Diffusive Behavior and the Modeling of Characteristic Times in Limit Order Executions
18 Pages Posted: 1 Feb 2007
Date Written: January 30, 2007
We present a study of the order book data of the London Stock Exchange for five highly liquid stocks traded during the calendar year 2002. Specifically, we study the first passage time of order book prices needed to observe a prescribed price change Delta, the time to fill (TTF) for executed limit orders and the time to cancel (TTC) for canceled ones. We find that the distribution of the first passage time decays asymptotically in time as a power law with an exponent L_FPT ~ 1.5. The median of the same quantity scales as Delta^1.6, which is different from the Delta^2 behavior expected for Brownian motion. The quantities TTF, and TTC are also asymptotically power law distributed with exponents L_TTF = 1.8-2.2 and L_TTC = 1.9-2.4, respectively. For the medians of the time to fill we observe a scaling proportional to Delta^1.4. We outline a simple model, which 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 a power L_LT. The model predicts L_TTF = L_TTC and we estimate from empirical data L_LT ~ 1.6. We verify that the observed results are not crucially dependent on simplifying assumptions.
Keywords: order books, double auction, random walk, econophysics
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