Diffusive behavior and the modeling of characteristic times in limit order executions

Abstract

We present an empirical study of the first passage time (FPT) of order book prices needed to observe a prescribed price change Δ, the time to fill (TTF) for executed limit orders and the time to cancel (TTC) for canceled orders in a double auction market. We find that the distribution of all three quantities decays asymptotically as a power law, but that of FPT has significantly fatter tails than that of TTF. Thus a simple first passage time model cannot account for the observed TTF of limit orders. We propose that the origin of this difference is the presence of cancelations. We outline a simple model that assumes that prices are characterized by the empirically observed distribution of the first passage time and orders are canceled randomly with lifetimes that are asymptotically power law distributed with an exponent λ_LT. In spite of the simplifying assumptions of the model, the inclusion of cancelations is sufficient to account for the above observations and enables one to estimate characteristics of the cancelation strategies from empirical data.

Keywords:

• Econophysics
• Limit order market
• Microstructure
• First passage time
• Censored data
• Time to fill

Acknowledgements

We would like to thank two anonymous referees for useful comments and suggestions. ZE is grateful to Jean-Philippe Bouchaud for discussions on the order book, to Michele Tumminello for advice on bootstrapping, and to Ingve Simonsen for help with the measurement of first passage times. The hospitality of l'Ecole de Physique des Houches and Capital Fund Management is also thankfully acknowledged. This work was supported by COST–STSM–P10–917 and OTKA T049238. FL and RNM acknowledge support from MIUR research project ‘Dinamica di altissima frequenza nei mercati finanziari’ and NEST-DYSONET 12911 EU project.

Notes

†The notion of competing risks applies to problems where one deals with several ‘risks’, i.e. random events, of which only the first can be observed (Bedford 2005). For example, limit orders are either executed or canceled and both events can be modeled by some random process. If an order is canceled, one can no longer directly observe what time it would have been eventually executed, and vice versa. Thus it is not possible to independently estimate either process without a bias, if one simply ignores information from the other.

†In our analyses, we removed the trading data for September 20, 2002. This is because, on that day, very unusual trading patterns were observed, including anomalous behavior of the bid–ask spread.

‡Many studies refer to this colloquially as the ‘upstairs’ market.

§In most of the literature the logarithm of the price is modeled, while throughout this paper we intentionally use the price itself. Our study is concerned with very small price changes on the order of the spread, when there is little difference between the two approaches. In our case it is important to keep bare prices, as stocks have a finite tick size (minimal price change). Taking bare prices enables us to classify the orders into discrete categories by price difference. The size of ticks depends on the stock, possible values being 1/4, 1/2 or 1 penny.

¶We repeated the statistical analysis with transaction time and observed a similar power law decay of the first passage time for large times. The value of the power law exponent turns out to be different for real time analysis and transaction time analysis. See appendix A for details.

∥This result is consistent with the Sparre–Andersen theorem (Redner 2001). Alternative descriptions obtained for the asymptotic time dependence of the FPT of Lévy flights which hypothesized a dependence of the distribution exponent from the index of the Lévy distribution, missed the fact that the method of images, which is extremely powerful in Gaussian diffusion, fails for Lévy flight processes (Chechkin et al. 2003). The behavior is of course more complex in the case of Lévy random processes described using a subordination scheme. In these cases the asymptotic behavior of the first passage time depends on the complete properties of the subordination procedure (Sokolov and Metzler 2004).

†Wyart et al. (2007) show that similar arguments give a very good approximation for the average shape of the order book.

‡Note that throughout the paper we use the language of buy orders, but analogous definitions can be given for sell orders. All measurements include both buy and sell orders.

†This is defined by first, for both quantities separately, replacing each observation by its rank in the sample (i.e. assigning 1 to the largest observation of first passage time, 2 to the second largest, etc., and then repeating the procedure for lifetimes). Then the usual cross-correlation coefficient is calculated for the ranks (Lee 2000).

‡The error bars were estimated by the bootstrapping procedure suggested by Schmid and Schmidt (2006) (for more details see references therein).