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Price dynamics in a Markovian limit order market
Rama Cont, Adrien De Larrard
TL;DR
The paper addresses how order flow shapes price dynamics in electronic limit order markets, where existing models face a trade-off between realism and tractability. It introduces a Markovian queueing model for best bid and ask queues, derives analytical price statistics and a diffusion limit, and compares key asymptotic insights with data. The model links volatility to order-arrival intensities relative to market depth while remaining a stylized first step toward realistic stochastic order-book models.
Problem
Price dynamics depend on order-book state, while empirical regularities are difficult to incorporate into a single tractable model.
Method
The paper models best bid and ask queues as a Markovian queueing system driven by market orders, limit orders and cancellations.
Results
The model analytically characterizes price-change durations, distributions, autocorrelations, conditional upward moves and diffusion-limit volatility.
Takeaways & Limitations
In frequently trading limit order markets, volatility increases with the ratio of order-arrival intensity to market depth, consistent with empirical comparisons.
Takeaways & Limitations
The model is stylized and assumes symmetry of the queue-size distribution for its diffusion-limit analysis.
Abstract
from arXiv · showhide
We propose and study a simple stochastic model for the dynamics of a limit order book, in which arrivals of market order, limit orders and order cancellations are described in terms of a Markovian queueing system. Through its analytical tractability, the model allows to obtain analytical expressions for various quantities of interest such as the distribution of the duration between price changes, the distribution and autocorrelation of price changes, and the probability of an upward move in the price, {\it conditional} on the state of the order book. We study the diffusion limit of the price process and express the volatility of price changes in terms of parameters describing the arrival rates of buy and sell orders and cancelations. These analytical results provide some insight into the relation between order flow and price dynamics in order-driven markets.
1. Introduction
The paper develops a tractable Markovian model of limit order book dynamics to connect order flow with price behavior. It analytically characterizes price-change statistics and volatility, including conditional behavior based on the order book state.
- Electronic order-driven markets centralize buy and sell orders in a limit order book, making price dynamics relevant to trading and price formation.
- Existing equilibrium models show that price evolution depends on order book state, while empirical studies document many difficult-to-integrate order book regularities.
- The paper proposes a simpler Markovian limit order market model that captures salient market- and limit-order dynamics while enabling analytical computation.
- The model focuses on interacting best bid and ask queues, with remaining book levels represented as a reservoir of next-to-best queue sizes.
- It analytically derives duration distributions, price-change distributions and autocorrelations, and the conditional probability of an upward price move.
- The model endogenously links durations and price changes, providing a step toward joint structural modeling of high-frequency prices and order flow.
- At longer horizons, diffusion-limit analysis expresses price-change volatility through buy, sell and cancellation arrival rates, with empirical comparisons supporting the model’s main insights.
2. A Markov model of limit order book dynamics
The model reduces the order book to interacting best-price queues whose events follow independent Poisson processes. Queue depletion moves prices by one tick and replenishes queues from stationary reservoir distributions, enabling analytical characterization of high-frequency quantities.
- 2.1. Level-1 representation of a limit order book: The reduced state records bid and ask prices together with the outstanding limit-buy and limit-sell queue sizes.
- 2.1. Level-1 representation of a limit order book: The spread is assumed to equal one tick, reducing the model’s dimensionality.
- 2.2. Order book dynamics: Limit orders increase the corresponding queue, whereas market orders and cancellations decrease it and therefore have the same state effect.
- 2.1. Level-1 representation of a limit order book: The model treats the remaining order-book levels as a reservoir, represented by the distribution of queue sizes at next-to-best prices.
- 2.2. Order book dynamics: When a queue is depleted, the price moves one tick and the newly exposed queue is drawn from f or ˜f; gaps would instead create larger jumps.
- 2.2. Order book dynamics: Limit orders arrive at rate λ, while market orders and cancellations jointly reduce queues at rate θ + µ.
- 2.2. Order book dynamics: The analytical framework characterizes conditional durations, upward-move probabilities, price-change dynamics and volatility in terms of order-flow statistics.
3. Analytical results
The model derives conditional analytical results for durations, price-direction probabilities, and price-change dependence from the observable order-book state. It also identifies heavy-tailed durations, state-dependent price moves, and negative short-lag autocorrelation under an asymmetry condition.
- 3.1. Duration until the next price change: The duration until the next price change is the first time either the bid or ask queue is depleted.The queue sizes evolve between price changes until the order-book process hits one of the axes.
- 3.1. Duration until the next price change: The conditional duration law has tail exponent 2 when λ < µ + θ and tail exponent 1 when λ = µ + θ.In the balanced case, the conditional expectation of τ is infinite whenever both initial queue sizes are positive.
- 3.2. Probability of an upward move: The probability of an upward price move is computed conditionally on the bid and ask queue sizes through a discrete harmonic boundary-value problem.The boundary conditions assign probability 0 when the bid queue is empty and 1 when the ask queue is empty.
- 3.2. Probability of an upward move: For a balanced order book, the conditional price-move probabilities are independent of the parameters describing order flow and agree well with Citigroup transition frequencies.Figure 4 compares the theoretical conditional probabilities with tick-by-tick data from June 26, 2008.
- 3.3. Dynamics of the price: Price-change autocovariance follows Cov(X1,Xk) = (2pcont −1)k−1, while queue asymmetry after an upward move implies negative high-frequency dependence.The model gives negative first-lag covariance when pcont < 1/2; increments are uncorrelated when pcont = 1/2.
4. Diffusion limit of the price process
The paper establishes diffusion limits for the price process under symmetric order-flow assumptions and relates long-horizon volatility to order-flow statistics and market depth. It also examines balanced and limit-order-depleted regimes, with empirical evidence supporting the predicted volatility relationship.
- Diffusion limit: Under symmetry of the post-move queue-size distribution, price increments are independent and the rescaled price converges to a diffusion process.The diffusion coefficient is computed from the statistics of the order flow driving the limit order book.
- Balanced order flow: In the balanced-flow case, the price exhibits diffusive behavior over time scales much larger than the millisecond-scale interval between order-book events.The diffusion coefficient uses the tick size, order-arrival parameters, and D(f), the geometric-average market-depth measure.
- Balanced order flow: Formula (13) expresses price-change variance using the tick size and order-flow quantities, yielding a volatility estimator that does not require observing prices.The relevant quantities include the limit-order arrival rate and D(f), which summarizes post-price-change bid and ask queue sizes.
- Balanced order flow: For balanced markets with the same tick size and best-quote arrival rate, greater next-to-best queue depth corresponds to lower price volatility.This comparison follows from the model’s relation between volatility and the ratio λ/D(f).
- Empirical test using high-frequency data: Across Dow Jones stocks, higher λ/D(f) is associated with higher 10-minute price-increment variance, with standard deviation increasing roughly proportionally to the ratio.Figure 5 compares the standard deviation of 10-minute price increments with λ/D(f); each point represents one stock and the red line is the best linear approximation.
- Case when market orders and cancelations dominate: When market orders and cancellations dominate limit orders, queues deplete faster, price changes become more frequent, and a diffusion limit remains available under different scaling.The resulting intermediate-frequency variance is expressed through the tick size and order-flow statistics, including the expected queue hitting time.