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 neededto observe a prescribed price change Delta, the time to fill (TTF) for executed limit orders andthe time to cancel (TTC) for canceled orders in a double auction market. We find that thedistribution of all three quantities decays asymptotically as a power law, but that of FPT hassignificantly fatter tails than that of TTF. Thus a simple first passage time model cannotaccount for the observed TTF of limit orders. We propose that the origin of this differenceis the presence of cancelations. We outline a simple model that assumes that prices arecharacterized by the empirically observed distribution of the first passage time and orders arecanceled randomly with lifetimes that are asymptotically power law distributed withan exponent lambdaLT. In spite of the simplifying assumptions of the model, the inclusionof cancelations is sufficient to account for the above observations and enables one to estimatecharacteristics of the cancelation strategies from empirical data.