A Correction Note for Price Dynamics in a Markovian Limit Order Market

Abstract

Abstract.A previously published and widely quoted paper, "Price Dynamics in a Markovian Limit Order Market," [SIAM J. Financial Math., 4 (2013), pp. 1-25] has provided analytic solutions for many interesting quantities for the price dynamics of the limit order book. However, for unbalanced order flow cases, one of its major results, the price increase probability conditional on the state of the limit order book, is found to be incorrect due to its citing an erroneous formula from another paper. We correct this error by proposing another analytical solution using the characteristic functions of probability distributions. The final analytical solution presented in this paper is of a simple form and easy to calculate.Key words. limit order book, random walk, price dynamics, hitting probability AMS subject classifications. 60J27, 60E10, 60K25DOI. 10.1137/16M10574371. Introduction. The limit order book (LOB) is a depository of resting limit orders that provides liquidity, allowing participants to trade in the market. Much effort has been put into trying to extract information from the LOB and predict the direction of price change in the near future.A recent paper [2] showed that several interesting conditional probabilities, including the direction of the next price change conditional on the state of the LOB, can be obtained analytically using a simplified model for the LOB based on a previous model proposed in [4]. This simplified model can capture the dynamics of market orders and limit orders and their influence on price dynamics in a more analytically tractable way. The paper brought new insights into the LOB, helped people understand its dynamics [1], [8], [6], and inspired applications based on it, for example, building optimal trading strategies [9].However, we find that one of its major results, Proposition 3, which provides the analytic solution to the conditional probability of price movement of the LOB when the order flow is unbalanced, is incorrect. This proposition itself is important because nowadays much attention has been focused on the study of the order flow imbalance (OFI) process, which is thought to be more sensitive to market information than price [3], [10]. Also, Proposition 3 gives the short term price dynamics of the LOB driven by this OFI process under the Markovian assumption in [2].A numerical and mathematical proof for this mistake is provided in this correction note,