Price dynamics in a Markovian limit order market (Q2873118)

The paper under review aims to analyze the price dynamics under the execution of market orders against outstanding limit orders and to study the diffusion limit of the price process and to express the volatility of the price changes in terms of parameters describing the arrival rates of buy and sell orders and cancellations. The equilibrium models of price formation in limit order markets have shown that the evolution of the price in such markets is rather complex and depends on the state of the order book. Empirical studies on limit order books provide an extensive list of statistical features of order book dynamics that are challenging to incorporate in a single model.NEWLINENEWLINEThe authors propose a Markovian model of a limit order market to capture salient features of the dynamics of market orders and limit orders and their influence on price dynamics. Empirical evidence indicates that most of the order flow is directed at the best bid and ask prices and the imbalance between the order flow at the bid and at the ask appears to be the main driver of price changes. Therefore the authors build a parsimonious model that the number of limit orders \((q_t^b, q_t^a)\) sitting at the bid and the ask represents a system of two interactive queues. Compared with econometric models of high frequency data, the model in this paper links durations and price changes in an endogenous manner and provides a step to a joint structural model of high frequency dynamics of price and order flow. Order arrivals and cancellations are very frequent and occur at the millisecond time scale, the metric of success is the volume-weighted average price (VWAP).NEWLINENEWLINESection 2 starts to build a Markov model (under the reduced-form model) of limit order book dynamics. All electronic trading venues also allow participants to place limit order pegged to the best available price, and market makers used these pegged orders to liquidate their inventories. The limit order book is described by a continuous-time process \(X_t= (s_t^b, q_t^b, q_t^a)\) with piecewise constant sample paths whose transitions correspond to the order book events. The duration \(T_i^a\) \((T_i^b)\) between two consecutive orders arriving at the ask (at the bid) is a sequence of independent random variables with exponential distribution with parameter \(\lambda + \theta +\mu\), and the size \(V_i^a\) \((V_i^b)\) of the associated change in queue size is a sequence of independent random variables with \(P(V_i^a=1)=\frac{\lambda}{\lambda + \theta + \mu}\) (\(P(V_i^b=1)=\frac{\lambda}{\lambda + \theta + \mu}\)). All these assumptions on \(T_i^a, T_i^b, V_i^a, V_i^b\) constitute a statistical simplification of the order flow. The price (\(s_t\), \(t>0\)) moves when the queue \(q_t = (q_t^b, q_t^a)\) hits one of the axes, the duration until the next price move is \(\tau = \sigma_a \wedge \sigma_b\), where \(\sigma_{a,b} =\inf\{T_1^{a, b} + \cdots + T_i^{a, b}: q_{T_1^{a, b} + \cdots + T_i^{a, b_-}}^{a, b} +V_i^{a, b} =0\}\).NEWLINENEWLINEProposition 1 in Section 3 gives the distribution of \(\tau\) conditioned on the initial state of the order book under the assumptions for the order flow. Proposition 2 gives the probability of price increases at the next price move, conditioned on having \(n\) orders on the bid side and \(p\) orders on the ask side. Proposition 3 extends Proposition 2 for an asymmetric order flow. A key quantity for the dynamics of the price is the probability of two successive price changes in the same direction. Proposition 4 describes this probability, and the nonobserved efficient price process is a martingale when the probability of two successive price changes in the same direction is 1/2.NEWLINENEWLINESection 4 analyzes the diffusion limit of the price process. Theorem 1 shows that the diffusion limit of the price process under \(\lambda = \theta +\mu\) (\(P(V_i^a=1)=0.5=P(V_i^b=1)\)) and relates the ``coarse-grained'' volatility of intraday returns at lower frequencies to the high-frequency arrival rates of orders. The authors use high-frequency data to do an empirical test. Theorem 2 shows the diffusion limit of the price process under \(\lambda < \theta +\mu\), corresponding to the case that the flow of market orders and cancellations dominates that of limit orders and leads to an expression of the variance \(\sigma^2 = \frac{\tau}{\tau_0 m (\lambda, \theta +\mu, f)} \delta^2\) of the price at a time scale \(\tau\gg \tau_0 \sim ms\), where \(m(\lambda, \theta +\mu, f)\) is the expected hitting time of the axes by the Markov queuing system \(q\) with parameters \((\lambda, \theta +\mu)\) and random initial condition with distribution \(f\). Various ramifications of the work are discussed in the last subsection 4.4, the conclusion.