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

We present an empirical study of the first passage time (FPT) 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 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 cancellations. 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 an exponent lambda_LT. In spite of the simplifying assumptions of the model, the inclusion of cancellations is enough to account for the above observations and enables one to estimate characteristics of the cancellation strategies from empirical data.