  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.
