Analysis of the limit order using jump process models

Abstract

We propose a discrete and a continuous limit order book models in trading of financial assets, employing compound Poisson processes and Brownian motions with drift, and derive the distributions of future best buy and best sell prices based on the current limit order book state. To analyze short-term price movement, we compute distribution of hitting time of limit order volume processes. We suggest a method to compute an efficient price, which is an underlying (semi)martingale process of an market price process, under the limit order book models by computing the expectations of future mid price. We prove the martingale property of the efficient price under the bid-ask symmetry condition. Moreover, we check the efficiency of our models using the market data. Finally, we suggest continuous processes with a self-exciting feature which can be applied to various models including limit order books, and computed its limit properties.