncoupled not-i.i.d. case of CTRW : SP L= L= Mean Exit Times and Survival Probability of Equity Assets

The uncoupled not-i.i.d. case of CTRW: MET Mean Exit Times of Equity Assets MET for the 20 stocks rescaled variables The observed data collapse is improved, although it is still not completely satisfactory 20 stocks